Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis. (English) Zbl 1111.34035

The authors consider the system \[ \dot z= f(t,z,0),\quad z\in \mathbb R^d, \tag{1} \] where dot denotes differentiation with respect to the time \(t\), the function \(f(\cdot,\cdot,\cdot)\) is continuous and \(f(t,z,\xi)=f(t+T,z,\xi)\) for all \(t\in \mathbb R_+\), \(\xi \in\mathbb R^\ell\) with \(\mathbb R_+=[0,\infty)\). They assume that (1) has a \(T\)-periodic solution \(z_*\) which is locally exponentially stable. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that for each sufficiently small \(\varepsilon\), the perturbed system has a continuum of periodic solutions in a small neighborhood of \(z_*\). The main result here is a theorem on the stability of this continuum.


34C25 Periodic solutions to ordinary differential equations
34C55 Hysteresis for ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
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