On Nonlinear Dynamics, Chaos and Complexities

Almost all evolving real systems emerging around us are nonlinear in nature and their dynamics are not as simple as in cases of linear systems. Principle of superposition is no more applicable to real system and, to study them, one must apply the recent rules and methodology suggested in nonlinear dynamics. Because of nonlinearity in nature, real systems show complexities in behavior while evolving and chaos is one such complexity. Principles of nonlinear dynamics can only help to understand complex and chaotic behaviors observed in any nonlinear system. Various tools, which have been discovered due to growing researches in this area, are helpful to understand phenomena of evolutions in real systems. In addition to the basic tools, (e.g., time series and phase plane graphs), some tools such as Lyapunov exponents (LCEs), topological entropies, correlation dimension, etc., have been suggested recently which help us to understand better the complexities and chaotic motion in a dynamical system. To distinguish regular and chaotic motion, some recent tools have also been discussed with their working limitations.

The present paper aims to explain evolutionary dynamics of nonlinear systems and to explain about complexities observed during the processes. Some specific models proposed for real systems would be discussed and calculations of LCEs, topological entropies, correlation dimensions have been obtained as a part of complexity measure. Numerically calculated results are displayed graphically with complete interpretation.