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Mathematical and Theoretical Physics

Research Article Volume 1 Issue 4

Some characteristics of the function methodology to describe the reinjection process in chaotic intermittency

Ezequiel del R o,2 Luis F Guti rrez Marcantoni,1 Sergio Elaskar1

1Departamento de Aeronautica, Universidad Nacional de Cordoba and CONICET, Argentina
2E.T.S.I. Aeronautica y Espacio, Universidad Politecnica de Madrid, Spain

Correspondence: Sergio Elaskar, Departamento de Aeronáutica, FCEFyN, Instituto de Estudios Avanzados en Ingenierııía y Tecnologıía, IDIT, Universidad Nacional de Córdoba and CONICET, Córdoba, Argentina, Tel +5435 1535 3800

Received: July 29, 2018 | Published: August 27, 2018

Citation: Elaskari S, del Río E, Gutiérrez Marcantoni L. Some characteristics of the M function methodology to describe the reinjection process in chaotic intermittency. Open Acc J Math Theor Phy. 2018;1(4):168-173 DOI: 10.15406/oajmtp.2018.01.00029

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Abstract

The M function methodology allows calculating the statistical properties of intermittency in a broad class of one-dimensional maps. It has shown to be very accurate in type I, II, III and V intermittencies; and it also includes the uniform reinjection as a particular case. This paper studies some properties of theM function methodology. We establish the conditions that a reinjection probability density function must verify to obtain m=1/2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyBaiabg2da9iaaigdacaGGVaGaaGOmaiaacYcaaaa@3BD8@ and we describe new pathological cases where the reinjection is not uniform, but the characteristic relation is the same that for uniform reinjection.

Keywords: m function methodology, reinjection process, chaotic intermittency, one-dimensional maps

Introduction

Intermittency is a traditional route to chaos, where a system evolves from regular behavior to chaotic behavior and trajectories alternate chaotic bursts and regular phases. The regular or laminar phases correspond to regions of pseudo-equilibrium or pseudo-periodic solutions, while the bursts are regions with chaotic evolution. In the early eighties, intermittency was classified, according to the system Floquet multipliers or the local Poincaré map eigenvalues, into three different types known as I, II and III.1‒4 Nevertheless, more recent advances have included other types such as V, X, on-off, in-out, ring and eyelet.5‒10 In engineering, biology, physics and chemistry there are several systems in which chaotic intermittency has been observed.11‒21 Furthermore, intermittency has been found in economics and medicine systems.22‒24 Consequently, a more accurate intermittency description might help to increase the knowledge about all these phenomena.

Intermittency behavior is defined by both the local map around the unstable or vanished fixed point and the reinjection mechanism.1,2,4 The reinjection mechanism maps trajectories from the chaotic zone to the laminar one, which is described by the reinjection probability density function (RPD). Accordingly, the accurate evaluation of the RPD function has a strong influence to describe the intermittency phenomenon correctly. Note that, the calculation of the RPD function from data series (experimental or numerical), is not a simple activity owing in no small amount of data needed and the statistical fluctuations involved in the numerical computations or experimental measurements. In consequence, many approaches have been utilized to describe the RPD function, where the most common one was a constant RPD (uniform reinjection).

A more general methodology to achieve the reinjection probability density function has been elaborated in the last decade, which is named the M function methodology. It includes the uniform reinjection as only a particular case. This methodology has shown to be very accurate for a wide class of maps showing type I, II, III and V intermittencies.4,25‒38 In this paper, we analyze some characteristics of this methodology. We have shown the M function methodology works very accurately for several maps, but there are pathological cases where it can only partially describe the intermittency reinjection process.

Evaluation of the RPD function

Let us consider a general one-dimensional map: x n+1 = F( x n ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaaeiiaiaad6gacqGHRaWkcaaIXaGaaeiiaiabg2da 9iaabccacaWGgbWdamaabmaabaWdbiaadIhacaqGGaGaamOBaaWdai aawIcacaGLPaaapeGaaiilaaaa@4429@ which shows chaotic intermittency. To describe the reinjection process, we should evaluate the RPD function, φ( x ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHgpGApaWaaeWaaeaapeGaamiEaaWdaiaawIcacaGLPaaa peGaaiilaaaa@3D09@ which determines the probability density that trajectories are reinjected into a point x inside the laminar interval.1,2,4

A new theoretical scheme to evaluate the intermittency reinjection process is the M function methodology, in which the RPD is indirectly obtained. First, the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function must be evaluated:4,25‒39

M( x )={ x ^ x τϕ( τ )dτ x ^ x ϕ( τ )dτ ,  if  x ^ x ϕ( τ )dτ0, 0,                otherwise, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaGabaWdaeaafaqabeGabaaabaWdbmaalaaapaqaa8qadaqfWa qab8aabaWdbiqadIhagaqcaaWdaeaapeGaamiEaaWdaeaapeGaey4k IipaaiaaykW7cqaHepaDcqaHvpGzdaqadaWdaeaapeGaeqiXdqhaca GLOaGaayzkaaGaamizaiabes8a0bWdaeaapeWaaubmaeqapaqaa8qa ceWG4bGbaKaaa8aabaWdbiaadIhaa8aabaWdbiabgUIiYdaacaaMc8 Uaeqy1dy2aaeWaa8aabaWdbiabes8a0bGaayjkaiaawMcaaiaadsga cqaHepaDaaGaaiilaiaacckacaGGGcGaamyAaiaadAgacaGGGcWaay bCaeqapaqaa8qaceWG4bGbaKaaa8aabaWdbiaadIhaa8aabaWdbiab gUIiYdaacqaHvpGzdaqadaWdaeaapeGaeqiXdqhacaGLOaGaayzkaa Gaamizaiabes8a0jabgcMi5kaaicdacaGGSaaapaqaa8qacaaIWaGa aiilaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaGPaVlaa cckacaGGGcGaam4BaiaadshacaWGObGaamyzaiaadkhacaWG3bGaam yAaiaadohacaWGLbGaaiilaaaaaiaawUhaaaaa@8FDC@  (1)

Where x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaaaaa@38E5@ is the lower boundary of reinjection. From the data series the x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaaaaa@38E5@ calculation is straightforward:

M( x ) 1 N J=1 N x j   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH fjcqdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamOtaaaadaGfWbqab8 aabaWdbiaadQeacqGH9aqpcaaIXaaapaqaa8qacaWGobaapaqaa8qa cqGHris5aaGaamiEamaaBaaajuaibaGaamOAaaqabaqcfaOaaiiOaa aa@483D@  (2)

