2019/10/28 by Yoshiki Kuramoto, Hiroya Nakao · 3 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Engineering · Mathematics · Physics and Astronomy · #Computer science #Coupling (piping) #Degrees of freedom (physics and chemistry) #Dimensional reduction #Dynamical systems theory #Engineering #Geometry #Mathematics #Nonlinear Dynamics and Pattern Formation #Photosynthetic Processes and Mechanisms #Physics #Quantum mechanics #Reduction (mathematics) #Statistical physics #Theoretical physics #nlin.AO #stochastic dynamics and bifurcation
paper · pdf · doi:10.1098/rsta.2019.0041
published as Phil. Trans. R. Soc. A 377: 20190041 (2019) · 20 pages, 2 figures
openalex publication_date 2019/10/28 · arxiv created 2019/10/30 · arxiv updated 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
An overview is given on two representative methods of dynamical reduction known as centre-manifold 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 . This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.