Dark curves and MSS sequence in a piece-wise linear system

Abstract

The intersection and tangency of dark curves embedded in bifurcation diagrams of nonlinear systems are closely related to various bifurcation behaviour and crises of chaotic attractors. In previous work, this has been limited to one-dimensional discrete mapping systems with specific function expressions, in which dark curves can be depicted by established dark-curve equations. A numerical algorithm for extracting dark curves is proposed by constructing return maps of chaotic time series, and, via a parabolic mapping system, the validity and error are analysed by comparing the dark curves simulated by the numerical method with those depicted by the dark-curve equations. On the basis of the proposed algorithm, the rich details of the dark curves in a piece-wise linear model are uncovered, and periodic windows and the positions of super-stable periodic orbits (SSPOs) are discussed via the tangency of the dark curves. Moreover, the symbolic sequences of these SSPOs and their order are investigated in the context of symbolic dynamics. It is found that the order of SSPOs in the piece-wise linear model is consistent with the universal MSS sequence found in unimodal mapping systems and named after the three founders.

Keywords:

• Piece-wise linear system
• Bifurcation
• Dark curves
• Periodic window
• Symbolic dynamics

AMS Subject Classification:

• 37M20
• 37G35
• 65P30

Acknowledgement

This work was supported by the National Natural Sciences of China (No. 10572011).