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.