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eISSN: 2574-8092

International Robotics & Automation Journal

Mini Review Volume 4 Issue 3

New generalized chaos-geometric and neural networks approach to nonlinear dynamics of the complex systems

Alexander V Glushkov, Vasily V Buyadzhi, Olga Yu Khetselius, Valentin B Ternovsky

Department of Mathematics, Odessa State Environmental University, Ukraine

Correspondence: Alexander V Glushkov, Department of Mathematics, Odessa State Environmental University, Ukraine, Tel +380-482-32673, Fax +380-482-32673

Received: January 27, 2018 | Published: May 10, 2018

Citation: Glushkov AV, Buyadzhi VV, Khetselius OY. New generalized chaos-geometric and neural networks approach to nonlinear dynamics of the complex systems. Int Rob Auto J. 2018;4(3):167-169. DOI: 10.15406/iratj.2018.04.00116

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Abstract

We present a new generalized approach to modeling nonlinear processes of chaotic systems based on the known concept of compact geometric attractors, chaos theory methods in effective realization plus implemented neural networks simulation algorithm. Using information on the phase space evolution of the nonlinear process in time and the neural networks simulation techniques can be considered as one of the fundamentally new approaches in the construction of global nonlinear prediction models for evolutionary dynamics of the complex chaotic systems and accurate description of the structure of the corresponding strange attractors.

Keywords: nonlinear dynamics, complex chaotic systems, compact geometric attractors, chaos theory methods, neural networks simulation

Introduction

Multiple physical, chemical, biological, technical, communication, robotic and automation and other systems (devices) demonstrate the typical complex chaotic behaviour. In many important situations typical dynamics of these systems is the world of strong nonlinearity. In principle, the most conventional direct approach to dynamics treating problem consists in building an explanatory model using an initial data and parameterizing sources and interactions between process properties. Unfortunately, such that kind of approach is realized with difficulties and its outcomes are insufficiently correct. In the past few decades different chaos and dynamical system theories and topology models have given many useful insights to understand the output data generated by the complex nonlinear systems1–12 especially when traditional linear models are incorrect. We are developing a new approach to modeling nonlinear processes of chaotic systems based on the known CGA concept, chaos theory methods plus implemented NNW algorithms. Using information on the phase space evolution of the nonlinear process in time and the NNW simulation techniques can be considered as one of the fundamentally new approaches in the construction of global nonlinear models of the most effective and accurate description of the structure of the corresponding attractor for studied complex system.

