Design of Limit-Cycle Oscillators with Prescribed Trajectories and Phase-Response Properties via Phase Reduction and Floquet Theory

We propose a method for designing stable limit-cycle oscillators with prescribed periodic trajectories and phase-response properties in general dimensions based on the phase reduction theory. The vector field of the oscillator is approximated by polynomials and their coefficients are optimized to satisfy required conditions. Linear stability of the periodic trajectory is ensured by imposing conditions on the eigenvalues of the monodromy matrix based on Floquet theory. We verify the validity of the proposed method by designing several types of oscillators with given properties. As an application, we design two oscillators with the same periodic trajectory but with different phase-response properties and show their distinct synchronization dynamics under the same periodic input.