Abstract

Emergence of chaos and complex behavior in real and physical systems has been discussed within the framework of nonlinear dynamical systems. The problems investigated include complexity of Child’s swing dynamics , chaotic neuronal dynamics (FHN model), complex Food-web dynamics, Financial model (involving interest rate, investment demand and price index) etc. Proper numerical simulations have been carried out to unravel the complex dynamics of these systems and significant results obtained are displayed through tables and various plots like bifurcations, attractors, Lyapunov exponents, topological entropies, correlation dimensions, recurrence plots etc. The significance of artificial neural network (ANN) framework for time series generation of some dynamical system is suggested.

Highlights

  • We investigate the dynamical complexity of several real physical systems

  • Beside the normal analysis used to understand the complex neuronal dynamics, say using Fitzhug-Nagumo model (FHN), recurrence plots (RPs) have been used along with the phase plot analysis and bifurcation diagram to picturise the transition of spike occurrence from periodic to quasi-periodic and chaotic oscillations in the presence of external periodic stimulation

  • Beside the application of bifurcation diagram, phase plot and Poincare surface of section technique, we introduce the idea of Lyapunov characteristic exponents (LCE), correlation dimension and topological entropy which provide further insight of a complex dynamical system

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Summary

Introduction

We investigate the dynamical complexity of several real physical systems. We use various tools e.g., phase plot, bifurcation diagram, Poincare surface of section and maps, Lyapunov exponent (LCE) etc., of theory of nonlinear dynamical system. We consider the problem of prey-predator system with Allee effect and introduce correlation dimension and topological entropy to characterize the fractal structure and the associated complexity in its dynamics. Beside the normal analysis used to understand the complex neuronal dynamics, say using Fitzhug-Nagumo model (FHN), recurrence plots (RPs) have been used along with the phase plot analysis and bifurcation diagram to picturise the transition of spike occurrence from periodic to quasi-periodic and chaotic oscillations in the presence of external periodic stimulation. Significance of multi-scale permutation entropy analysis to characterize nonlinear dynamical complexity of real system is suggested while analyzing a financial system involving interest rate, investment demand and interest rates. We describe the utility of time series generation of dynamical variables of chaotic system, such as Lorentz system, using artificial neural network

Pendulum motion
Problem of Swing oscillation
Regular and Chaotic motion of the swing
Complexity in prey-predator system with Allee effect
Discrete prey-predator model
Bifurcation diagrams
Numerical simulations
Lyapunov exponents (LCEs)
Correlation dimension
Topological entropies
Recurrence plot
Regular and Chaotic neuronal dynamics
Basic dynamics of FHN model
FHN neuron in the presence of external periodic electrical stimulation
Chaotic dynamics in finance model
Chaotic financial model
Conclusion
Full Text
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