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 center-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.

20 pages, 2 figures

References in corpus (7)

• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Applied Koopmanism
• Inference of Time-Evolving Coupled Dynamical Systems in the Presence of Noise
• Collective Phase Sensitivity
• Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems
• Phase-amplitude descriptions of neural oscillator models
• Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless non-identical case

Cited by in corpus (30)

• Data-driven transient lift attenuation for extreme vortex gust-airfoil interactions
• Turing patterns on discrete topologies: from networks to higher-order structures
• Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory
• Dynamical phase transitions in the nonreciprocal Ising model
• Nonlinear phase-amplitude reduction of delay-induced oscillations
• Enlarged Kuramoto Model: Secondary Instability and Transition to Collective Chaos
• Synchrony for weak coupling in the complexified Kuramoto model
• Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data
• Linking fast and slow: the case for generative models
• Global topological synchronization of weighted simplicial complexes
• Theory of phase reduction from hypergraphs to simplicial complexes: a general route to higher-order Kuramoto models
• Optimal coupling functions for fast and global synchronization of weakly coupled limit-cycle oscillators
• Designing two-dimensional limit-cycle oscillators with prescribed trajectories and phase-response characteristics
• A definition of the asymptotic phase for quantum nonlinear oscillators from the Koopman operator viewpoint
• Pinning control of chimera states in systems with higher-order interactions
• Analytical Phase Reduction for Weakly Nonlinear Oscillators
• Phase and amplitude description of complex oscillatory patterns in reaction-diffusion systems
• Phase-Amplitude Reduction and Optimal Phase Locking of Collectively Oscillating Networks
• Phase autoencoder for rapid data-driven synchronization of rhythmic spatiotemporal patterns
• Global Topological Dirac Synchronization
• Two distinct transitions in a population of coupled oscillators with turnover: desynchronization and stochastic oscillation quenching
• Gaussian Process Phase Interpolation for estimating the asymptotic phase of a limit cycle oscillator from time series data
• Sparse optimization of mutual synchronization in collectively oscillating networks
• Partial synchronization and community switching in phase-oscillator networks and its analysis based on a bidirectional, weighted chain of three oscillators
• RASER MRI: Magnetic Resonance Images formed Spontaneously exploiting Cooperative Nonlinear Interaction
• Definition and data-driven reconstruction of asymptotic phase and amplitudes of stochastic oscillators via Koopman operator theory
• Energy Transfer and Coherence in Coupled Oscillators with Delayed Coupling: A Classical Picture for Two-Level Systems
• Asymptotic phase and amplitude for classical and semiclassical stochastic oscillators via Koopman operator theory
• Synchronization of nonlinearly coupled Stuart-Landau oscillators on networks
• On the efficiency of pairwise Hamiltonian control to desynchronize the higher-order Kuramoto model