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Correlation dimension and topological entropy in discrete maps
, Prasad S., Mohanty R.K.
Published in
Volume: 6
Issue: 81-84
Pages: 4197 - 4209
The state of chaos exhibited in nonlinear system may appear through various roots. However, for mostly one dimensional system, it appears through period doubling roots. Phenomena of bifurcations, drawn by varying certain parameter of the system, explain clearly regular as well as chaotic evolution of the system. As a measure of chaos, the most suitable tools to be considered are: Lyapunov characteristic exponents (LCE) and topological entropies. Though the plots of both of the LCE's and topological entropies are similar, but both have certain limitations. LCE would not work for systems having relativistic considerations but the topological entropy can work. In fact, topological entropy can nicely provide the measure of complexity of the system in the sense that the more complexity in the system means more topological entropy it will have. A chaotic system exhibits a chaotic set, strange attractor, having fractal property. Calculation of correlation dimension is required to obtain the dimension of such attractor. The work presented here explains the appearance of chaos through bifurcation in some nonlinear one dimensional discrete system. The stability of the steady state, (i.e. fixed points), have been examined for each system and then plots of Lyapunov exponents and topological entropies for their evolution have been obtained. Then, the calculations have been extended to find the correlation dimensions chaotic sets, chaotic attractors, seen for different systems. Graphical results reveal some interesting information.
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