Exploring Parameter Spaces in Dynamical Systems

TL;DR

This paper introduces a fast algorithm for analyzing high-dimensional parameter spaces in dynamical systems, enabling efficient exploration of features like limit cycle amplitude with rigorous error control.

Contribution

The paper presents a novel, efficient algorithm for numerical analysis of dynamical systems' parameter spaces, with rigorous error analysis and practical applications in model-data comparison.

Findings

Algorithm effectively analyzes limit cycle amplitude across seven parameters.

Provides rigorous error bounds for numerical analysis.

Demonstrates utility in model validation against experimental data.

Abstract

The parameter space of dynamical systems arising in applications is often found to be high-dimensional and difficult to explore. We construct a fast algorithm to numerically analyze "quantitative features" of dynamical systems depending on parameters. Using a classical problem from mathematical ecology as an example, we demonstrate how to apply the algorithm to investigate the amplitude of a limit cycle depending on seven parameters. We stress the practical value of the algorithm but we also provide a rigorous error analysis to justify the overall strategy. Our approach turns out to be particularly useful in the case of comparing experimental data to a model defined by differential equations and to investigate whether the equations can approximate the modeled system.