Abstract
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. In this paper, we develop a new mathematical framework in study of chaotic synchronization of discrete-time dynamical systems. In the novel drive-response discrete-time 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.
Highlights
Population dynamics can be modeled through the continuous-time system and the discrete-time system
Synchronization in population dynamics can lead to arising complex dynamics and understanding the synchronization of oscillations is crucially important in this area
In 1990, two Lorenz systems with the property of sensitive dependence on the initial conditions could be synchronized starting from different initial states
Summary
Population dynamics can be modeled through the continuous-time system and the discrete-time system. Carroll, described a coupling method which constructs a real set of chaotic synchronization circuits [13]. They have applied this common signal to several well-known continuous-time dynamical systems such as Lorenz and Rossler and they claimed. We explain that how this coupling method can be applied on a general discrete-time dynamical system to get a complete synchronization. The long term analysis through bifurcation diagrams and time-series analysis exhibit that this drive-response system which reveals complex dynamics including cascade of period doubling to chaotic solutions, for smaller synchronization threshold, get completely synchronized
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