Where the reinjection points { x,j } j=1 N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaGadaWdaeaaieWapeGaa8hEaiaacYcacaWFQbaacaGL7bGa ayzFaaWdamaaDaaajuaibaWdbiaa=PgacqGH9aqpcaaIXaaapaqaa8 qacaWFobaaaaaa@40D1@ must be sorted from the lowest to the highest,25‒33 i.e. x j x j+1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbmqcfaieaa aaaaaaa8qacaWF4bWdamaaBaaajuaibaWdbiaa=PgaaKqba+aabeaa peGaeyizImQaa8hEa8aadaWgaaqcfasaa8qacaWFQbGaey4kaSIaaG ymaaWdaeqaaKqbakaac6caaaa@41D6@

Previous studies have shown a linear M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ for a wide class of maps with type I, II, III and V intermittencies:4,25‒39

M( x )={ m( x x ^ )+x,       if      x ^ xc, 0,                 otherwhise,   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaGabaWdaeaafaqabeGabaaabaWdbiaad2gadaqadaWdaeaape GaamiEaiabgkHiTiqadIhagaqcaaGaayjkaiaawMcaaiabgUcaRiaa dIhacaGGSaGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaadMgacaWGMbGaaiiOaiaacckacaGGGcGaaiiOaiaacckaceWG 4bGbaKaacqGHKjYOcaWG4bGaeyizImQaam4yaiaacYcaa8aabaWdbi aaicdacaGGSaGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaWGVbGaamiDaiaadIgacaWGLbGaamOCaiaa dEhacaWGObGaamyAaiaadohacaWGLbGaaiilaaaacaGGGcaacaGL7b aaaaa@7C85@ (3)

c is the limit of the laminar interval.

In the previous equation the main parameter is m( 0,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyicI48aaeWaa8aabaWdbiaaicdacaGGSaGaaGym aaGaayjkaiaawMcaaaaa@3E1B@  (the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function slope) which is determined by the nonlinear map. When M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ is a linear function, it has been shown the RPD function is a power law:4,25-26,28,31,38,39

ϕ( x )=b( α ) ( x x ^ ) α ,     with    α= 2m1 1m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHvpGzdaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaiab g2da9iaadkgadaqadaWdaeaapeGaeqySdegacaGLOaGaayzkaaWaae Waa8aabaWdbiaadIhacqGHsislceWG4bGbaKaaaiaawIcacaGLPaaa daahaaqcfasabeaacqaHXoqyaaqcfaOaaiilaiaacckacaGGGcGaai iOaiaacckacaGGGcGaae4DaiaabMgacaqG0bGaaeiAaiaabckacaqG GcGaaeiOaiaacckacqaHXoqycqGH9aqpdaWcaaWdaeaapeGaaGOmai aad2gacqGHsislcaaIXaaapaqaa8qacaaIXaGaeyOeI0IaamyBaaaa aaa@6031@  (4)

Where b( α )= α+1 ( c x ^ ) ( α+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGIbWaaeWaa8aabaWdbiabeg7aHbGaayjkaiaawMcaaiab g2da9maalaaapaqaa8qacqaHXoqycqGHRaWkcaaIXaaapaqaa8qada qadaWdaeaapeGaam4yaiabgkHiTiqadIhagaqcaaGaayjkaiaawMca a8aadaahaaqcfasabeaajuaGpeWaaeWaaKqbG8aabaWdbiabeg7aHj abgUcaRiaaigdaaiaawIcacaGLPaaaaaaaaaaa@4B2F@ is the normalization parameter.

From Eq. (4) two parameters are only needed to describe the RPD function: m and x ^ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaacaGGSaaaaa@3995@ which are calculated from M( x ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaGG Uaaaaa@3C01@  m is the slope of M( x ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaGG Saaaaa@3BFF@ and it verifies M( x ^ ) = x ^ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiqadIhagaqcaaGaayjkaiaawMca aiaacckacqGH9aqpceWG4bGbaKaapaGaaiilaaaa@3F55@ i.e. it allows to calculate x ^ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaacaGGUaaaaa@3997@ Then, the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function stores the map nonlinear information.4,35,36

Finally, note that uniform reinjection ( α = 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaqadaWdaeaapeGaeqySdeMaaiiOaiabg2da9iaacckacaaI WaaacaGLOaGaayzkaaaaaa@3F27@ is only a particular case of the new theoretical formulation when m=1/2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyypa0JaaGymaiaac+cacaaIYaGaaiOlaaaa@3CAC@

Some characteristic of the M(x) function

In this section some properties of the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function are studied. We have special interest to describe the behavior of ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHvpGzcaGGOaGaamiEaiaacMcaaaa@3BF6@ and M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ functions, mainly when the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function slope is m=1/2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyypa0JaaGymaiaac+cacaaIYaGaaiOlaaaa@3CAC@

In the following Theorem, will expose two conditions that the RPD must verify to obtain m=1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyypa0JaaGymaiaac+cacaaIYaaaaa@3BFA@ in Eq. (3).

Theorem:

If the reinjection probability density function, φ( x ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHgpGApaWaaeWaaeaapeGaamiEaaWdaiaawIcacaGLPaaa caGGSaaaaa@3CF9@ satisfies the following conditions:

  1. ϕ( x ^ )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHvpGzcaGGOaGabmiEayaajaGaaiykaiabgcMi5kaaicda aaa@3E87@
  2. dϕ( x ) dx | x= x ^   is bounded MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabcaWdaeaapeWaaSaaa8aabaWdbiaadsgacqaHvpGzdaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamizaiaadI haaaaacaGLiWoapaWaaSbaaeaapeGaamiEaiabg2da9iqadIhagaqc aaWdaeqaa8qacaGGGcGaaiiOaiaadMgacaWGZbGaaiiOaiaadkgaca WGVbGaamyDaiaad6gacaWGKbGaamyzaiaadsgaaaa@505C@

Then the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function can be approximated as M( x )=x/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH 9aqpcaWG4bGaai4laiaaikdaaaa@3EC1@ for points close enough to x = x ^ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bGaaiiOaiabg2da9iqadIhagaqcaiaac6caaaa@3CBE@

Proof:

