Measuring Chaos: Topological Entropy and Correlation Dimension in Discrete Maps

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[Abstract] Nonlinear systems may exhibit chaos during evolution and at the state of chaos one sees the emergence of certain chaotic attractors. Chaotic set appears during the phenomena of bifurcations while varying certain parameter. The Lyapunov characteristic exponents (LCE) and topological entropies are both suitable tools for description of the transition to chaos. Plots of both of these indicators are similar as for identification of chaos. But LCE has limitation that it would not work for systems having relativistic considerations. However, the topological entropy is considered as a nice way to measure the complexity of a system such as chaos in the sense that the more complexity in the system means more topological entropy it will have. In the present work, appearance of chaos through bifurcation in some one dimensional discrete nonlinear systems have been considered and plots of Lyapunov exponents and topological entropy for such evolution have been obtained. Then, the calculation of correlation dimensions of such chaotic sets, chaotic attractors, have been carried out. Graphical results reveal some interesting informations.