534 research outputs found

    Periodic impact behavior of a class of Hamiltonian oscillators with obstacles

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    AbstractIn this paper, we study the existence of harmonic and subharmonic solutions of a class of non-smooth Hamiltonian systems, then apply its results to the vibration problems{−x″=q(x)|xâ€Č|2+g(t)xâ€Č+f(t),x(t)>0,xâ€Č(t0−)=−xâ€Č(t0+),ifx(t0)=0. Infinitely many harmonic and subharmonic bouncing solutions are always obtained if q(x) satisfies some coercive conditions

    Subharmonic solutions of sublinear second order systems with impacts

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    AbstractWe mainly consider subharmonic bouncing solutions of sublinear second order systems with an obstacle based on variational method

    Twist maps in low regularity and non-periodic settings

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    Symplectic twist maps already appeared in the works of Poincaré. They emerge naturally as discretizations of certain low-dimensional Hamiltonian systems and offer a nice handle for studying their dynamics. The associated theory had experienced a tremendous boost from the discoveries made by Kolmogorow, Arnold and Moser in the 1960's, and by Aubry and Mather in the 1980's. In this thesis, we will substantiate the usefulness of studying twist maps also in non-periodic and low regularity settings, that is in situations where the classical results are not applicable. In particular, we will concentrate on perturbative methods for near-integrable systems. One typical feature of such systems is the existence of ``approximate first integrals'', so called adiabatic invariants, and their presence usually has strong consequences for the dynamics. Here, we derive growth rates for a large class of non-periodic twist maps depending on the regularity assumptions. As an application, the Fermi-Ulam ping-pong is considered, where the possible growth in velocity is linked to the number of bounded derivatives of the forcing function. Moreover, in systems with adiabatic invariants and almost periodic time-dependence the underlying compact structure enables one to use a generalization of Poincaré's recurrence theorem. By harnessing this fact, we prove that in such systems the set of initial condition leading to escaping orbits typically has measure zero. This is again demonstrated using the ping-pong model. Other applications are found in the Littlewood boundedness problem, where we consider a periodically forced piecewise linear oscillator together with its discontinuous limit case, and also a super-linear oscillator with an almost periodic forcing term. These systems are given by differential equations and thus the mentioned results imply also the Poisson stability of almost every solution. Even in the periodic case, these insights represent valuable contributions due to the low regularity assumptions necessary to obtain them

    Global dynamics of a vibro-impacting linear oscillator

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    International audienceThe steady state, vibro-impacting responses of one dimensional, harmonically excited, linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation. The steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction. Unlike the free, harmonically excited oscillator, the phase space of a vibro-impacting system may be inhabited by many attracting sets. For example, there are sub-harmonic, multi-impact, periodic orbits and chaotic, steady state responses. In order to build a qualitative understanding of vibro-impact response, an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges. Use is made of the Poincaré section or stroboscopic mapping technique, essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer. The qualitative understanding gained from the analysis and simulations is discussed in an engineering context
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