# Prescribed energy connecting orbits for gradient systems

## TL;DR

This paper proves the existence of energy-specific connecting orbits in gradient systems with certain potential functions, classifying their types and demonstrating applications to classical mechanical models.

## Contribution

It introduces an energy constrained variational method to establish the existence of connecting orbits in gradient systems with general potentials, including applications to double-well, Duffing, and pendulum systems.

## Key findings

- Existence of bounded connecting solutions with prescribed energy.
- Classification of solutions into brake, heteroclinic, and homoclinic types.
- Convergence results for families of solutions in specific potential cases.

## Abstract

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 $(q_c)$.