To prove this hypothesis, the slope of the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function for x close to x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaaaaa@38E5@ can be derived from Eq. (1):

lim x x ^ dM( x ) dx = lim x x ^ xϕ( x ) x ^ x ϕ( τ )dτϕ( x ) x ^ x τϕ( τ )dτ ( x ^ x ϕ( τ )dτ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWdaeaapeGaamiEaiab gkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWdaeaapeGaamizaiaad2 eadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamiz aiaadIhaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaa WdaeaapeGaamiEaiabgkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWd aeaapeGaamiEaiabew9aMnaabmaapaqaa8qacaWG4baacaGLOaGaay zkaaWaaubmaeqapaqaa8qaceWG4bGbaKaaa8aabaWdbiaadIhaa8aa baWdbiabgUIiYdaacqaHvpGzdaqadaWdaeaapeGaeqiXdqhacaGLOa GaayzkaaGaamizaiabes8a0jabgkHiTiabew9aMnaabmaapaqaa8qa caWG4baacaGLOaGaayzkaaWaaubmaeqapaqaa8qaceWG4bGbaKaaa8 aabaWdbiaadIhaa8aabaWdbiabgUIiYdaacqaHepaDcqaHvpGzdaqa daWdaeaapeGaeqiXdqhacaGLOaGaayzkaaGaamizaiabes8a0bWdae aapeWaaeWaa8aabaWdbmaavadabeWdaeaapeGabmiEayaajaaapaqa a8qacaWG4baapaqaa8qacqGHRiI8aaGaeqy1dy2aaeWaa8aabaWdbi abes8a0bGaayjkaiaawMcaaiaadsgacqaHepaDaiaawIcacaGLPaaa paWaaWbaaeqabaWdbiaaikdaaaaaaaaa@8299@  (5)

Which by the L’Hôpital rule, gives

lim x x ^ dM( x ) dx = lim x x ^ ϕ( x ) x ^ x ϕ( τ )dτ+x ϕ ( x ) x ^ x ϕ( τ )dτ ϕ ( x ) x ^ x τϕ( τ )dτ 2ϕ( x ) x ^ x ϕ( τ )dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWdaeaapeGaamiEaiab gkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWdaeaapeGaamizaiaad2 eadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamiz aiaadIhaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaa WdaeaapeGaamiEaiabgkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWd aeaapeGaeqy1dy2aaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaada qfWaqab8aabaWdbiqadIhagaqcaaWdaeaapeGaamiEaaWdaeaapeGa ey4kIipaaiabew9aMnaabmaapaqaa8qacqaHepaDaiaawIcacaGLPa aacaWGKbGaeqiXdqNaey4kaSIaamiEaiqbew9aMzaafaWaaeWaa8aa baWdbiaadIhaaiaawIcacaGLPaaadaqfWaqab8aabaWdbiqadIhaga qcaaWdaeaapeGaamiEaaWdaeaapeGaey4kIipaaiabew9aMnaabmaa paqaa8qacqaHepaDaiaawIcacaGLPaaacaWGKbGaeqiXdqNaeyOeI0 Iafqy1dyMbauaadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaamaa vadabeWdaeaapeGabmiEayaajaaapaqaa8qacaWG4baapaqaa8qacq GHRiI8aaGaeqiXdqNaeqy1dy2aaeWaa8aabaWdbiabes8a0bGaayjk aiaawMcaaiaadsgacqaHepaDa8aabaWdbiaaikdacqaHvpGzdaqada WdaeaapeGaamiEaaGaayjkaiaawMcaamaavadabeWdaeaapeGabmiE ayaajaaapaqaa8qacaWG4baapaqaa8qacqGHRiI8aaGaeqy1dy2aae Waa8aabaWdbiabes8a0bGaayjkaiaawMcaaiaadsgacqaHepaDaaaa aa@96E6@  (6)

The last expression results:

lim x x ^ dM( x ) dx = lim x x ^ [ 1 2 + 1 2 x ϕ ( x ) ϕ( x ) 1 2 ϕ ( x ) ϕ( x ) x ^ x τϕ( τ )dτ x ^ x ϕ( τ )dτ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWdaeaapeGaamiEaiab gkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWdaeaapeGaamizaiaad2 eadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamiz aiaadIhaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaa WdaeaapeGaamiEaiabgkziUkqadIhagaqcaaWdaeqaa8qadaWadaWd aeaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaey4kaS YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaWaaSaaa8aabaWd biaadIhacuaHvpGzgaqbamaabmaapaqaa8qacaWG4baacaGLOaGaay zkaaaapaqaa8qacqaHvpGzdaqadaWdaeaapeGaamiEaaGaayjkaiaa wMcaaaaacqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaa aadaWcaaWdaeaapeGafqy1dyMbauaadaqadaWdaeaapeGaamiEaaGa ayjkaiaawMcaaaWdaeaapeGaeqy1dy2aaeWaa8aabaWdbiaadIhaai aawIcacaGLPaaaaaWaaSaaa8aabaWdbmaavadabeWdaeaapeGabmiE ayaajaaapaqaa8qacaWG4baapaqaa8qacqGHRiI8aaGaeqiXdqNaeq y1dy2aaeWaa8aabaWdbiabes8a0bGaayjkaiaawMcaaiaadsgacqaH epaDa8aabaWdbmaavadabeWdaeaapeGabmiEayaajaaapaqaa8qaca WG4baapaqaa8qacqGHRiI8aaGaeqy1dy2aaeWaa8aabaWdbiabes8a 0bGaayjkaiaawMcaaiaadsgacqaHepaDaaaacaGLBbGaayzxaaGaaG PaVdaa@86ED@  (7)

Applying the L’Hôpital rule only on the last term, we can obtain:

lim x x ^ dM( x ) dx = lim x x ^ [ 1 2 + 1 2 x ϕ ( x ) ϕ( x ) 1 2 x ϕ ( x ) ϕ( x ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaCbeae aaqaaaaaaaaaWdbiGacYgacaGGPbGaaiyBaaWdaeaapeGaamiEaiab gkziUkqadIhagaqcaaWdaeqaa8qadaWcaaWdaeaapeGaamizaiaad2 eadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamiz aiaadIhaaaGaeyypa0ZdamaaxababaWdbiGacYgacaGGPbGaaiyBaa WdaeaapeGaamiEaiabgkziUkqadIhagaqcaaWdaeqaa8qadaWadaWd aeaapeWaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaGaey4kaS YaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaikdaaaWaaSaaa8aabaWd biaadIhacuaHvpGzgaqbamaabmaapaqaa8qacaWG4baacaGLOaGaay zkaaaapaqaa8qacqaHvpGzdaqadaWdaeaapeGaamiEaaGaayjkaiaa wMcaaaaacqGHsisldaWcaaWdaeaapeGaaGymaaWdaeaapeGaaGOmaa aadaWcaaWdaeaapeGaamiEaiqbew9aMzaafaWaaeWaa8aabaWdbiaa dIhaaiaawIcacaGLPaaaa8aabaWdbiabew9aMnaabmaapaqaa8qaca WG4baacaGLOaGaayzkaaaaaaGaay5waiaaw2faaaaa@6B88@  (8)

Because  ϕ( x )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGGcGaeqy1dy2aaeWaa8aabaWdbiaadIhaaiaawIcacaGL PaaacqGHGjsUcaaIWaaaaa@3FEA@ and dϕ( x ) dx | x= x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabcaWdaeaapeWaaSaaa8aabaWdbiaadsgacqaHvpGzdaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdaeaapeGaamizaiaadI haaaaacaGLiWoapaWaaSbaaKqbGeaapeGaamiEaiabg2da9iqadIha gaqcaaqcfa4daeqaaaaa@4532@ is bounded, we obtain 1/2 for the limit given in Eq. 6.

