Periodic orbits for double regularization of piecewise smooth systems with a switching manifold of codimension two
In this paper, the author investigates the dynamical behaviors of a class of piecewise smooth dynamical systems defined in \(\mathbb{R}^n\). Let \(x=(x_1, \cdots, x_n)\in \mathbb{R}^n\). The system has a codimension 2 switching manifold \(\Sigma\) which is an intersection of two hyperplanes \(\Sigma_1:=\{x\in \mathbb{R}^n: \ x_1=0\}\) and \(\Sigma_2:=\{x\in \mathbb{R}^n: \ x_2=0\}\). The author first introduces the double regularized system with two small parameters \(\varepsilon\) and \(\eta\) of the original piecewise smooth system. Then the relation between periodic orbits of the piecewise smooth system and periodic orbits of its double regularized system is studied. By applying the implicit function theorem and geometric singular perturbation theory, the author proves that if the piecewise smooth system has an asymptotically stable sliding periodic orbit \(\gamma\), then under some conditions, the regularized system has also a unique, asymptotically stable periodic orbit \(\gamma_{\varepsilon\eta}\) in a neighborhood of \(\gamma\). Moreover, \(\gamma_{\varepsilon\eta}\) converges to \(\gamma\) as both \(\varepsilon\) and \(\eta\) converge to 0.
scientific article; zbMATH DE number 7461168
scientific article