**1. Introduction**

In this chapter, we investigate the dynamical complexity of several real physical systems. We present our analysis of various problems considered here and present results graphically based on actual numerical simulation for various system. We revisit the analysis of complexity of nonlinear pendulum dynamics and its application to unravel the complex oscillations observed in a swing pumped by a child. For the analysis, 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. Next 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. Further, 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 also suggested while analyzing a financial system involving interest rate, investment demand and interest rates. Finally, we describe the utility of time series generation of dynamical variables of chaotic system, such as Lorentz system, using artificial neural network.