This Theorem generalizes previous developments in which x ^ = x 0 =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qaceWG4bGbaKaacqGH9aqpcaWG4bWdamaaBaaajuaibaWdbiaa icdaa8aabeaajuaGpeGaeyypa0JaaGimaiaacYcaaaa@3F2D@ where x 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaGimaaaa@3C6A@ is the vanished or unstable fixed point.4,26 For the Theorem, the LBR ( x ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaGGOaGabmiEayaajaGaaiykaaaa@3A3E@ can be any point inside the laminar interval, being the vanished or unstable fixed point x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaaaaa@3A9A@ only a particular case.

If the conditions (1) and (2) are verified, then for a small laminar interval ( c 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aaqaaaaaaaaaWdbiaadogacaqGGaGaeyOKH4QaaGimaaWdaiaawIca caGLPaaapeGaaiilaaaa@3E62@ the RPD can be uniform, and the classical theory is valid. However, there could be RPDs satisfying the two conditions and not having uniform reinjection; we call these RPDs as pathological cases.

Pathological case for type II intermittency

Type II intermittency starts from a subcritical Hopf bifurcation.2,4 Then, two complex-conjugate eigenvalues of the system leave the unit circle. When a control parameter (ε) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiikaiabew7aLjaacMcaaaa@3A06@ is less than a critical value, the map has a stable fixed point. On the other hand, if the control parameter rises above this threshold, a bifurcation leads to a new chaotic attractor. The old attractor remains as a subset of the new attractor.

In several previous studies the Theorem of Section 3 has been verified for type I, II, III and V intermittencies with and without lower boundary of reinjection.4,25‒38

In this section, we study the intermittency process for a map showing type II intermittency. The map is an extension of those studied25,40 and it is given by:

F( x )={ F 1 ( x )=( 1ε )xa x 3 ,  x< x r F 2 ( x )=w ( x x r 1 x r ) γ 2 ,  x> x r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacqGH 9aqpdaGabaWdaeaafaqabeGabaaabaWdbiaadAeapaWaaSbaaKqbGe aapeGaaGymaaqcfa4daeqaa8qadaqadaWdaeaapeGaamiEaaGaayjk aiaawMcaaiabg2da9maabmaapaqaa8qacaaIXaGaeyOeI0IaeqyTdu gacaGLOaGaayzkaaGaamiEaiabgkHiTiaadggacaWG4bWdamaaCaaa beqcfasaa8qacaaIZaaaaKqbakaacYcacaGGGcGaaiiOaiaadIhacq GH8aapcaWG4bWdamaaBaaajuaibaWdbiaadkhaa8aabeaaaKqbagaa peGaamOra8aadaWgaaqcfasaa8qacaaIYaaajuaGpaqabaWdbmaabm aapaqaa8qacaWG4baacaGLOaGaayzkaaGaeyypa0Jaam4Damaabmaa paqaa8qadaWcaaWdaeaapeGaamiEaiabgkHiTiaadIhapaWaaSbaaK qbGeaapeGaamOCaaWdaeqaaaqcfayaa8qacaaIXaGaeyOeI0IaamiE a8aadaWgaaqcfasaa8qacaWGYbaapaqabaaaaaqcfa4dbiaawIcaca GLPaaapaWaaWbaaeqabaWdbiabeo7aNnaaBaaajuaibaGaaGOmaaqc fayabaaaaiaacYcacaGGGcGaaiiOaiaadIhacqGH+aGpcaWG4bWdam aaBaaajuaibaWdbiaadkhaaKqba+aabeaaaaaapeGaay5Eaaaaaa@74C3@  (9)

x r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaadkhaaKqba+aabeaaaaa@3AD7@ is obtained from F 1 ( x r )=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaaBaaajuaibaWdbiaaigdaa8aabeaajuaGpeWa aeWaa8aabaWdbiaadIhapaWaaSbaaKqbGeaapeGaamOCaaqcfa4dae qaaaWdbiaawIcacaGLPaaacqGH9aqpcaaIXaGaaiilaaaa@41A1@ and 0  w  1. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaIWaGaaiiOaiabgsMiJkaacckacaWG3bGaaiiOaiabgsMi JkaacckacaaIXaGaaiOlaaaa@42F5@  The map has a fixed point x 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaaicdaaKqba+aabeaapeGa eyypa0JaaGimaaaa@3C6A@  which is stable for 2 < ε < 0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqGHsislcaaIYaGaaiiOaiabgYda8iaacckacqaH1oqzcaGG GcGaeyipaWJaaiiOaiaaicdacaGGUaaaaa@432C@ It becomes unstable for ε > 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzcaGGGcGaeyOpa4JaaiiOaiaaicdaaaa@3D89@ and type II intermittency can happen. The laminar interval is [ 0, c ]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaaMc8+aamWaa8aabaWdbiaaicdacaGGSaGaaiiOaiaadoga aiaawUfacaGLDbaacaGGUaaaaa@3F9C@  In Eq. (9), F 1 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaaBaaajuaibaWdbiaaigdaa8aabeaajuaGpeWa aeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3D1E@ function is the typical local map for type II intermittency and the nonlinear function, F 2 ( x ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGgbWdamaaBaaajuaibaWdbiaaikdaaKqba+aabeaapeWa aeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaacaGGSaaaaa@3DCF@ generates the reinjection process. Therefore, the exponents γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzdaWgaaqcfasaaiaaigdaaeqaaaaa@3A89@  and γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzdaWgaaqcfasaaiaaikdaaKqbagqaaaaa@3B18@ will drive the reinjection mechanism.

To numerically analyze type II intermittency phenomenon for this map, we consider the following parameters: γ 1 =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzdaWgaaqcfasaaiaaigdaaeqaaKqbakabg2da9iaa ikdacaGGSaaaaa@3D89@ γ 1 =0.87, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHZoWzdaWgaaqcfasaaiaaigdaaKqbagqaaiabg2da9iaa icdacaGGUaGaaGioaiaaiEdacaGGSaaaaa@3FBC@ w=1/2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG3bGaeyypa0JaaGymaiaac+cacaaIYaGaaiilaaaa@3CB4@ ε=0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaH1oqzcqGH9aqpcaaIWaGaaiOlaiaaicdacaaIWaGaaGym aaaa@3E20@ and N=100000, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGobGaeyypa0JaaGymaiaaicdacaaIWaGaaGimaiaaicda caaIWaGaaiilaaaa@3EBE@ and we apply the M function methodology to calculate the RPD function. Figure 1 displays the results. Red points represent the numerical data and blue lines the theoretical results calculated using Eqs. (2) and (4).

