Simple program to detect Smale horseshoes automatically

This program shows how to find Smale horseshoes in the Duffing, Hénon, Ikeda, and Lozi maps.

Descriptions
Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on

Smale’s famous example, provide a rigorous way to study chaos in dynamical

systems. This program is written according to the algorithm proposed in the

paper, which has submitted to DDSN “An algorithm to automatically detect Smale

horseshoes”. The program includes the codes for the four examples in that paper

(the Duffing map, the Hénon map, the Ikeda map, and the Lozi map).

How does the program work?
The file “FindPeirod.m” is used to get the period obits and find the saddle

points, as well as the eigenvalues and eigenvectors of the Jacobian at those

points. You may pick one, copy it into “HorseshoeFindAuto_demo.m” .

The file “HorseshoeFindAuto_demo.m” demonstrates the step of the algorithm in

that paper. For an animation set the variable “Anm” to be 1 second.

Key words: Chaos, Smale horseshoe, chaotic maps, topological horseshoe