Periodic Solutions of Discontinuous Duffing Equations
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Periodic Solutions of Discontinuous Duffing Equations Fangfang Jiang1 Received: 22 September 2019 / Accepted: 20 September 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we investigate the multiplicity problem of periodic solutions for a class of periodically forced Duffing equations allowing for discontinuities. By using a generalized form of the Poincaré–Birkhoff theorem due to Ding (Proc Am Math Soc 88:341–346, 1983), we demonstrate that the discontinuous equation has an infinite number of periodic solutions with large amplitude. Keywords Discontinuity · Duffing equation · Poincaré-Birkhoff theorem , multiplicity
1 Introduction There has been a wide study on periodic solution problem of second order Duffing equation (1.1) x + g(x) = e(t), because of its physical significance and applications of some mathematical techniques on it, such as Poincaré–Birkhoff theorem, variational method and topological degree theory. By giving assumptions to g, such as superlinearity, sublinearity, semilinearity and so on, scholars have obtained many interesting results, see for example [1,2,7– 11,13,14,24,27,30,32–35] and the references therein. In addition, there are many practical applications which can be modeled by differential equations containing discontinuous terms such as state-dependent jumps in control theory, abrupt changes at certain instants, or discontinuity in the vector field. So a natural question rises: if (1.1) possesses some type of discontinuities, what happens to the periodic solutions of the equation? As we know, discontinuous differential equations play an important role in many practical applications such as control theory, dynamical contact problems and nonlinear oscillators. It attracts a lot of attentions and many researchers studied the relevant
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Fangfang Jiang [email protected] School of Science, Jiangnan University, Wuxi 214122, China 0123456789().: V,-vol
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theory. Recently, there are a few studies on the second order Duffing equations with discontinuities (i.e. impulsive effects or discontinuous vector field), see [3–5,20,23,28,29] for example and the references therein. In [3–5], the authors studied the following oscillating model of elastic beam 1 = e(t), x +x 1− √ a2 + x 2
it can be smooth or discontinuous depending on the value of parameter a, i.e. the smooth dynamics appears for a > 0, while the discontinuous dynamical behavior occurs if a = 0. When g(x) in (1.1) is substituted by a jump of the restoring force, the authors [23] studied a discontinuous piecewise linear differential equation of the form (1.2) x + x + asign(x) = e(t). By KAM theory, they showed that when a > 0 is large sufficiently, all solutions of (1.2) are bounded, and it has infinitely many periodic solutions. In this paper, we address the similar question: whether there are many periodic solutions. We demonstrate the result by investigating a class of periodically forced second order nonlinear Duffing equations with discontinuous vector field, where th
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