Figure 1 Type II intermittency. M(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eaca GGOaGaamiEaiaacMcaaaa@3A0E@ and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMj aacIcacaWG4bGaaiykaaaa@3B04@ functions for γ 1 =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGymaaqabaqcfaOaeyypa0JaaGOmaiaacYcaaaa@3C97@ γ 2 =0.87, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGOmaaqcfayabaGaeyypa0JaaGimaiaac6cacaaI 4aGaaG4naiaacYcaaaa@3ECB@ w=0.5, ε=0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew7aLj abg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIXaaaaa@3D2E@ and N=100000. Red points represent the numerical data, and the blue line the theoretical results calculated using Eqs. 2 and 4.

From Figure 1(A) we can observe the theoretical RPD cannot approach accurately the numerical RPD. Beside this, the numerical RPD verifies the conditions ϕ( x ^ )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacqaHvpGzdaqadaWdaeaapeGabmiEayaajaaacaGLOaGaayzk aaGaeyiyIKRaaGimaaaa@3ED7@ and bounded dϕ dx | x= x ^ ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaabcaWdaeaapeWaaSaaa8aabaWdbiaadsgacqaHvpGza8aa baWdbiaadsgacaWG4baaaaGaayjcSdWdamaaBaaabaWdbiaadIhacq GH9aqpceWG4bGbaKaaa8aabeaacaGG7aaaaa@4291@ therefore following the previous theorem the M( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGnbWaaeWaa8aabaWdbiaadIhaaiaawIcacaGLPaaaaaa@3B4F@ function slope must be m=1/2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyypa0JaaGymaiaac+cacaaIYaGaaiOlaaaa@3CAC@ Using the date shown in Figure 1(B) we can calculate m0.5060.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyyrIaKaaGimaiaac6cacaaI1aGaaGimaiaaiAda cqGHfjcqcaaIWaGaaiOlaiaaiwdaaaa@4101@ in agreement with the theory.

Following Section 2, if m0.5, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWGTbGaeyyrIaKaaGimaiaac6cacaaI1aGaaiilaaaa@3CD9@ the RPD must be constant (uniform reinjection). However, the numerical data in Figure 1(B) (red points) do not correspond to uniform reinjection. Therefore, it is a pathological case for the M function methodology; then the theoretical RPD (calculated using the M function methodology) does not show high accuracy with the numerical data.

To analyze the intermittency behavior for this pathological test, we study the map at points previous to reinjection (pre-reinjection points).39 If the reinjection points are x n+1 =F( x n ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaad6gacqGHRaWkcaaIXaaa juaGpaqabaWdbiabg2da9iaadAeadaqadaWdaeaapeGaamiEa8aada Wgaaqcfasaa8qacaWGUbaapaqabaaajuaGpeGaayjkaiaawMcaaiaa cYcaaaa@43B5@ we study the map derivative at x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaad6gaaKqba+aabeaaaaa@3AD4@ points; because they have influence on the RPD function.35 The derivative of map (9) can be written as:

dF( x ) dx ={ d F 1 ( x ) dx =( 1+ε )+3a x 2 ,                                                                       x x r d F 2 ( x ) dx = w ( 1 x r ) γ 1 γ 1 ( x x r ) γ 1 1  + 1w ( 1 x r ) γ 2 γ 2 ( x x r ) γ 2 1 ,                  x> x r    MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadsgacaWGgbWaaeWaa8aabaWdbiaadIha aiaawIcacaGLPaaaa8aabaWdbiaadsgacaWG4baaaiabg2da9maace aaeaqabeaadaWcaaWdaeaapeGaamizaiaadAeapaWaaSbaaKqbGeaa peGaaGymaaqcfa4daeqaa8qadaqadaWdaeaapeGaamiEaaGaayjkai aawMcaaaWdaeaapeGaamizaiaadIhaaaGaeyypa0ZaaeWaa8aabaWd biaaigdacqGHRaWkcqaH1oqzaiaawIcacaGLPaaacqGHRaWkcaaIZa GaamyyaiaadIhapaWaaWbaaKqbGeqabaWdbiaaikdaaaqcfaOaaiil aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaaccka caGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOai aacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGa aiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckaca GGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaamiEaiabgs MiJkaadIhapaWaaSbaaKqbGeaapeGaamOCaaWdaeqaaaqcfayaa8qa daWcaaWdaeaapeGaamizaiaadAeapaWaaSbaaKqbGeaapeGaaGOmaa qcfa4daeqaa8qadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWd aeaapeGaamizaiaadIhaaaGaeyypa0ZaaSaaa8aabaWdbiaadEhaa8 aabaWdbmaabmaapaqaa8qacaaIXaGaeyOeI0IaamiEa8aadaWgaaqc fasaa8qacaWGYbaajuaGpaqabaaapeGaayjkaiaawMcaa8aadaahaa qcfasabeaapeGaeq4SdCwcfa4damaaBaaajuaibaWdbiaaigdaa8aa beaaaaaaaKqba+qacqaHZoWzpaWaaSbaaKqbGeaapeGaaGymaaWdae qaaKqba+qadaqadaWdaeaapeGaamiEaiabgkHiTiaadIhapaWaaSba aKqbGeaapeGaamOCaaqcfa4daeqaaaWdbiaawIcacaGLPaaapaWaaW baaKqbGeqabaWdbiabeo7aNLqba+aadaWgaaqcfasaa8qacaaIXaaa paqabaWdbiabgkHiTiaaigdaaaqcfaOaaiiOaiabgUcaRmaalaaapa qaa8qacaaIXaGaeyOeI0Iaam4DaaWdaeaapeWaaeWaa8aabaWdbiaa igdacqGHsislcaWG4bWdamaaBaaajuaibaWdbiaadkhaaKqba+aabe aaa8qacaGLOaGaayzkaaWdamaaCaaabeqaa8qacqaHZoWzpaWaaSba aKqbGeaapeGaaGOmaaWdaeqaaaaaaaqcfa4dbiabeo7aN9aadaWgaa qcfasaa8qacaaIYaaapaqabaqcfa4dbmaabmaapaqaa8qacaWG4bGa eyOeI0IaamiEa8aadaWgaaqcfasaa8qacaWGYbaapaqabaaajuaGpe GaayjkaiaawMcaa8aadaahaaqabKqbGeaapeGaeq4SdCwcfa4damaa BaaajuaibaWdbiaaikdaa8aabeaapeGaeyOeI0IaaGymaaaajuaGca GGSaGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaa cckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaai iOaiaacckacaGGGcGaamiEaiabg6da+iaadIhapaWaaSbaaKqbGeaa peGaamOCaaqcfa4daeqaa8qacaGGGcGaaiiOaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8oaaiaawUhaaaaa@18C8@  (10)

