Antimonotonicity, Chaos and Multidirectional Scroll Attractor in Autonomous ODEs Chaotic System

Three-dimensional autonomous ordinary differential equations (ODEs) are the simplest and most important chaotic systems in nonlinear dynamics. In fact, they have been applied in many fields. In this paper, a systematic methodology for analyzing complex behavior of the ODEs chaotic system, as one of the ODEs chaotic systems, the improved TCS which satisfies the condition <inline-formula> <tex-math notation="LaTeX">$a_{12}a_{21} = 0$ </tex-math></inline-formula>, is proposed. It is dissipative, chaos, symmetric, antimonotonicity and can generate multiple directional (<inline-formula> <tex-math notation="LaTeX">$M\times N\times L$ </tex-math></inline-formula>) scroll attractors. Then, bifurcation diagrams, Lyapunov exponents, time series, Poincare sections, and Hausdroff dimensions are analyzed by setting the parameters and initial value. More interestingly, antimonotonicity (named reverse period-doubling bifurcation) and coexisting bifurcations are also reported. Finally, the results of theoretical analyses may be verified by electric experimental.