Prescribed energy connecting orbits for gradient systems

Francesca Alessio, Piero Montecchiari, Andres Zuniga
2019 Discrete and Continuous Dynamical Systems. Series A  
We are concerned with conservative systemsq = ∇V (q), q ∈ R N for a general class of potentials V ∈ C 1 (R N ). Assuming that a given sublevel set {V ≤ c} splits in the disjoint union of two closed subsets V c − and V c + , for some c ∈ R, we establish the existence of bounded solutions qc to the above system with energy equal to −c whose trajectories connect V c − and V c + . The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are
more » ... resent in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of ∇V on ∂V c ± . 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 (qc). 2010 Mathematics Subject Classification. Primary: 34C25, 34C37, 49J40, 49J45. (q(t)) > c + h ρ,M for any t ∈ (σ n , τ n ) and so, by Remark 3, we conclude J c,(σn,τn) (q) ≥ 2h ρ,M |q(τ n ) − q(σ n )| = 2h ρ,M ρ, for all n ∈ N. But then J c (q) ≥ ∞ n=1 J c,(σn,τn) (q) = +∞, thus contradicting the assumption J c (q) < +∞. Moreover, by (9) we obtain the following concentration result Lemma 2.4. There existsr ∈ (0, ρ0 2 ) so that for any r ∈ (0,r), there exist L r > 0, ν r > 0, in such a way that for any q ∈ Γ c satisfying: dist (q(0), V c ) ≥ ρ 0 , q L ∞ (R,R N ) ≤ R and J c (q) ≤ m c + ν r , one has
doi:10.3934/dcds.2019200 fatcat:2hxrwvnsxnb4nhxjcc75bvdkqm