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Prescribed energy connecting orbits for gradient systems

  • * Corresponding author: Andres Zuniga

    * Corresponding author: Andres Zuniga

The third author is supported by a public grant overseen by the French National Research Agency (ANR) as part of the Investissement d'Avenir project, reference ANR-10-LABX-0098, LabEx SMP, and also supported by the project EFI ANR-17-CE40-0030 of the ANR

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  • 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}) $.

    Mathematics Subject Classification: Primary: 34C25, 34C37, 49J40, 49J45.


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  • Figure 1.  Possible configurations in Duffing like systems

    Figure 2.  Possible configurations in pendulum like systems

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