Application of Population Dynamics in Chaos Synchronization in Discrete-Time Dynamical Systems

In this paper, we develop a new mathematical framework in study of chaotic synchronization of
discrete-time dynamical systems. Study of chaotic synchronization as a fundamental phenomenon in
the nonlinear dynamical systems theory has been recently raised many interests in science,
engineering, and technology. Population dynamics can be modeled through the continuous-time
system and the discrete-time system. However, when population size is small or that population does
not overlap, discrete-time system is more appropriate to use. In the novel drive-response discretetime
dynamical system which has been coupled using convex link function, we introduce a
synchronization threshold which passes that makes the drive-response system lose complete
coupling and synchronized behaviors. We provide the application of this type of coupling in
synchronized cycles of well-known Ricker model. This model displays a rich cascade of complex
dynamics from stable fixed point and cascade of period-doubling bifurcation to chaos. We also
numerically verify the effectiveness of the proposed scheme and demonstrate how this type of
coupling makes this chaotic system and its corresponding coupled system starting from different initial
conditions, quickly get synchronized. In chaotic regime, for larger values of synchronization threshold
s, closer to one, we could not get a complete synchronization. But, for smaller synchronization
threshold s, closer to zero, we have shown that two systems are in complete synchronization even
though the dynamics is chaotic.