We study the large-time behaviour of the nonlinear oscillatorwhere and is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case with real, . We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.
We study the large-time behaviour of the nonlinear oscillator where m, k>0 and f is a monotone real function representing nonlinear friction. We are interested in understanding the long-time effect of a nonlinear damping term, with special attention to the model case with α real, A>0. We characterize the existence and behaviour of fast orbits, i.e., orbits that stop in finite time.
Let (P,Q) be a C1 vector field defined in a open subset U ⊂ R2. We call a null divergence factor a C1 solution V (x, y) of the equation P ∂V/∂x + Q ∂V/ ∂y = ( ∂P/∂x + ∂Q/∂y ) V. In previous works it has been shown that this function plays a fundamental role in the problem of the center and in the determination of the limit cycles. In this paper we show how to construct systems with a given null divergence factor. The method presented in this paper is a generalization of the classical Darboux method...
The period function of a planar parameter-depending Hamiltonian system is examined. It is proved that, depending on the value of the parameter, it is either monotone or has exactly one critical point.
We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation , when is arbitrary and or when . The proof uses upper and lower solutions and the Leray–Schauder degree.
Sufficient conditions on the existence of periodic solutions for semilinear differential inclusions are given in general Banach space. In our approach we apply the technique of the translation operator along trajectories. Due to recent results it is possible to show that this operator is a so-called decomposable map and thus admissible for certain fixed point index theories for set-valued maps. Compactness conditions are formulated in terms of the Hausdorff measure of noncompactness.
The Poincaré-Bendixson Theorem and the development of the theory are presented - from the papers of Poincaré and Bendixson to modern results.