Phase synchronization dynamics in coupled limit cycles with distributed natural frequencies are explored. A synchronization tree from free oscillations to local clustering and global phase locking is found. A desynchronization-induced transition to chaos is shown. Near the onset of various phase synchronization points, a simultaneous quantized stick-slip feature of the phases of oscillators is observed on the desynchronization side and heuristically interpreted in terms of a heteroclinic path instability.

The spatiotemporal dynamics of a damped sine-Gordon chain with sinusoidal nearest-neighbor couplings driven by a constant uniform force are discussed. The velocity characteristics of the chain versus the external force are shown. Dynamics in the high- and low-velocity regimes are investigated. It is found that in the high-velocity regime, the dynamics is dominated by rotating modes, the velocity shows a branching bifurcation feature, while in the low-velocity regime, the velocity exhibits steplike dynamical transitions, broken by the destruction of strong resonances.

The kink dynamics of the damped Frenkel-Kontorova (discrete sine-Gordon) chain driven by a constant external force is investigated. Resonant steplike transitions of the average velocity occur due to the competition between the moving kinks and their radiated phasonlike modes. A mean-field consideration is introduced to give a precise prediction of the resonant steps. Slip-stick motion and spatiotemporal dynamics on those resonant steps are discussed. Our results can be applied to studies of the fluxon dynamics of one-dimensional Josephson-junction arrays and ladders, dislocations, tribology, and other fields.

The ergodic property of the hard-ball systems is investigated numerically by comparing the microcanonical single-particle distributions and the corresponding numerical results in two- and three-dimensional cases. A Boltzmann-entropy-like quantity H(nc) is defined and its evolution is discussed. The deviation of this entropy from the equilibrium value obeys a power decay law in the long time limit, and the origin of this tail is explored heuristically.

A mechanism responsible for the directed transport of symmetrically coupled lattices in a symmetric periodic potential is proposed. Under the action of an external wave that breaks the spatiotemporal symmetry and introduces inhomogeneities of the lattice, a net unidirectional current can be observed, originating from a collaboration of the lattice (periodic potential and mutual interactions) and the wave (amplitude, frequency, and phase shifts). Mode-locking induced resonant steps are observed and predicted. The directed transport can be optimized and furthermore controlled by suitably adjusting the parameters of the system.

Directed collective motion in a circular array of unidirectionally coupled oscillators with symmetric potential is obtained numerically in the absence of external bias. This striking feature is interpreted as the effect of the spontaneous breaking of temporal symmetry of the coupling. It is revealed that a proper match of various control parameters is important in generating an optimal coherent global transport. Noise-sustained directed transport is also observed, and the related stochastic resonance in an autonomous system is identified..

Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems.

Irreversible processes of few hard-ball (N) mechanical systems are investigated numerically and compared with the theoretical results of quasistatic processes. The thermodynamic second law is valid for N≥2 for both equilibrium and nonequilibrium systems if the average ensemble of the large number of identical systems is taken.

The relation between generalized synchronization and phase synchronization is investigated. It was claimed that generalized synchronization always leads to phase synchronization, and phase synchronization is a weaker form than generalized synchronization. We propose examples that generalized synchronization can be weaker than phase synchronization, depending on parameter misfits. Moreover, generalized synchronization does not always lead to phase synchronization.

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated. Some novel collective phenomena are revealed. At the onset of instability of the phase-locking state, simultaneous phase slips of all oscillators and quantized phase shifts in these phase slips are observed. By increasing the coupling, a bifurcation tree from high-dimensional quasiperiodicity to chaos to quasiperiodicity and periodicity is found. Different orders of phase synchronizations of chaotic oscillators and chaotic clusters play the key role for constructing this tree structure.