Figure 3 shows the Eq. (10) (red line) inside the interval [ x r 0.682, F 2 1 ( c )0.726473 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qadaWadaWdaeaapeGaamiEa8aadaWgaaqcfasaa8qacaWGYbaa juaGpaqabaWdbiabgwKiajaaicdacaGGUaGaaGOnaiaaiIdacaaIYa GaaiilaiaadAeapaWaa0baaKqbGeaapeGaaGOmaaWdaeaapeGaeyOe I0IaaGymaaaajuaGdaqadaWdaeaapeGaam4yaaGaayjkaiaawMcaai abgwKiajaaicdacaGGUaGaaG4naiaaikdacaaI2aGaaGinaiaaiEda caaIZaaacaGLBbGaayzxaaaaaa@5079@ which contains all the pre-reinjection points, i.e x n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaad6gaaKqba+aabeaaaaa@3AD4@ points previous to reinjection. Moreover, this figure shows the derivative of the map used in25,40 with γ=0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaaaa@3BBE@ (blue line) and γ=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaigdaaaa@3A4F@ (green line):

F 1 ( x )={ F 1 ( x )=( 1+ε )x+a x 3 ,                      x x r d F 2 ( x ) dx = ( x x r 1 x r ) γ ,                        x> x r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOra8aadaWgaaqcfasaa8qacaaIXaaapaqabaqcfa4dbmaa bmaapaqaa8qacaWG4baacaGLOaGaayzkaaGaeyypa0Zaaiqaaqaabe qaaiaadAeapaWaaSbaaKqbGeaapeGaaGymaaWdaeqaaKqba+qadaqa daWdaeaapeGaamiEaaGaayjkaiaawMcaaiabg2da9maabmaapaqaa8 qacaaIXaGaey4kaSIaeqyTdugacaGLOaGaayzkaaGaamiEaiabgUca RiaadggacaWG4bWdamaaCaaajuaibeqaa8qacaaIZaaaaKqbakaacY cacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGc GaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaamiEaiabgsMi JkaadIhapaWaaSbaaKqbGeaapeGaamOCaaqcfa4daeqaaaqaa8qada WcaaWdaeaapeGaamizaiaadAeapaWaaSbaaKqbGeaapeGaaGOmaaqc fa4daeqaa8qadaqadaWdaeaapeGaamiEaaGaayjkaiaawMcaaaWdae aapeGaamizaiaadIhaaaGaeyypa0ZaaeWaa8aabaWdbmaalaaapaqa a8qacaWG4bGaeyOeI0IaamiEa8aadaWgaaqcfasaa8qacaWGYbaaju aGpaqabaaabaWdbiaaigdacqGHsislcaWG4bWdamaaBaaajuaibaWd biaadkhaaKqba+aabeaaaaaapeGaayjkaiaawMcaa8aadaahaaqcfa sabeaapeGaeq4SdCgaaKqbakaacYcacaGGGcGaaiiOaiaacckacaGG GcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacc kacaGGGcGaaiiOaiaacckacaGGGcGaaiiOaiaacckacaGGGcGaaiiO aiaacckacaGGGcGaaiiOaiaacckacaWG4bGaeyOpa4JaamiEa8aada Wgaaqcfasaa8qacaWGYbaajuaGpaqabaaaa8qacaGL7baaaaa@A769@  (11)

x r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaadkhaa8aabeaaaaa@3A4A@ verifies F 1 ( x r )=1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOra8aadaWgaaqcfasaa8qacaaIXaaapaqabaqcfa4dbmaa bmaapaqaa8qacaWG4bWdamaaBaaajuaibaWdbiaadkhaaKqba+aabe aaa8qacaGLOaGaayzkaaGaeyypa0JaaGymaiaacYcaaaa@40D0@ and x 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaaIWaaapaqabaqcfa4dbiab g2da9iaaicdaaaa@3B98@ is a fixed point of the map, which is stable for 2<ε<0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeyOeI0IaaGOmaiabgYda8iabew7aLjabgYda8iaaicdacaGG Uaaaaa@3DCA@

Note some differences between the three curves. The derivative of Eq. (9) (red line) has lower values regarding the derivative of Eq. (11) (blue line) –which is important for points close to x r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaieaaaaaa aaa8qacaWG4bWdamaaBaaajuaibaWdbiaadkhaa8aabeaaaaa@3A4A@ – and the second derivative of map (9) goes from negative to positive values inside the interval [0.682, 0.726473]. This derivative behavior generates a different reinjection process between maps (9) and (11).

The RPD for γ=0.4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaaaa@3BBE@ is given in Figure 2(B), note as x x ^ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEaiabgkziU+aaceWG4bGbaKaacaGGSaaaaa@3BBC@ the derivative dF( x )/dx, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaiabgkziUkabg6HiLkaacYcaaaa@4206@ which implies ϕ(x)0. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqy1dyMaaiikaiaadIhacaGGPaGaeyOKH4QaaGimaiaac6ca aaa@3E7D@ Also, for γ=0.4, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaiaacYcaaaa@3C6E@ the derivative dF( x )/dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaaaa@3DF8@ decreases inside the laminar interval.41‒47 On the other hand, for Eq. (9), the derivative F( x )/dx| x x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaqGaa8aabaWdbiaadAeadaqadaWdaeaapeGaamiEaaGaayjk aiaawMcaaiaac+cacaWGKbGaamiEaaGaayjcSdWdamaaBaaabaWdbi aadIhacqGHsgIRpaGabmiEayaajaaabeaaaaa@430A@  does not tend to infinite, but it takes higher values than in other points in the laminar interval. This result might seem contradictory with the analytical limit of Eq. (10) which tends to infinite when x x ^ . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEaiabgkziU+aaceWG4bGbaKaacaGGUaaaaa@3BBE@ This contradiction can be explained because Figure 3 is obtained by a discrete process where there are a large but finite number of reinjected points, and not by a continuous process with infinite number of reinjected points (like as of Eq. (10)).

Figure 2 Type II intermittency. M(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaad2eaca GGOaGaamiEaiaacMcaaaa@3A0E@ and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMj aacIcacaWG4bGaaiykaaaa@3B04@ functions for γ=0.4, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaiaacYcaaaa@3C6D@ ε=0.0001, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew7aLj abg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIWaGaaGymaiaacYca aaa@3E98@ c = 0.1 and N = 20000. Red points represent the numerical data, and the blue line the theoretical results calculated using Eqs. (2) and (4).

