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Article Dans Une Revue Discrete and Continuous Dynamical Systems - Series A Année : 2019

Prescribed energy connecting orbits for gradient systems

Résumé

We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed subsets $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$, for some $c\in\mathbb{R}$, we establish the existence of bounded solutions $q_c$ to the above system with energy equal to $-c$ whose trajectories connect $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of $\nabla V$ on $\partial\mathcal{V}^c_{\pm}$. Next, we illustrate applications of the existence result to double-well potentials $V$, and for potentials associated to systems of Duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions $(q_c)$.

Dates et versions

hal-01990860 , version 1 (23-01-2019)

Identifiants

Citer

Francesca Alessio, Piero Montecchiari, Andres Zuniga. Prescribed energy connecting orbits for gradient systems. Discrete and Continuous Dynamical Systems - Series A, 2019, 39 (8), pp.4895. ⟨10.3934/dcds.2019200⟩. ⟨hal-01990860⟩
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