The visualization of periodic orbits in 1D bimodal maps

Abstract

The word-lifting technique (WLT) is a powerful tool in the study of symbolic dynamics. An admissible super-stable period sequence (SSKS) determines a solvable system of nonlinear equations, two issues often rise in probing the numerical solution: one is the selection of initial values, another is the convergence condition of its corresponding system of nonlinear equations, sometimes, the solution we obtained is not the genuine one. In this paper, we proposed a forward-iteration validation, which can judge if the solution obtained from the WLT is the genuine solution, even it has met the numerical convergence condition; second, an empirical method for the selection of initial values is given, while about 2.94 million admissible doubly super-stable kneading sequences (DSSKS) had been batch calculated, the corresponding parameters were verified to be right; third, we have designed a computer program based on MATLAB GUI, for an arbitrary DSSKS in the admissible sets is selected, its numerical iteration orbit will be drawn in real time, the visualization increases the ease of understanding of knowledge of symbolic dynamics in 1D maps.