Calculus of Variations and Geometric Measure Theory

F. Alessio - P. Montecchiari - A. Zuniga

Prescribed-energy connecting orbits for gradient systems

created by zuniga on 31 Jan 2019
modified on 11 Jul 2020


Published Paper

Inserted: 31 jan 2019
Last Updated: 11 jul 2020

Journal: Discrete Cont. Dyn. Syst. Series A
Volume: 39
Number: 8
Pages: 4895--4928
Year: 2019
Doi: 10.3934/dcds.2019200

ArXiv: 1901.06951 PDF
Links: Journal Link


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 $(q_{c})$.

Keywords: Variational methods, heteroclinic orbits, conservative systems, energy constraints, brake orbits, homoclinic orbits, convergence of solutions