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<p>We introduce a non-iterative, derivative-free control-based continuation (CBC) algorithm for tracking branches of limit cycles in autonomous systems. The algorithm employs three control layers: a derivative feedback controller for stabilization, a phase-locked loop for phase and frequency synthesis, and an arclength controller to enable path-following. Theoretical analysis via the method of averaging yields system-independent tuning rules that guarantee stability and allow the controller to follow stable and unstable branches, including through fold bifurcations. To address invasiveness caused by higher-order harmonics, the method is extended with an adaptive filter. The algorithm is validated numerically on a generalized Van der Pol oscillator.</p>

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Theoretical analysis of a derivative free control based continuation algorithm with path following capability for autonomous systems.

  • 期刊论文

  • 发表日期：

    2025年8月26日

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  • Etienne Gourc,
  • Romain Caron,
  • Fabrice Silva,
  • Christophe Vergez,
  • Bruno Cochelin

摘要

We introduce a non-iterative, derivative-free control-based continuation (CBC) algorithm for tracking branches of limit cycles in autonomous systems. The algorithm employs three control layers: a derivative feedback controller for stabilization, a phase-locked loop for phase and frequency synthesis, and an arclength controller to enable path-following. Theoretical analysis via the method of averaging yields system-independent tuning rules that guarantee stability and allow the controller to follow stable and unstable branches, including through fold bifurcations. To address invasiveness caused by higher-order harmonics, the method is extended with an adaptive filter. The algorithm is validated numerically on a generalized Van der Pol oscillator.