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Delay stabilization of periodic orbits in coupled oscillator systems.
B Fiedler1, V Flunkert, P Hövel
1Institut für Mathematik I, Free Universität Berlin, Arnimallee 2-6, 14195 Berlin, Germany.
Summary
This study stabilizes unstable anti-phase orbits in diffusively coupled oscillators using non-invasive delay coupling. A nonlinearity condition is identified as necessary and sufficient for successful stabilization in super- and subcritical Hopf normal forms.
Area of Science:
- Nonlinear dynamics
- Coupled oscillator systems
- Complex systems
Background:
- Diffusively coupled oscillators often exhibit unstable anti-phase orbits.
- Hopf normal form describes systems near a supercritical or subcritical Hopf bifurcation.
- Stabilizing unstable states is crucial for controlling complex systems.
Purpose of the Study:
- To investigate the stabilization of unstable anti-phase orbits in diffusively coupled oscillators.
- To introduce and analyze the effect of non-invasive delay coupling.
- To derive conditions for successful stabilization in super- and subcritical Hopf normal forms.
Main Methods:
- Analysis of diffusively coupled oscillators in Hopf normal form.
- Introduction of non-invasive time-delayed coupling.
- Derivation and proof of a necessary and sufficient condition on oscillator nonlinearity.
- Review of existing literature on delayed feedback stabilization.
Main Results:
- A condition on oscillator nonlinearity is established for stabilizing anti-phase orbits.
- This condition is proven to be necessary and sufficient for both super- and subcritical cases.
- Successful stabilization is demonstrated through numerical simulations.
- Previous findings on stabilizing odd-number orbits via time-delayed feedback are contextualized.
Conclusions:
- Non-invasive delay coupling effectively stabilizes unstable anti-phase orbits in coupled oscillators.
- The derived nonlinearity condition provides a clear criterion for parameter selection.
- The findings extend the understanding of control strategies in nonlinear dynamical systems.
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