As the derivative F( x )/dx| x x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaqGaa8aabaWdbiaadAeadaqadaWdaeaapeGaamiEaaGaayjk aiaawMcaaiaac+cacaWGKbGaamiEaaGaayjcSdWdamaaBaaabaWdbi aadIhacqGHsgIRpaGabmiEayaajaaabeaaaaa@430A@ reaches higher values than in other points inside the laminar interval for map (9), then the RPD acquires lower and non-zero values close to x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qcaaaa@37F3@ (Figure 1(B)). As x increases, dF( x )/dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaaaa@3DF8@  decreases and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqy1dyMaaiikaiaadIhacaGGPaaaaa@3B24@ increases until dF( x )/dx=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaiabg2da9iaaicdaaaa@3FB8@ (inlection point), from this point dF( x )/dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaaaa@3DF8@ grows very softly and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqy1dyMaaiikaiaadIhacaGGPaaaaa@3B24@ decreases very softly too. The inflection point is x in 0.707, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaWGPbGaamOBaaWdaeqaaKqb a+qacqGHfjcqcaaIWaGaaiOlaiaaiEdacaaIWaGaaG4naiaacYcaaa a@408A@ it can be calculated by:

x in  =  x r +( 1 x r )  ( 1w w γ 1 ( 1 γ 2 ) γ 2 ( γ 1 1 ) ) 1 γ 1 γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaWGPbGaamOBaaWdaeqaaKqb a+qacaGGGcGaeyypa0JaaiiOaiaadIhapaWaaSbaaKqbGeaapeGaam OCaaqcfa4daeqaa8qacqGHRaWkdaqadaWdaeaapeGaaGymaiabgkHi TiaadIhapaWaaSbaaKqbGeaapeGaamOCaaqcfa4daeqaaaWdbiaawI cacaGLPaaacaGGGcWaaeWaa8aabaWdbmaalaaapaqaa8qacaaIXaGa eyOeI0Iaam4DaaWdaeaapeGaam4DaaaadaWcaaWdaeaapeGaeq4SdC 2damaaBaaajuaibaWdbiaaigdaa8aabeaajuaGpeWaaeWaa8aabaWd biaaigdacqGHsislcqaHZoWzpaWaaSbaaKqbGeaapeGaaGOmaaWdae qaaaqcfa4dbiaawIcacaGLPaaaa8aabaWdbiabeo7aN9aadaWgaaqc fasaa8qacaaIYaaajuaGpaqabaWdbmaabmaapaqaa8qacqaHZoWzpa WaaSbaaKqbGeaapeGaaGymaaWdaeqaaKqba+qacqGHsislcaaIXaaa caGLOaGaayzkaaaaaaGaayjkaiaawMcaa8aadaahaaqabeaapeWaaS aaa8aabaWdbiaaigdaa8aabaWdbiabeo7aN9aadaWgaaqcfasaa8qa caaIXaaapaqabaqcfa4dbiabgkHiTiabeo7aN9aadaWgaaqcfasaa8 qacaaIYaaajuaGpaqabaaaaaaaaaa@6D5D@ (12)

Finally, we highlight that for uniform reinjection dF( x )/dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamizaiaadAeadaqadaWdaeaapeGaamiEaaGaayjkaiaawMca aiaac+cacaWGKbGaamiEaaaa@3DF8@ is constant (green line in Figure 3).

Figure 3 Type II intermittency. The derivative of maps (9) and (11). The parameters for map 9 are γ 1 =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaGOmaiaacYcaaaa@3C97@ γ 2 =0.87, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGOmaaqcfayabaGaeyypa0JaaGimaiaac6cacaaI 4aGaaG4naiaacYcaaaa@3ECB@ w=0.5, ε=0.001 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew7aLj abg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIXaaaaa@3D2E@ and N=100000. The parameters for map 11 are γ=0.4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaaaa@3BBD@ (blue) and γ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaigdaaaa@3A4E@ (green), ε=0.001. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew7aLj abg2da9iaaicdacaGGUaGaaGimaiaaicdacaaIXaGaaiOlaaaa@3DE0@ Red points correspond to map 9. Blue and green points correspond to map (11) γ=0.4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaicdacaGGUaGaaGinaaaa@3BBD@ with and γ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNj abg2da9iaaigdaaaa@3A4E@ respectively. Figure (B) is a zoom of Figure (A).

The characteristic relation is an important expression to determine the intermittency behavior. Using the M function methodology, this relation for type II intermittency is:4

l ¯ α ε β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadYgaga qeaabaaaaaaaaapeGaeqySdeMaaGPaVlabew7aL9aadaahaaqcfasa beaapeGaeyOeI0IaeqOSdigaaaaa@3FDD@ (13)

Where the critical exponent β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdigaaa@38A7@ is given:25

β=  3α2 31 = 3( 1m )1 ( 31 )( 1m )   the same thing MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaeyypa0JaaiiOamaalaaapaqaa8qacaaIZaGaeyOe I0IaeqySdeMaeyOeI0IaaGOmaaWdaeaapeGaaG4maiabgkHiTiaaig daaaGaeyypa0ZaaSaaa8aabaWdbiaaiodadaqadaWdaeaapeGaaGym aiabgkHiTiaad2gaaiaawIcacaGLPaaacqGHsislcaaIXaaapaqaa8 qadaqadaWdaeaapeGaaG4maiabgkHiTiaaigdaaiaawIcacaGLPaaa daqadaWdaeaapeGaaGymaiabgkHiTiaad2gaaiaawIcacaGLPaaaaa GaaiiOaiaacckacaWG0bGaamiAaiaadwgacaGGGcGaam4Caiaadgga caWGTbGaamyzaiaacckacaWG0bGaamiAaiaadMgacaWGUbGaam4zaa aa@628D@  (14)

and   l ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiiOa8aaceWGSbGbaKaaaaa@393A@ is the average laminar length, which can be calculated using:

I ¯ = x 0 c x 0 +c ϕ( x )l( x,c )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGPaVlqadMeagaqeaiabg2da9maapedabaGaeqy1dy2aaeWa a8aabaWdbiaadIhaaiaawIcacaGLPaaacaWGSbWaaeWaa8aabaWdbi aadIhacaGGSaGaam4yaaGaayjkaiaawMcaaiaadsgacaWG4baajuai baGaamiEaKqba+aadaWgaaqcKvaG=haapeGaaGimaaWdaeqaaKqbG8 qacqGHsislcaWGJbaabaGaamiEaKqba+aadaWgaaqcKvaG=haapeGa aGimaaWdaeqaaKqbG8qacqGHRaWkcaWGJbaajuaGcqGHRiI8aaaa@55F0@  (15)

l( x, c ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiBamaabmaapaqaa8qacaWG4bGaaiilaiaacckacaWGJbaa caGLOaGaayzkaaaaaa@3D58@ is the laminar length for each point in the laminar interval, which measures the number of iterations inside the laminar interval.2,4

