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Published on: February 28, 2016
Piecewise linear models for the quasiperiodic transition to chaos
David K. Campbell1, Roza Galeeva, Charles Tresser
1Physics Department, University of Illinois, 1110 W. Green St., Urbana, Illinois 61801UMPA, ENS, 46 Allee d'Italie, 69364 Lyon Cedex 07, FranceIBM P.O. Box 218, Yorktown Heights, New York 10598Mathematics Department, University of North Dakota, Grand Forks, North Dakota 58202-8376.
Chaos (Woodbury, N.Y.)
|June 1, 1996
Summary
This study introduces solvable models for mode locking and chaos transitions. Researchers solved key problems for circle maps, offering insights into complex dynamical systems.
Area of Science:
- Dynamical Systems
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Mode locking and the quasiperiodic transition to chaos are complex phenomena in dynamical systems.
- Smooth circle maps present analytical challenges in understanding these transitions.
- Piecewise linear models offer a tractable approach to studying complex dynamics.
Purpose of the Study:
- To analytically and computationally study two families of piecewise linear circle maps.
- To provide solvable models for mode locking and quasiperiodic transitions to chaos.
- To address fundamental questions about circle maps that remain unanswered for smooth families.
Main Methods:
- Formulation of two families of piecewise linear degree one circle maps.
- Analytical and computational investigations of these map families.
- Characterization of rotation intervals, topological entropy, and attractor structures.
Main Results:
- Complete solutions for prescribed rotation intervals in the studied map families.
- Identification of boundaries between zero and positive topological entropy.
- Detailed description of attractor structures and bifurcations for one family.
Conclusions:
- Piecewise linear circle maps serve as valuable, solvable models for complex dynamical phenomena.
- Results offer insights into the behavior of smooth circle maps and related physical systems.
- The study provides a foundation for further research in nonlinear dynamics and chaos theory.
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