Journal ArticleDOI
Unbounded Solutions in Non‐Smooth Dynamical Systems at Resonance
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In this paper, the authors consider periodically forced non-smooth dynamical systems at resonance, described by differential inclusions, and show that all solutions are unbounded in the (x)-phase plane, if the first Fourier coefficient of the forcing is large compared to a certain quantity related to the nonlinearity.Abstract:
We consider periodically forced non-smooth dynamical systems at resonance, described by differential inclusions, and we show that analogously to the case of ODEs all solutions are unbounded in the (x)-phase plane, if the first Fourier coefficient of the forcing is large compared to a certain quantity related to the nonlinearity.read more
Citations
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Journal ArticleDOI
Boundedness in Nonlinear Oscillations at Resonance
TL;DR: In this paper, the boundedness of all the solutions for the semilinear Duffing equation with the Lazer-Leach condition was studied, where p ( t ) is a smooth 2 π -periodic function and the perturbation φ ( x ) is bounded.
Journal ArticleDOI
On the Application of KAM Theory to Discontinuous Dynamical Systems
TL;DR: In this article, it was shown that all solutions are bounded for a > 0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case.
Book ChapterDOI
Non-Smooth Dynamical Systems: An Overview
Markus Kunze,Tassilo Küpper +1 more
TL;DR: In this article, the authors review mathematical results on non-smooth dynamical systems, i.e., systems that incorporate effects of friction and/or impacts, and show that these systems can be modeled as convex systems.
Journal ArticleDOI
Boundedness and unboundedness of solutions for reversible oscillators at resonance
TL;DR: In this article, the authors considered the problem of boundedness of solutions for the oscillator x'' + f(x)x' + n2x + (x) = p(t).
Journal ArticleDOI
On Lyapunov Exponents for Non-Smooth Dynamical Systems with an Application to a Pendulum with Dry Friction
TL;DR: In this paper, the authors considered dynamical systems from mechanics for which, due to some non-smooth friction effects, Oseledets' Multiplicative Ergodic Theorem cannot be applied canonically to define Lyapunov exponents.
References
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Journal ArticleDOI
Dynamical behaviour of a friction oscillator with simultaneous self and external excitation
TL;DR: In this article, it is shown how the very robust limit cycles of stick-slip vibrations can be broken up by a harmonic disturbance, based on a simple model of a friction oscillator with simultaneous self and external excitation, and the resulting bifurcation behavior and the routes to chaos are investigated for a wide range of system parameters.
Journal ArticleDOI
Unbounded solutions of semilinear equations at resonance
José Miguel Alonso,Rafael Ortega +1 more
TL;DR: In this paper, a forced harmonic oscillator at resonance with a nonlinear perturbation is considered and a sharp condition for the existence of unbounded motions is obtained for the case of a semilinear vibrating string.
Journal ArticleDOI
On the Application of KAM Theory to Discontinuous Dynamical Systems
TL;DR: In this article, it was shown that all solutions are bounded for a > 0 sufficiently large, and that there are infinitely many periodic and quasiperiodic solutions in this case.
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