scholarly journals Computing Complex Horseshoes by Means of Piecewise Maps

2018 ◽  
Vol 28 (12) ◽  
pp. 1830039
Author(s):  
Álvaro G. López ◽  
Álvar Daza ◽  
Jesús M. Seoane ◽  
Miguel A. F. Sanjuán
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
New Method ◽  
Nonlinear Dynamical ◽  
Systematic Procedure ◽  
Wada Property ◽  
Piecewise Functions

A systematic procedure to numerically compute a horseshoe map is presented. This new method uses piecewise functions and expresses the required operations by means of elementary transformations, such as translations, scalings, projections and rotations. By repeatedly combining such transformations, arbitrarily complex folding structures can be created. We show the potential of these horseshoe piecewise maps to illustrate several central concepts of nonlinear dynamical systems, as for example, the Wada property.

An Unravelling Algorithm for Global Analysis of Dynamical Systems: An Application of Cell-to-Cell Mappings

1980 ◽  
Vol 47 (4) ◽  
pp. 940-948 ◽  
Author(s):  
C. S. Hsu ◽  
R. S. Guttalu
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
New Method ◽  
Asymptotically Stable ◽  
Periodic Motions ◽  
Nonlinear Dynamical ◽  
Effective Manner

A new method is offered here for global analysis of nonlinear dynamical systems. It is based upon the idea of constructing the associated cell-to-cell mappings for dynamical systems governed by point mappings or governed by ordinary differential equations. The method uses an algorithm which allows us to determine in a very effective manner the equilibrium states, periodic motions and their domains of attraction when they are asymptotically stable. The theoretic base and the detail of the method are discussed in the paper and the great potential of the method is demonstrated by several examples of application.

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