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Asymptotically stable heteroclinic cycles in discrete-time \(\mathbb{Z}_4\)-equivariant cubic dynamical systems. (English) Zbl 1478.37030

Summary: We consider a discrete-time \(\mathbb{Z}_4\)-equivariant cubic dynamical system depending upon a real parameter. The system under consideration is a particular case of a discrete analogue to the principal \(\mathbb{Z}_4\)-equivariant differential equations. We rigorously prove that the system has an asymptotically stable heteroclinic cycle, relatively to an open subset of a compact subspace of the plane, for values of the parameter in the interval \(\left(0,\frac{1}{2}\right]\). We explore properties of the omega-limit sets for the points attracted by the heteroclinic cycle.

MSC:

37B25 Stability of topological dynamical systems
37C75 Stability theory for smooth dynamical systems
37C81 Equivariant dynamical systems
39A30 Stability theory for difference equations

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