设计具有预定轨迹与相位响应特性的二维极限环振荡器
Designing two-dimensional limit-cycle oscillators with prescribed trajectories and phase-response characteristics
AI总结：
基于相位降维理论，提出一种通过凸优化设计具有预定周期轨迹和相位响应特性的二维极限环振荡器的方法，并验证了其有效性及应用价值。
摘要：
我们提出了一种基于相位降维理论设计具有预定周期轨迹和相位响应特性的二维极限环振荡器的方法，该理论对弱扰动的极限环振荡器给出了简明描述，并广泛应用于同步动力学分析。我们开发了一种设计具有稳定极限环的向量场的算法，该极限环具有给定的形状以及给定的相位敏感度函数。极限环振荡器的向量场由多项式近似，其系数通过凸优化进行估计。通过引入 Floquet 指数的上限来确保极限环的线性稳定性。通过设计几种类型的二维现有和人工振荡器，在数值上验证了所提方法的有效性。作为应用，我们首先设计了一个具有人工星形周期轨迹的极限环振荡器，并演示了全局同步锁定。然后，我们设计了一个具有人工高谐波相位敏感度函数的极限环振荡器，并演示了由高频周期输入引起的多稳态同步锁定。
英文摘要：
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.
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