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Simultaneous border-collision and period-doubling bifurcations.
1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.
Chaos (Woodbury, N.Y.)
|October 2, 2009
Summary
This study reveals how period-doubling bifurcations emerge from border-collision bifurcations in piecewise-smooth maps. Conditions are identified for the onset of chaos in these complex dynamical systems.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Bifurcation Theory
Background:
- Piecewise-smooth systems exhibit complex behaviors, including bifurcations.
- Understanding the interplay of different bifurcation types is crucial for predicting system dynamics.
- Border-collision and period-doubling bifurcations are fundamental phenomena in nonlinear dynamics.
Purpose of the Study:
- To investigate the codimension-two simultaneous occurrence of border-collision and period-doubling bifurcations.
- To analyze the conditions under which these bifurcations interact in piecewise-smooth, continuous maps.
- To classify the local dynamics and identify conditions leading to chaos in one-dimensional cases.
Main Methods:
- Analysis of a general piecewise-smooth, continuous map.
- Identification of nondegeneracy conditions for bifurcation interactions.
- Classification of local dynamics for one-dimensional maps.
Main Results:
- A locus of period-doubling bifurcations emanates nontangentially from a border-collision bifurcation locus under sufficient nondegeneracy.
- Period-doubled solutions exhibit border-collision bifurcations along a specific curve originating from the codimension-two point.
- Complete classification of local dynamics and conditions for chaos are provided for one-dimensional maps.
Conclusions:
- The interaction between border-collision and period-doubling bifurcations is characterized.
- The study provides a framework for understanding complex dynamics in piecewise-smooth systems.
- Specific conditions are established that lead to chaotic behavior in simplified one-dimensional models.
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