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

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  • Yoshiki Kuramoto
    Department of Physics, Kyoto University, Kyoto 606-8502, Japan (emeritus)
  • Hiroya Nakao
    Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan

抄録

<jats:p> An overview is given on two representative methods of dynamical reduction known as <jats:italic>centre-manifold reduction</jats:italic> and <jats:italic>phase reduction</jats:italic> . 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 <jats:italic>reduction</jats:italic> . </jats:p> <jats:p>This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.</jats:p>

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