Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

Abstract

Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the Van der Pol and Willamowski-Rössler oscillators with real or complex Floquet exponents are analyzed as examples.It is demonstrated that the proposed methods can achieve much faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.

Sparse optimization of mutual synchronization in collectively oscillating networks

“…For non-weak internetwork coupling, the phase reduction becomes inaccurate and the present results may not apply. Recently, a phase-amplitude reduction theory that considers deviations of the system state from the limit cycle due to non-weak perturbations has been developed [39,40,62,63], and it has been used to devise non-weak periodic inputs to achieve fast entrainment [64,65]. Such a phase-amplitude framework would also be applicable to coupled networks.…”

Asymptotic phase and amplitude for classical and semiclassical stochastic oscillators via Koopman operator theory

“…By focusing only on the asymptotic phase and amplitude, we can perform phase-amplitude reduction (or isochronisostable reduction) of a limit-cycle oscillator [12,14,18,19], in which we reduce the dimensionality of the system dynamics from N to 2 and approximately describe it by a simple set of two-dimensional phase and amplitude equations. The phase equation has been extensively used for the analysis of weakly coupled limit-cycle oscillators [1][2][3][4][5][6], and the amplitude equation has also been used recently for the analysis and control of limit-cycle oscillators [18][19][20][21]24].…”

“…Moreover, they have shown that the (asymptotic) amplitude and isostables, which characterize deviation of the system state from the limit cycle and extends the Floquet coordinates [13,15,16] to the nonlinear regime, can be introduced naturally in terms of the Koopman eigenfunctions associated with the Floquet exponents with non-zero real parts [10][11][12][13][14]17]. By using the asymptotic phase and amplitude functions, we can derive a reduced description of limit-cycle oscillators, which is useful for the analysis and control of synchronization dynamics of limit-cycle oscillators [18][19][20][21][22][23][24]. The theory can also be generalized to delay-differential systems [25] and spatially extended systems [26].…”