Synchronization-Desynchronization Transition Dynamics In Coupled Nonlinear Oscillators

Synchronization is a collective behavior in which two or more distinct units spontaneously reach cotrdination due to their mutual coupling interaction.It occars ubiquitous in many temporally rhythmic systems ranging from physics,chemistry,biology to social sciences.The studies on the physical mechanism of synchronization and its performances in different physical backgrounds have been the focus of nonlinear dynamics in the past decades.Many different dynamical models have been introduced to study its behavior.A simplified physical model is the phase oscillator model,of which the Kuramoto model is the most successful.In the coupled phase models,synchronization is represented as the selforganized ordering of a group of non-uniform phase oscillators with different natural frequencies behavior through nonlinear coupling.Even when frequency synchronization of coupled oscillators is achieved,the collective phase dynamics still exhibit rich behaviors,and these phase orderings and spatial pattern behaviors are related to many practical applications,so they have important research significance.The present dissertation focuses on both temporally continuous and discrete coupled oscillator models.We systematically explore various issues related to the multi-stable phase-ordered states and synchronizationdesynchronization transitions of coupled nonlinear oscillator systems.We analyze the existence and stability conditions of twisted states in locally coupled oscillator systems by studying the stable regions and critical boundaries of twisted states under different local coupling weight functions.The synchronization-desynchronization phase transition and its mechanisms of coupled nonlinear oscillator systems under different complex network topologies are discussed,and the stabilities of synchronous and twisted states are analyzed.A coupled circle map model is proposed,and theoretical and numerical analysis of the transitions from desynchronization to synchronization and back to desynchronization are studied,revealing the significant differences of the collective dynamics between time-discrete coupled oscillator systems(maps)and timecontinuous dynamical systems.The main achievements and innovations of the paper are summarized as follows:(1)This paper focuses on studying the stability and transition mechanisms of q-twisted states in a locally coupled phase oscillator system under the influence of different coupling weight functions.Using the spatial continuity scheme,we analytically derived the stability conditions of twisted states with different wave numbers q,and proposed the analysis of the stability twisted states based on the concavity and convexity characters of the weight coupling function to determine the stability of various twisted states in the system.The stable regions of twisted states were obtained for different forms of weight coupling functions such as the uniform function,the exponential function,and the Fermi function,and the phase diagrams and critical lines were given.Our study showed that for the case of the uniform coupling weight function,stable and unstable regions appear alternately in the parameter plane,and the size of the stable region gradually decreases with the increase of the relative coupling distance r,indicating a weak stability of the higher-order q-twisted states.For the exponential coupling weight function,due to the rapid decay of the spatial coupling weight function with increasing exponent β,higher-order qtwisted states exhibit a weak stability.With the increase of the effective temperature and the Fermi surface,the stability of higher-order even-q twisted modes is weakened,while the stability of odd-q mode twisted states(except for the q=1 mode)is strengthened,while a lower effective temperature and smaller Fermi surface may enhance the stability of more twisted modes.(2)The synchronization and desynchronization transitions in a network of coupled oscillators.It is pointed out the error brought by a of finite integral time steps in numerical integrations of the differential equations of coupled oscillators under strong coupling may lead to the loss of synchronization.Further analysis shows that the mechanism behind this instability is the accumulation of errors in numerical integrations with finite integration steps,which in turn leads to the loss of stability of the synchronized state.Our extensive re-calculations show that as the integration time step decreases,the critical coupling strength for desynchronization increases and diverges with decreasing the integral step in a power-law form,numerically indicating that the synchronized state is locally stable,which is supported by theoretical analysis.Furthermore,strictly speaking,the synchronized state in a coupled oscillator network is locally stable with a finite basin of attraction,depending on the initial conditions.This implies that other metastable ordered states,such as twisted states,can coexist with the synchronized mode.These twisted states can be locally stable on sparse networks,but lose their stability as the network becomes denser,which is consistent with the results obtained in(1).However,our studies show that even when the oscillators are heterogeneous,e.g.,when they have different natural frequencies,multiple stable twisted states can still be observed.(3)The synchronization dynamics of a coupled phase oscillator differential dynamical system model,using coupled circle mappings as its discrete form,have been studied in various network topologies.Despite the simplicity of the mathematical form of this model,we discovered a large number of interesting synchronization behaviors that differ from those of continuous-time models due to time discretization.By changing the coupling strength between the coupled circle map oscillators,multiple synchronization and desynchronization transitions of phase and frequency occur.The mechanism behind these transitions is explained using meanfield methods,revealing the collective cascade bifurcations of coupled circle map oscillators.The synchronization behavior of the coupled circle map also demonstrates significant differences between differential dynamics and discrete-time mappings.