Symmetry restoration in a class of forced oscillators
The author considers families of differential equations \(\dot x=f(x,\lambda,t)\), \(x\in\mathbb{R}^n\). The vector field \(f\) is assumed to be \(T\)-periodic in \(t\). So, in canonical manner, a Poincaré-map \(P\) can be assigned to those systems. Moreover, a particular symmetry is required: \(f(x,\lambda,t)=-f(-x,\lambda,t+(1/2)T)\). Finally, the notion of symmetry restoring bifurcation of \(\omega\)-limit sets of \(P\) has been introduced. The author studies this kind of bifurcation in several examples. The argumentation is based on numerical investigations and the corresponding plots of attractors. In all examples, a collision of a nonsymmetric set and a symmetric set takes place at the critical parameter value. The differences lie in the properties of the limit sets that are involved. Both chaotic and nonchaotic ones are considered. The description of the examples is very short. However, the present paper is based on the PhD thesis of the author where more detailed information can be found.
scientific article; zbMATH DE number 1813049