Abstract
In this paper, we study a drive-response discrete-time dynamical system which has been coupled using convex functions and we introduce a synchronization threshold which is crucial for the synchronizing procedure. We provide one application of this type of coupling in synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from stable fixed point to periodic orbits, quasi periodic orbits and chaos. We explain how this way of coupling makes these two chaotic systems starting from very different initial conditions, quickly get synchronized. We investigate the qualitative behavior of GNB model and its synchronized model using time series analysis and its long time dynamics by the help of bifurcation diagram.
Highlights
The study of long time behavior of a dynamical systems was based on the examples of ordinary differential equations with regular solutions and those solutions which remained in a bounded region of the phase space could be divided into two different types based on their local behavior: first, a stable equilibrium point and second, a periodic oscillation
We developed a drive-response system by defining a convex continuous link function which maps the orbits of the drive system into the orbits of its coupled system and keeps the same qualitative dynamics
We provided a new concept in chaos synchronization, called, synchronization threshold, which means that the solutions of drive and response system diverge from each other and lose the complete synchronization properties when they pass the threshold
Summary
The study of long time behavior of a dynamical systems was based on the examples of ordinary differential equations with regular solutions and those solutions which remained in a bounded region of the phase space could be divided into two different types based on their local behavior: first, a stable equilibrium point and second, a periodic (or quasi-periodic) oscillation. Bernd Blasius and Lewi Stone worked on a chaotic UPCA foodweb model and they claimed that the spatio-temporal structures associated with phase synchronization have important implications for conservation ecology. They proposed that even though perturbation of a local patch population can bring them to the brink of extinction, the periodicity of spatial phase synchronization can help to buffer the endangered population by colonizers. Using this convex function, we drive the response system which inherits all the complex qualitative dynamics of GNB model and mimics that certain properties of the motion which is shared between them. We demonstrate the complex dynamics of GNB model and its coupled system by conducting some time series and bifurcation analysis
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