Designing two-dimensional limit-cycle oscillators with prescribed trajectories and phase-response characteristics

TL;DR

This paper introduces a method to design two-dimensional limit-cycle oscillators with specific trajectories and phase response properties using phase reduction theory and convex optimization, verified through numerical examples.

Contribution

It presents a novel algorithm for designing oscillators with prescribed shapes and phase sensitivities, ensuring stability and enabling targeted synchronization behaviors.

Findings

Successfully designed oscillators with star-shaped trajectories

Achieved multistable entrainment with high-harmonic phase sensitivity

Validated the method through numerical simulations

Abstract

We propose a method for designing two-dimensional limit-cycle oscillators with prescribed periodic trajectories and phase response properties based on the phase reduction theory, which gives a concise description of weakly-perturbed limit-cycle oscillators and is widely used in the analysis of synchronization dynamics. We develop an algorithm for designing the vector field with a stable limit cycle, which possesses a given shape and also a given phase sensitivity function. The vector field of the limit-cycle oscillator is approximated by polynomials whose coefficients are estimated by convex optimization. Linear stability of the limit cycle is ensured by introducing an upper bound to the Floquet exponent. The validity of the proposed method is verified numerically by designing several types of two-dimensional existing and artificial oscillators. As applications, we first design a limit-cycle oscillator with an artificial star-shaped periodic trajectory and demonstrate global entrainment. We then design a limit-cycle oscillator with an artificial high-harmonic phase sensitivity function and demonstrate multistable entrainment caused by a high-frequency periodic input.