To study the characteristic relation, we have carried out several numerical tests. Table 1 shows m and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdegaaa@38A5@ for different ε. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyTduMaaiOlaaaa@395F@  From the table, we can observe that α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdegaaa@38A5@ does not depend on the control parameter, ε. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqyTduMaaiOlaaaa@395F@

From Table 1, we have found α0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdeMaeyyrIaKaaGimaaaa@3A92@ and m1/2; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGPaVlaad2gacqGHfjcqcaaIXaGaai4laiaaikdacaGG7aaa aa@3D9F@ then the theoretical evaluation for the characteristic relation exponent, given by Eq. (14), is β1/2. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaeyyrIaKaaGymaiaac+cacaaIYaGaaiOlaaaa@3CB6@ To verify this exponent, we calculate l ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadYgaga qeaaaa@37EF@ for the tests included in Table 1. The results are exposed in Figure 4, which shows the numerical characteristic relation for the map (9). Red points are the numerical data, and the blue line is its linear interpolation. The slope of this straight line is approximately -0.495, which is very close to the theoretical exponent β=1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaeyypa0JaaGymaiaac+cacaaIYaaaaa@3BD7@ predicted using the M function methodology.4,25 Another consequence of the previous Theorem is that the traditional values of the characteristic relation exponent, β=1/2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaeyypa0JaaGymaiaac+cacaaIYaGaaiilaaaa@3C87@ can be obtained not only for uniform reinjection but also for any RPD holding the conditions (1) and (2). Figure 4 shows this behavior. Therefore, the test studied in this subsection does not have a uniform RPD, but its characteristic relation is equal to those obtained with uniform reinjection.

ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew7aLb aa@388D@

m

α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg7aHb aa@3885@

0.05

0.508

0.035

0.01

0.507

0.032

0.005

0.504

0.0189

0.001

0.506

0.024

0.0005

0.504

0.0188

0.0001

0.506

0.024

0.00001

0.505

0.0229

0.000001

0.505

0.0229

Table 1 Values of m, for different Parameters: w=0.5 and c=0.1

Figure 4 Type II intermittency. Characteristic relation for γ 1 =2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaGOmaiaacYcaaaa@3C97@ γ 1 =0.87, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGymaaqcfayabaGaeyypa0JaaGimaiaac6cacaaI 4aGaaG4naiaacYcaaaa@3ECA@ and w=0.5. Red points are the numerical average laminar length, and the blue line the linear interpolation.

Analysis and conclusion

In this paper, we have analyzed a theorem describing some properties of the M(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaacIcacaWG4bGaaiykaaaa@3A2E@ and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqy1dyMaaiikaiaadIhacaGGPaaaaa@3B24@ functions. We have been able to identify numerically two reinjection mechanisms satisfying this theorem. One of them corresponds to uniform reinjection process, and the M function methodology captures and describes it very well. Pathological cases like Eq. (9) give the other one. For these cases, the M(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamytaiaacIcacaWG4bGaaiykaaaa@3A2E@ function slope verifies m=1/2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyBaiabg2da9iaaigdacaGGVaGaaGOmaiaacYcaaaa@3BD8@ but there is not uniform reinjection, and the M unction methodology would seem to introduce errors in the RPD evaluation, but it calculates the characteristic relation correctly.

The map derivative for pathological cases at points previous to reinjection, x n x n+1 =F( x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaWGUbaajuaGpaqabaWdbiab gkHiTiaadIhapaWaaSbaaKqbGeaapeGaamOBaiabgUcaRiaaigdaa8 aabeaajuaGpeGaeyypa0JaamOramaabmaapaqaa8qacaWG4bWdamaa BaaajuaibaWdbiaad6gaa8aabeaaaKqba+qacaGLOaGaayzkaaaaaa@462A@ where x n+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaWGUbGaey4kaSIaaGymaaqc fa4daeqaaaaa@3B9E@ are the reinjected points–shows a different behavior regarding to the non-pathological cases (Figure 3). For pathological behavior, the map derivative does not tend to infinity close to x ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadIhaga qcaaaa@37F3@ which is produced by the discrete process and the finite number of reinjected points. Another aspect to highlight is that inside the laminar interval the second derivative goes from negative to positive values.

TheM function methodology has theoretically established that the characteristic relation exponent, β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaaiilaaaa@3957@ depends on α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdegaaa@38A5@ using Eqs. (13) and (14). Here, we have found numerically that the exponent, β, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqOSdiMaaiilaaaa@3957@ depends on α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdegaaa@38A5@ in the same way for both pathological cases and uniform reinjection processes. The characteristic relation for all cases with α0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeqySdeMaeyyrIaKaaGimaiaac6cacaaI1aaaaa@3C03@ results:

l ¯ α ε β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakqadYgaga qeaabaaaaaaaaapeGaeqySdeMaaGPaVlabew7aL9aadaahaaqcfasa beaapeGaeyOeI0IaeqOSdigaaaaa@3FDD@  (16)

It is a valuable property because the characteristic relation is used to describe the intermittency process. Therefore, when we obtain from experimental or numerical data a characteristic relation like Eq. (16), we have to study other parameters to determine the intermittency behavior, because it can be a process with uniform reinjection or a pathological case.

In some previous papers, we have verified that the M function methodology works accurately in several maps with different intermittency types. Besides, here we have shown there are pathological cases where this methodology partially describes the intermittency reinjection process. For these cases, theM function methodology captures very well de characteristic relation, but it could introduce errors in the RPD function evaluation. This phenomenon occurs because a discrete reinjection process around x 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaaIWaaajuaGpaqabaWdbiab g2da9iaaicdaaaa@3B98@ is studied; this process has a large but finite number of reinjected points. In this case none of the two exponents γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaGPaVlabeo7aNbaa@3A38@ in Eq. (9)( γ 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4SdC2aaSbaaKqbGeaacaaIXaaabeaajuaGcqGH9aqpcaaI Yaaaaa@3C07@ and γ 2 =0.87 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaeq4SdC2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaI WaGaaiOlaiaaiIdacaaI3aaaaa@3E3B@ ) dominate the reinjection mechanism. However, if we study a process with an infinite number of reinjected points, the RPD will tend to zero for x 0 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVye9Vfeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamiEa8aadaWgaaqcfasaa8qacaaIWaaajuaGpaqabaWdbiab g2da9iaaicdaaaa@3B98@ as shown Eq. (10), the Theorem of Section 4 would not be applied, and the M function methodology would work correctly.

Acknowledgements

This work was supported by CONICET, Universidad Nacional de Córdoba, and Universidad Politécnica de Madrid.

Conflict of interest

The author declares that there is no conflict of interest.

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