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Local and global bifurcations in 3D piecewise smooth discontinuous maps.
Mahashweta Patra1, Sayan Gupta2, Soumitro Banerjee3
1Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, Indiana 47405, USA.
Chaos (Woodbury, N.Y.)
|March 23, 2021
Summary
This study analyzes bifurcation phenomena in 3D discontinuous maps using piecewise linear approximations. It reveals conditions for periodic orbits and demonstrates complex bifurcations, including hyperchaos.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Mathematical Physics
Background:
- Bifurcation phenomena are crucial in understanding complex systems.
- Discontinuous maps present unique challenges in dynamical analysis.
- Previous studies often focused on continuous systems.
Purpose of the Study:
- To analyze bifurcation phenomena in three-dimensional discontinuous maps.
- To provide analytical conditions for the existence of periodic orbits.
- To illustrate complex bifurcations and chaotic behaviors.
Main Methods:
- Utilizing piecewise linear approximation near map borders.
- Analytical calculation of periodic orbit existence conditions.
- Numerical simulations to demonstrate bifurcation scenarios.
Main Results:
- Existence conditions for periodic orbits were analytically derived.
- Demonstrated discontinuous bifurcations involving stable/saddle fixed points and period-2 cycles.
- Illustrated occurrences of multiple attractor bifurcation and hyperchaos.
Conclusions:
- Piecewise linear approximation is effective for analyzing 3D discontinuous maps.
- Discontinuous systems exhibit rich and complex bifurcation behaviors.
- The study contributes to understanding chaos and hyperchaos in discrete dynamical systems.
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