Stability and Bifurcations in 2D Spatiotemporal Discrete Systems

In this work, we give theoretical and numerical analyses for local bifurcations of 2D spatiotemporal discrete systems of the form [Formula: see text] where [Formula: see text] is a real nonlinear function, [Formula: see text] and [Formula: see text] are two independent integer variables, representing respectively a spatial coordinate and the time. On the basis of the spectral theory, we derive the conditions under which the local bifurcations such as flip and fold occur at the fixed points for some parameter values. As a case-study, a quite complex system, [Formula: see text]D spatiotemporal dynamic given by two coupled logistic maps, named [Formula: see text]D logistic coupled maps ([Formula: see text]D-LCM) is considered. The proposed map provides a reliable experimental and theoretical basis for identifying some cases of local bifurcations.

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Dynamics and Global Bifurcations in Two Symmetrically Coupled Non-Invertible Maps

Soula

Jahanshahi

Al-Barakati

et al. 2023

Mathematics

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The theory of critical curves determines the main characteristics of a discrete dynamical system in two dimensions. One important property that has garnered recent attention is the problem of chaos synchronization, along with the location of its chaotic attractors, basin boundaries, and bifurcation mechanisms. Varying the parameters of the maps reveals the instrumental role that these curves play, where the bifurcation leads to complex topological structures of the basins occurs by contact with the basin boundaries, resulting in the appearance or disappearance of some components of the basin. This study focuses on the properties of a discrete dynamical system consisting of two symmetrically coupled non-invertible maps, specifically those with an invariant one-dimensional submanifold (or one-dimensional maps). These maps exhibit a complex structure of basins with the coexistence of symmetric chaotic attractors, riddled basins, blow-out, on-off intermittency, and, most significantly, the appearance of chaotic synchronization with a correlation between all the characteristics. The numerical method of critical curves can be used to demonstrate a wide range of dynamic scenarios and explain the bifurcations that lead to their occurrence. These curves play a crucial role in a system of two symmetrically coupled maps, and their significance will be discussed.

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Stability and Bifurcations in 2D Spatiotemporal Discrete Systems

Cited by 3 publications

References 23 publications

Dynamical behavior and bifurcations in a two-dimensional discrete chaotic system with a rational fraction

Dynamical behavior and bifurcations in a two-dimensional discrete chaotic system with a rational fraction

Bifurcations in 2D Spatiotemporal Maps

Dynamics and Global Bifurcations in Two Symmetrically Coupled Non-Invertible Maps

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