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Resonance in Preisach systems. (English) Zbl 1010.34038

The aim of the paper is to make a first step towards the investigation of large amplitude oscillations in Preisach systems outside the convexity domain. As a model example a simple hysteretic oscillator governed by a second-order ODE of the form \[ \ddot w(t) + u(t) = \psi (t), \quad w = u + P[u], \] is studied. (Here, \(P\) is a Preisach operator, \(\psi \in L^{\infty }(0, \infty)\) is a given function and \(t \geq 0\) is the time variable.)
An asymptotic condition (a relation between \(\psi \) and the Preisach measure) is established which is sufficient for the boundedness of every solution \(u\) (theorem 2.2.). Theorem 2.3 states that every bounded solution tends to \(0\) as \(t \to \infty \) provided \(\lim _{t \to \infty } \psi (t) = 0\), and the operator \(P\) does not degenerate to \(0\) in any neighbourhood of the origin. Further, a result concerning the (qualitative) optimality of the conditions in theorem 2.2 and the result saying that \(t^{-\frac 12}\) is the precise bound for the decay rate of \(\psi (t)\) under which every solution remains bounded for each choice of the data independently of the operator \(P\) are proved.

MSC:

34C55 Hysteresis for ordinary differential equations
82D40 Statistical mechanics of magnetic materials
34D40 Ultimate boundedness (MSC2000)
34C11 Growth and boundedness of solutions to ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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