Synchronization in directionally coupled systems: Some rigorous results
This paper concerns dynamical systems \((x,y)\in \mathbb{R}^m\times \mathbb{R}^n\mapsto (x',y')=F(x,y) \in\mathbb{R}^m \times\mathbb{R}^n\), which are generated by unidirectional coupling of the type \(x'=f(x)\), \(y'=g(x,y)\), where \(f\) is a homeomorphism and \(g\) is continuous. It is assumed that there exists a ball of dissipation \(B\subset\mathbb{R}^m\times\mathbb{R}^n\) and that master-slave synchronization occures in \(B\). Let \({\mathcal A}=\cap^\infty_{\ell =0}F^\ell(B)\) be the maximal attractor in \(B\), and denote by \(P_1:\mathbb{R}^m \times\mathbb{R}^n \to \mathbb{R}^m\) and \(P_2:\mathbb{R}^m \times\mathbb{R}^n \to\mathbb{R}^n\) the canonical projections. It is proved that there exists a map \(h:P_1({\mathcal A}) \to\mathbb{R}^n\) with \({\mathcal A}=\text{graph}(h)\), \(h\circ P_1\circ F^\ell= P_2\circ h\circ F^\ell\) and that \(h\) is Hölder continuous or even Lipschitz continuous (depending on the coupling strength). In a second part the authors prove almost everywhere synchronization for coupling of two identical maps of the interval that are neither continuous nor invertible.
Hölder continuous synchronization function
scientific article; zbMATH DE number 1736541
scientific article