A toolbox for finding horseshoes in 2D maps

This GUI programe presents an efficient method for finding topological horseshoes in chaotic systems

A toolbox for finding horseshoes 2D maps

By Qingdu Li,
Institute for Nonlinear Systems, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China.
Email: [e-mail]

The Update on 2011.11.23
Add an example for continuous dynamical systems, i.e. the 3D Sprott chaotic system.
Add a paper about how to use this toolbox
Add a paper which finds hyperchaos with this toolbox.

As one of the most important results in chaotic dynamics, topological horseshoe theory provides a powerful tool in rigorous studies of chaos in dynamical systems, such as giving the topological entropy, verifying the existence of chaos, showing the structure of chaotic attractors, revealing the mechanism inside of chaotic phenomena and so on. However, this theory is not applicable for most readers because it is hard to find a topological horseshoe.

HSTOOL is a MATLAB toolbox trying to make it easy to find a horseshoe. The program is written according the method proposed in [1]. The toolbox utilizes graphical user interface. Readers only need to write an m-function for their chaotic maps (or Poincar'e maps for time-continuous system). Beside this, almost all operations for finding a horseshoe are mouse clicks, so it is user-friendly. For example, in order to take a polygon (D_i), one first clicks the left button of the mouse on each vertex one by one, then clicks the right button to end the operation. With this tool, it become very efficient to find a horseshoe in a common 2D chaotic map. For most
application, it is just a work of making several clicks with a mouse.

The current version of HSTOOL is the first version. The source codes with a demo animation are available via Email to the author.

Suggestions and questions from users are most welcome for the author to update the toolbox in the future.

Reference
[1] Li, Q. D. & Yang, X. S. [2010] "A Simple Method for Finding Topological Horseshoes," International Journal of Bifurcation and Chaos 20, 467-478.
[2] Li, Q. D., Yang, X. S. & Chen, S. [2011] "Hyperchaos in a Spacecraft Power System," International Journal of Bifurcation and Chaos 21, 1719-1726.
[3] Li, Q. D., Chen, S. & Zhou, P. [2011] "Horseshoe and Entropy in a Fractional-Order Unified System," Chinese Physics B 20.