Synchronization in networks of initially independent dynamical systems

The dynamical system becomes initial-dependent when nonlinear quadratic term is considered, which the attractor can be switched between chaotic and periodical states by resetting the initial values even the parameters are fixed. Standard dynamical analysis and Hamilton energy are calculated to confirm the dynamics dependence on the initial setting. Feedback-based initial setting is applied to find the synchronization dependence on selection of initial values for the memory variable z. The nonlinear quadratic term z²y can suppress the oscillation of variable y via negative feedback, thus periodic oscillation can be triggered to tame another oscillator under bidirectional coupling, then periodic synchronization can be reached. Furthermore, the synchronization approach and pattern selection are considered on the network, and the factor of synchronization is calculated to find the synchronization dependence on coupling intensity. It is found that the network synchronization can be enhanced when noise-like disturbance is applied to reset the memory variable z, the potential mechanism is that the local kinetics is effectively adjusted to trigger periodic stimulus on some nodes thus the collective behaviors are controlled to become consensus.