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

TL;DR

The paper addresses designing two-dimensional limit-cycle oscillators with prescribed trajectory shapes and phase-response characteristics to tailor synchronization behavior. It combines phase reduction theory with a convex, polynomial-vector-field design, enforcing trajectory accuracy via discretized constraints and phase response via the adjoint equation, while guaranteeing stability through a Floquet-exponent bound. The key contributions include a practical convex optimization framework that yields vector fields matching target trajectories and phase sensitivity functions, demonstrated on both existing oscillators (van der Pol, FitzHugh–Nagumo) and artificial targets (star-shaped trajectories and high-harmonic PSFs) leading to global and multistable entrainment. This approach enables controlled synchronization design for engineering systems, robotics, and bio-inspired applications, with potential extensions to higher-dimensional oscillators.

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.

Figures (12)

• Figure 1: Periodic trajectories and PSFs of the designed oscillator and the original vdP oscillator. (a) Periodic trajectories. (b,c) Velocities on the periodic trajectories. (b) $x_{1}$ component, (c) $x_{2}$ component. (d,e) PSFs. (d) $x_{1}$ component, (e) $x_{2}$ component. In each graph, the red line shows the designed functional form and the black dotted line shows the original one, respectively.
• Figure 2: Vector fields of the designed oscillator and the original vdP oscillator. (a) designed, (b) original.
• Figure 3: Periodic trajectories and PSFs of the designed oscillator and the original FHN oscillator. (a) Periodic trajectories. (b,c) Velocities on the periodic trajectories. (b) $x_{1}$ component, (c) $x_{2}$ component. (d,e) PSFs. (d) $x_{1}$ component, (e) $x_{2}$ component. In each graph, the red line shows the designed functional form and the black dotted line shows the original one, respectively.
• Figure 4: Vector fields of the designed oscillator and the original FHN oscillator. (a) designed, (b) original.
• Figure 5: Periodic trajectory and PSFs of the designed oscillator and the prescribed functional forms. (a) Periodic trajectories. (b,c) Velocities on the periodic trajectories. (b) $x_{1}$ component, (c) $x_{2}$ component. (d,e) PSFs. (d) $x_{1}$ component, (e) $x_{2}$ component. In each graph, the red line shows the designed functional form and the black dotted line shows the original one, respectively.