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.
Part of the book: A Collection of Papers on Chaos Theory and Its Applications
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.