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On the local stability of limit cycles
Fathei Ali1, Michael Menzinger
1Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
Limit cycle stability arises from competing decay and growth of perturbations. This study reveals how phase-dependent stability affects oscillator responses to disturbances.
Area of Science:
- Dynamical systems theory
- Nonlinear dynamics
- Theoretical physics
Background:
- Limit cycles represent stable periodic behaviors in dynamical systems.
- Orbital stability is determined by the average behavior of perturbations over a cycle.
- Understanding local stability variations is crucial for predicting system responses.
Purpose of the Study:
- To investigate the coexistence of attractive and repulsive phases on limit cycles.
- To analyze local rates of phase space expansion and contraction.
- To explore how non-uniform local stability influences oscillator responses to perturbations.
Main Methods:
- Analysis of perturbations on limit cycles in a moving reference frame.
- Examination of generalized Bonhoeffer-van-der-Pol and Rossler models.
- Study of systems near various bifurcation points.
Main Results:
- Demonstrated the interplay between local stability and instability phases.
- Quantified the rates of phase space volume changes along limit cycles.
- Showcased phase-dependent responses of oscillators to external perturbations.
Conclusions:
- Non-uniform local stability is a key factor in limit cycle oscillator behavior.
- The phase of an oscillator significantly impacts its response to perturbations.
- This research provides insights into the complex dynamics of nonlinear systems.