Rate-induced phenomena in dynamical systems with attracting limit cycles
George Chappelle1, Martin Rasmussen1
1Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom.
Chaos (Woodbury, N.Y.)
|July 7, 2025
Summary
Rapidly changing parameters in dynamical systems can cause unexpected behavior. This study reveals rate-induced phase sensitivity, leading to finite-time unpredictability and interacting with rate-induced tipping.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Mathematical Physics
Background:
- Investigating dynamical systems with time-dependent parameters is crucial for understanding complex behaviors.
- Previous work established rate-induced phenomena in systems with limit cycle attractors.
- Understanding how parameter change speed influences system dynamics is an ongoing challenge.
Purpose of the Study:
- To extend the study of rate-induced phenomena to continuous-time planar dynamical systems with limit cycle attractors.
- To discover and characterize new phenomena arising from rapid parameter changes.
- To analyze the interplay between newly discovered phenomena and established concepts like rate-induced tipping.
Main Methods:
- Analysis of continuous-time planar dynamical systems.
- Mathematical modeling of systems with time-dependent external parameters.
- Extension of existing theoretical frameworks for rate-induced phenomena.
Main Results:
- Discovery of rate-induced phase sensitivity, a novel phenomenon.
- Demonstration that rapid parameter change can induce finite-time unpredictability.
- Observation of significant interactions between rate-induced phase sensitivity and rate-induced tipping.
Conclusions:
- Rapid parameter variations can lead to unexpected and unpredictable dynamics in systems with limit cycles.
- Rate-induced phase sensitivity represents a new mechanism for finite-time unpredictability.
- The interaction between phase sensitivity and tipping warrants further investigation for a comprehensive understanding of system behavior.
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