Generalized chaos-geometric approach to complex system dynamics

The basic idea of the construction of our approach to prediction of chaotic properties of complex systems is in the use of the traditional concept of a CGA in which evolves the measurement data, plus the NNW algorithm implementation. Let us consider some scalar measurements s( n )=s( t 0 +nΔt )=s( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada qadaqaaiaad6gaaiaawIcacaGLPaaacqGH9aqpcaWGZbWaaeWaaeaa caWG0bWaaSbaaKqbGeaacaaIWaaajuaGbeaacqGHRaWkcaWGUbGaeu iLdqKaamiDaaGaayjkaiaawMcaaiabg2da9iaadohadaqadaqaaiaa d6gaaiaawIcacaGLPaaaaaa@4914@ , where t 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadshada WgaaqcfasaaiaaicdaaKqbagqaaaaa@396B@ is the start time, Δt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabfs5aej aadshaaaa@393A@ is the time step, and n is the number of the measurements. The main task is to reconstruct phase space using as well as possible information contained in s( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada qadaqaaiaad6gaaiaawIcacaGLPaaaaaa@3A4F@ . To do it, the method of using time-delay coordinates by Packard et al.3 can be used, the direct using lagged variables s( n+τ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadohada qadaqaaiaad6gacqGHRaWkcqaHepaDaiaawIcacaGLPaaaaaa@3CF6@  (here τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes8a0b aa@38A0@ is some integer to be defined) results in a coordinate system where a structure of orbits in phase space can be captured. A set of time lags is used to create a vector in d dimensions, y( n )=[ s( n ),s( n+τ ),s( n+2τ ),...,s( n+( d1 )τ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadMhada qadaqaaiaad6gaaiaawIcacaGLPaaacqGH9aqpdaWadaqaaiaadoha daqadaqaaiaad6gaaiaawIcacaGLPaaacaGGSaGaam4Camaabmaaba GaamOBaiabgUcaRiabes8a0bGaayjkaiaawMcaaiaacYcacaWGZbWa aeWaaeaacaWGUbGaey4kaSIaaGOmaiabes8a0bGaayjkaiaawMcaai aacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam4CamaabmaabaGaamOB aiabgUcaRmaabmaabaGaamizaiabgkHiTiaaigdaaiaawIcacaGLPa aacqaHepaDaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@5CBE@ , the required coordinates are provided. Here the dimension d is the embedding dimension, d E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqcfasaaiaadweaaKqbagqaaaaa@396B@ . To determine the proper time lag at the beginning one should use the known method of the linear ACF C L ( δ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadgeaca WGdbGaamOraiaadoeadaWgaaqcfasaaiaadYeaaKqbagqaamaabmaa baGaeqiTdqgacaGLOaGaayzkaaaaaa@3ED8@ and look for that time lag where C L ( δ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeada WgaaqaaKqzadGaamitaaqcfayabaWaaeWaaeaacqaH0oazaiaawIca caGLPaaaaaa@3D7F@  first passes through 0.4 The alternative additional approach is provided by the AMI method as an approach with so called nonlinear concept of independence. The further next step is to determine the embedding dimension, d E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqcfasaaiaadweaaKqbagqaaaaa@396B@ and correspondingly to reconstruct a Euclidean space R d MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada ahaaqabKqbGeaacaWGKbaaaaaa@38EB@ large enough so that the set of points d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqaaKqzadGaamyqaaqcfayabaaaaa@3A67@ can be unfolded without ambiguity. The dimension d E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqcfasaaiaadweaaKqbagqaaaaa@396B@ must be greater, or at least equal, than a dimension of attractor, d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqaaKqzadGaamyqaaqcfayabaaaaa@3A67@ , i.e. d E > d A MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqcfasaaiaadweaaKqbagqaaiabg6da+iaadsgadaWgaaqcfasa aiaadgeaaKqbagqaaaaa@3CFF@ . To reconstruct the attractor dimension and to study the signatures of chaos in a time series, one could use such methods as the CIA by Grassberger and Procaccia5 or the FNN one by Kennel et al.6 The principal question of studying any complex chaotic system is to build the corresponding prediction model and define how predictable is a chaotic system. The new element of our approach is using the NNW algorithm in forecasting nonlinear dynamics of chaotic systems.9,10 In terms of the neuro-informatics and neural networks theory the process of modelling the evolution of the system can be generalized to describe some evolutionary dynamic neuro-equations. Imitating the further evolution of a system within NNW simulation with the corresponding elements of the self-study, self- adaptation, etc., it becomes possible to significantly improve the prediction of its evolutionary dynamics. The fundamental parameters to be computed are the Kolmogorov entropy (and correspondingly the predictability measure as it can be estimated by the Kolmogorov entropy), the LE, the KYD etc. The LE is usually defined as asymptotic average rates and they are related to the Eigen values of the linearized dynamics across the attractor. Naturally, the knowledge of the whole LE allows determining other important invariants such as the Kolmogorov entropy and the attractor's dimension. The Kolmogorov entropy is determined by the sum of the positive LE. The estimate of the dimension of the attractor is provided by the Kaplan and Yorke conjecture d L =j+ i=1 n λ i / | λ j+1 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadsgada WgaaqcfasaaiaadYeaaKqbagqaaiabg2da9iaadQgacqGHRaWkdaae WbqaamaalyaabaGaeq4UdW2aaSbaaKqbGeaacaWGPbaajuaGbeaaae aadaabdaqaaiabeU7aSnaaBaaajuaibaGaamOAaiabgUcaRiaaigda aKqbagqaaaGaay5bSlaawIa7aaaaaKqbGeaacaWGPbGaeyypa0JaaG ymaaqaaiaad6gaaKqbakabggHiLdaaaa@4EA5@ , where j is such that i=1 j λ i >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqahaba Gaeq4UdW2aaSbaaKqbGeaacaWGPbaabeaaaeaacaWGPbGaeyypa0Ja aGymaaqaaiaadQgaaKqbakabggHiLdGaeyOpa4JaaGimaaaa@41E4@ and i=1 j+1 λ i <0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCe9Ff0Jb9YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaqahaba Gaeq4UdW2aaSbaaKqbGeaacaWGPbaabeaaaeaacaWGPbGaeyypa0Ja aGymaaqaaiaadQgacqGHRaWkcaaIXaaajuaGcqGHris5aiabgYda8i aaicdaaaa@437D@ and the LE are taken in descending order. In Figure 1 we present the flowchart of the combined chaos-geometric and neural networks computational approach to nonlinear analysis and prediction of dynamics of any complex system.8–17

Figure 1 Flowchart of the combined chaos-geometric approach and NNW to nonlinear analysis and prediction of chaotic dynamics of the complex systems (structures, devices).

Physiological activities and emotional induction

There are several physiological activities that can allow the determination of emotion beyond the face, voice and body gestures:

Electro-myographic activity (EMG)

In particular, EMG makes it possible to measure the electrical activity of the muscles via electrodes placed on the face. Several studies have shown that EMG signals provide an objective measure for the emotion recognition.6

Heart rate (ECG)

It defines the number of heartbeats (heartbeats) per unit of time, usually in beats per minute (BPM). It is generally associated with activation of the autonomic nervous system (ANS)7 itself related to the emotion treatment.8 Thus, the heart rate variation can be associated with different emotions.

Skin temperature (SKT)

The body controls the internal temperature by balancing heat production and heat loss. Heat production is achieved through muscle contraction, metabolic activity and vasoconstriction of the skin blood vessels. The activation of this indicator varies according to the emotion considered and the subjects, which induces a form of complex response making it possible to distinguish different emotions.

Respiratory frequency (FR)

Central nervous system

The central nervous system (CNS) is composed of the brain, cerebellum, brain stem and spinal cord. The brain activities of the CNS play a prominent role in the emotion recognition.

Acknowledgements

None.

Conflict of interest

The author declares there is no conflict of interest.

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