On the concept of dynamical reduction: the case of coupled oscillators

Abstract

An overview is given on two representative methods of dynamical reduction known as centermanifold reduction and phase reduction. These theories are presented in a somewhat more unified fashion than the theories in the past. The target systems of reduction are coupled limit-cycle oscillators. Particular emphasis is placed on the remarkable structural similarity existing between these theories. While the two basic principles, i.e. (i) reduction of dynamical degrees of freedom and (ii) transformation of reduced evolution equation to a canonical form, are shared commonly by reduction methods in general, it is shown how these principles are incorporated into the above two reduction theories in a coherent manner. Regarding the phase reduction, a new formulation of perturbative expansion is presented for discrete populations of oscillators. The style of description is intended to be so informal that one may digest, without being bothered with technicalities, what has been done after all under the word reduction. arXiv:1910.13775v1 [nlin.AO] 30 Oct 2019 II. CENTER-MANIFOLD REDUCTIONWe begin with the center-manifold reduction for a single oscillator, and then proceed to multi-oscillator systems. It will be seen how the two elements of reduction, namely, the reduction of dynamical degrees of freedom and transformation of the evolution equation to a canonical form, are incorporated into a systematic perturbation theory. A. Reduction of a single free oscillatorThe situation of our concern is the neighborhood of the Hopf bifurcation of an n-dimensional dynamical systeṁ X = F(X) with fixed point X = 0. We separate F into an unperturbed part and perturbation, where the former

Phase-synchronization properties of laminar cylinder wake for periodic external forcings

“…Most control or reduction techniques conventionally employed for fluid flows rely on the linearization of the flow around a steady or quasisteady state, which makes them underperform when the deviation from the bifurcation point is relatively large (Bewley 2001). Approaches such as Floquet theory (Herbert, Bertolotti & Santos 1987) or phase-reduction theory (Kuramoto & Nakao 2019) enable the extension of analysis to periodic flows, as they pave a path towards active flow control with respect to time-varying base flows.…”

Nonlinear phase-amplitude reduction of delay-induced oscillations

“…Finally, before proceeding, we note that there are also other formulations of phase or phaseamplitude reduced equations for analyzing higher-order effects of perturbations on limit cycles described by ODEs, such as non-pairwise phase interactions [23], higher-order phase reduction [49], nonlinear phase coupling function [50], and higher-order approximations of coupling functions [41], which can capture more detailed aspects of synchronization than the lowest-order phase equation.…”

“…The phase reduction theory is a standard mathematical framework for characterizing response properties of weakly perturbed limit-cycle oscillators and analyzing their synchronization dynamics via dimensionality reduction [18][19][20][21][22][23][24]. Recently, the phase reduction theory has been extended also to DDEs exhibiting stable limit-cycle oscillations, which requires non-trivial mathematical generalization because DDEs are infinite-dimensional dynamical systems [25,26].…”

Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences

“…The theoretical contributions are developed sequentially in order of increasing complexity, starting from basic formulations and moving on towards new applications. To set the context, this part starts with the review article by Kuramoto & Nakao [76] on the concept of dynamical reduction theory for coupled oscillators. Their approach places particular emphasis on the remarkable structural similarity that exists between centre-manifold reduction and phase reduction methods.…”