Abstract

Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.

Highlights

  • Henri Poincaré, (1892–1908), [1], was first to acknowledge the possible existence of chaos in nonlinear systems while studying a 3-body problem comprising Sun, Moon and Earth

  • He noticed the dynamics of the system turned to be sensitive towards initial conditions, which was later termed as chaos

  • Lorenz provided the foundation of chaos theory and inspired a fundamental reappraisal of systems of nonlinearity in many disciplines of science, engineering, biological and medical sciences, atmospheric science, economics, social sciences and where not? In our everyday life, chaos happened frequently in various form like cyclone, tsunami, tornado, epidemics/pandemics etc

Read more

Summary

Introduction

Henri Poincaré, (1892–1908), [1], was first to acknowledge the possible existence of chaos in nonlinear systems while studying a 3-body problem comprising Sun, Moon and Earth. He noticed the dynamics of the system turned to be sensitive towards initial conditions, which was later termed as chaos. His results based on theoretical analysis and he could not demonstrate it because computers were not available at that time. Lorenz provided the foundation of chaos theory and inspired a fundamental reappraisal of systems of nonlinearity in many disciplines of science, engineering, biological and medical sciences, atmospheric science, economics, social sciences and where not? Systematic studies in various areas resulting in numerous articles on chaos and nonlinear dynamics appeared in many well-reputed scientific journals, [3–19]

A Collection of Papers on Chaos Theory and Its Applications
Dynamics of laser map
Dynamics of biological red cells model
Continuous Volterra-Petzoldt Model
Chaos control technique
Applications
Food chain model
Controlling Chaos in 2-D Burger’s Map xnþ1 1⁄4 ð1 À aÞ xn À y2n ynþ1 1⁄4 ð1 þ bÞ yn þ xn yn (20)
Controlling Chaos in Volterra-Petzoldt Map
Discussions
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call