  The physics of critical phenomena in a many-body system far from thermal
equilibrium is an interesting and important issue to be addressed both
experimentally and theoretically. The trapped cold atoms have been actively
used as a clean and versatile simulator for classical and quantum-mechanical
systems, deepening understanding of the many-body physics behind. Here we
review the nonlinear and collective dynamics in a periodically modulated
magneto-optically trapped cold atoms. By temporally modulating the intensity of
the trapping lasers with the controlled phases, one can realize two kinds of
nonlinear oscillators, the parametrically driven oscillator and the resonantly
driven Duffing oscillator, which exhibit the dynamical bistable states. Cold
atoms behave not only as the single-particle nonlinear oscillators, but also as
the coupled oscillators by the light-induced inter-atomic interaction, which
leads to the phase transitions far from equilibrium in a way similar to the
phase transition in equilibrium. The parametrically driven cold atoms show the
ideal mean-field symmetry-breaking transition, and the symmetry is broken with
respect to time translation by the modulation period. Such a phase transition
results from the cooperation and competition between the inter-particle
interaction and the fluctuations, which lead to the nonlinear switching of
atoms between the vibrational states, and the experimentally measured critical
characteristics prove it as the ideal mean-field transition class. On the other
hand, the resonantly driven cold atoms that possess the coexisting periodic
attractors exhibit the kinetic phase transition analogous to the discontinuous
gas-liquid phase transition in equilibrium, and interestingly the global
interaction between atoms causes the shift of the phase-transition boundary.
