An Alternative Approach for Identifying Nonlinear Dynamics of the Cascade Logistic-Cubic System

Abstract

The 0-1 test for chaos, which is a simple binary method, has been widely used to detect the nonlinear behaviors of the non-cascade chaotic dynamics. In this paper, the validity checks of the 0-1 test for chaos to the popular cascade Logistic-Cubic (L-C) system is conducted through exploring the effects of sensitivity parameters. Results show that the periodic, weak-chaotic, and strong-chaotic states of the cascade L-C system can be effectively identified by the introduced simple method for detecting chaos. Nevertheless, the two sensitivity parameters, including the frequency ω and the amplitude α, are critical for the chaos indicator (i.e., the median of asymptotic growth rate, Km) when the cascade dynamic is detected by the method. It is found that the effect of α is more sensitive than that of ω on Km regarding the three dynamical states of the cascade L-C system. Meanwhile, it is recommended that the three states are identified according to the change of K with α from zero to ten since the periodic and weak-chaotic states cannot be identified when the α is greater than a certain constant. In addition, the modified mean square displacement Dc*(n) fails to distinguish its periodic and weak-chaotic states, whereas it can obviously distinguish the above two and strong-chaotic states. This work is therefore invaluable to gaining insight into the understanding of the complex nonlinearity of other different cascade dynamical systems with indicator comparison.