Study on Complete Chaos Synchronization Using Contraction Mapping Theorem

Many researchers in various fields have studied chaotic behaviours and synchronised cycles of population models. We use a recently developed method to synchronise a discrete-time dynamical system in this paper. We are able to find a threshold to completely synchronise a dynamical system using this coupling approach, and we will apply this result to a quadratic population model. Depending on the parameter values, this model reveals various types of dynamics ranging from stable equilibrium to periodic behaviour and chaos. Traditionally, synchronization was based upon periodic signals. This rigorous method assists us in successfully synchronising the chaotic attractors of the original system and its coupled one. Finally, we will test the analytic results using various numerical tools such as the mean phase difference and mean amplitude difference, time series, and bifurcation diagrams for various threshold and parameter values.