Bifurcation of relative periodic solutions in symmetric systems with hysteretic constitutive relations

Dmitrii Rachinskii; Dmitrii Rachinskii

doi:10.3934/mine.2025004

Mathematics in Engineering

2025, Volume 7, Issue 2: 61-95. doi: 10.3934/mine.2025004

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Bifurcation of relative periodic solutions in symmetric systems with hysteretic constitutive relations

Dmitrii Rachinskii ^,

Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Rd, Richardson, TX 75080, USA

Received: 14 September 2024 Accepted: 25 February 2025 Published: 19 March 2025

We consider a differential system coupled to a hysteresis operator of Preisach type. It is assumed that the system is equivariant with respect to an action of the group $ \Gamma\times S^1 $ (where $ \Gamma $ is a finite group) in the phase space. Moreover, there is a branch of symmetric relative equilibria. We develop an application of the equivariant twisted topological degree, which detects branches of relative periodic solutions bifurcating from the relative equilibrium at an equivariant Hopf bifurcation point. These branches are classified according to their symmetric properties. The general theorem is illustrated with an example, where equations of motion of an $ S_5\times S^1 $-equivariant electromechanical system are coupled with the Prandtl–Ishlinskii hysteresis operator; this operator models the stress-strain constitutive relation of an elastoplastic spring. Hysteresis operators are non-smooth but can be differentiable at particular points. At the same time, applications of the equivariant degree require the vector field to be differentiable at the bifurcation point. To satisfy this requirement, we construct $ \Gamma\times S^1 $-vector fields, for which the zero set consists of the relative equilibria and relative periodic solutions of the system with the hysteresis operator, and ensure the differentiability at the zeros corresponding to the relative equilibria. This construction is the main technical contribution of the paper.
Citation: Dmitrii Rachinskii. Bifurcation of relative periodic solutions in symmetric systems with hysteretic constitutive relations[J]. Mathematics in Engineering, 2025, 7(2): 61-95. doi: 10.3934/mine.2025004

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Abstract

We consider a differential system coupled to a hysteresis operator of Preisach type. It is assumed that the system is equivariant with respect to an action of the group $ \Gamma\times S^1 $ (where $ \Gamma $ is a finite group) in the phase space. Moreover, there is a branch of symmetric relative equilibria. We develop an application of the equivariant twisted topological degree, which detects branches of relative periodic solutions bifurcating from the relative equilibrium at an equivariant Hopf bifurcation point. These branches are classified according to their symmetric properties. The general theorem is illustrated with an example, where equations of motion of an $ S_5\times S^1 $-equivariant electromechanical system are coupled with the Prandtl–Ishlinskii hysteresis operator; this operator models the stress-strain constitutive relation of an elastoplastic spring. Hysteresis operators are non-smooth but can be differentiable at particular points. At the same time, applications of the equivariant degree require the vector field to be differentiable at the bifurcation point. To satisfy this requirement, we construct $ \Gamma\times S^1 $-vector fields, for which the zero set consists of the relative equilibria and relative periodic solutions of the system with the hysteresis operator, and ensure the differentiability at the zeros corresponding to the relative equilibria. This construction is the main technical contribution of the paper.

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