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Routes to complex dynamics in a ring of unidirectionally coupled systems
P Perlikowski1, S Yanchuk, M Wolfrum
1Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany. przemyslaw.perlikowski@p.lodz.pl
Chaos (Woodbury, N.Y.)
|April 8, 2010
Summary
Coupling autonomous Duffing oscillators in a ring creates complex dynamics, including chaos and hyperchaos. Unstable periodic solutions drive emergent spatiotemporal structures and phenomena like the Eckhaus effect.
Area of Science:
- Nonlinear dynamics
- Complex systems
- Chaos theory
Background:
- Autonomous Duffing oscillators exhibit simple equilibrium dynamics individually.
- Coupling oscillators can lead to emergent complex behaviors.
- Understanding transitions in coupled systems is crucial for predicting their dynamics.
Purpose of the Study:
- To investigate the dynamical transitions in a ring of unidirectionally coupled autonomous Duffing oscillators.
- To analyze the influence of coupling on system behavior, ranging from periodic to hyperchaotic.
- To explore the role of unstable periodic solutions in generating complex spatiotemporal patterns.
Main Methods:
- Analytical investigation of coupled Duffing oscillator dynamics.
- Numerical simulations for small and large numbers of oscillators.
- Experimental validation using an electronic circuit implementation.
Main Results:
- Coupling induces transitions from trivial dynamics to periodic, quasiperiodic, chaotic, and hyperchaotic states.
- Unstable periodic solutions are key to the formation of chaotic rotating waves and spatiotemporal structures.
- The Eckhaus effect is observed in systems with a large number of oscillators.
Conclusions:
- The study reveals rich dynamical behaviors in coupled Duffing oscillator rings.
- Unstable periodic solutions play a critical role in the emergence of complex phenomena.
- Experimental results confirm the analytical and numerical findings, validating the model.
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