Bounded solutions for a forced bounded oscillator without friction

Some results on the existence of forced oscillations in mechanical systems

We present sufficient conditions for the existence of forced oscillations in non-autonomous mechanical systems. Previously, similar results were obtained for systems with friction. Presented results hold both for systems with and without friction. Some examples are given.

Boundedness in forced isochronous oscillators

In this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator $$x''+V'(x)+g(x)=f(t),$$ where $V$ is a so-called $T$-isochronous potential, the perturbation $g$ is assumed to be bounded, and the $2π$-periodic function $f(t)$ is smooth. Using the resonant small twist theorem and averaged small twist theorem established by Ortega, we will prove the boundedness of all solutions for the above forced isochronous oscillator in the resonant and non-resonant cases under some reasonable assumptions, respectively.

Harmonic oscillators at resonance, perturbed by a non-linear friction force

This note is an addendum to the results of P.O. Frederickson and A.C. Lazer [1], and A.C. Lazer [4] on periodic oscillations, with linear part at resonance. We show that a small modification of the argument in [4] provides a more general result. It turns out that things are different for the corresponding Dirichlet boundary value problem.

Attractiveness of periodic orbits in parametrically forced systemswith time-increasing friction

We consider dissipative one-dimensional systems subject to a periodic force and study numerically how a time-varying friction affects the dynamics. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on th…

受迫振子周期解的反最大值原理

An antimaximum principle for periodic solutions of a forced oscillator

针对带正的2π周期强迫项的线性振子方程，研究其周期解与正解的存在性关系，通过建立该问题与凸分析分离问题的联系，证明了周期解存在则正解存在的反最大值原理。

Remarks on forced oscillations in some systems with gyroscopic forces

In the paper we study the existence of a forced oscillation in two Lagrange systems with gyroscopic forces: a spherical pendulum in a magnetic field and a point on a rotating closed convex surface. We show how it is possible to prove the existence of forced oscillations in these systems provided the systems move in the presence of viscous friction.