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Published on: September 11, 2019
Snap-back repellers and chaotic attractors.
Laura Gardini1, Fabio Tramontana
1Department of Economics and Quantitative Methods, University of Urbino, Urbino, Italy. laura.gardini@uniurb.it
Summary
Snap-back repellers, when present, can lead to robust chaotic attracting sets in two-dimensional maps. This study explores their association in both continuous and discontinuous piecewise-linear maps.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Snap-back repellers are points in dynamical systems where homoclinic orbits exist.
- These repellers are known to generate complex system behaviors.
- Understanding their role is crucial for characterizing chaos.
Purpose of the Study:
- To investigate the association between snap-back repellers and robust chaotic attracting sets.
- To analyze this phenomenon in two-dimensional piecewise-linear maps.
- To examine both continuous and discontinuous map forms.
Main Methods:
- Analysis of two-dimensional piecewise-linear maps in canonical form.
- Identification and characterization of homoclinic orbits.
- Demonstration of snap-back repeller existence and behavior.
Main Results:
- Snap-back repellers are demonstrated to be linked with robust chaotic attracting sets, not exclusively chaotic repellers.
- Examples are provided for both continuous and discontinuous piecewise-linear maps.
- The study confirms the broader role of snap-back repellers in generating complex dynamics.
Conclusions:
- Snap-back repellers contribute to the formation of robust chaotic attracting sets in the studied map types.
- The findings apply to both continuous and discontinuous piecewise-linear dynamical systems.
- This research expands the understanding of chaos generation mechanisms in nonlinear systems.
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