Unbounded Solutions to Systems of Differential Equations at Resonance

Abstract

We deal with a weakly coupled system of ODEs of the type x j + n 2 j x j + h j (x 1 ,. .. , x d) = p j (t), j = 1,. .. , d, with h j locally Lipschitz continuous and bounded, p j continuous and 2π-periodic, n j ∈ N (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms h 1 ,. .. , h d are assumed.

Unbounded solutions to a system of coupled asymmetric oscillators at resonance

“…The first, quite natural, possibility that we present is a semi-perturbative result (cf. Corollary 4.3), dealing with the case in which the L ∞ -norms of the coupling terms φ 1 , φ 2 are not too big: it is worth noticing that this provided a genuinely asymmetric (non-quantitative) generalization of a result obtained in [3] for coupled linear oscillators. Other results, more global in nature but focusing on specific choices for the parameters a i , b i or the forcing terms p i , are given by Corollary 4.6 and Corollary 4.7.…”

“…where p 1,n is as in (4.7). Let us observe that this situation is not covered by the results in [3]. with p 2,n as in (4.7), there exist two infinite measure sets E ± ⊂ R 2 × R 2 such that:…”

“…In spite of this extensive bibliography, the existence of unbounded solutions for systems of coupled oscillators seems to be an essentially unexplored topic. To the best of our knowledge, the only available results are the one contained in the recent paper [3], dealing however with systems of equations looking like weakly coupled perturbations of linear oscillators (i.e. a i = b i = n 2 i for i = 1, 2) and not being applicable to the more general setting of (1.1).…”

“…Let us analyze the result of Corollary 4.3 in the symmetric linear case a i = b i = n 2 , i = In this situation, in the recent paper[3] the existence of unbounded solutions has been proved under the assumption4|φ i (+∞)| < | p i,n | 2 , i = 1, 2,(4.6)wherep i,n =2π 0 p i (t)e int dt (4.7) (see Theorem 3.1 in [3]). The assumption |φ i (+∞)| < φ * (i = 1, 2), with φ * = φ * (a 1 , b 1 , p 1 , p 2 ), in Corollary 4.3 is then on the same spirit of (4.6).…”

Unbounded Solutions to a System of Coupled Asymmetric Oscillators at Resonance

“…In spite of this extensive bibliography, the existence of unbounded solutions for systems of coupled oscillators seems to be an essentially unexplored topic. To the best of our knowledge, the only available results are the ones contained in the recent paper [3], dealing however with systems of equations looking like weakly coupled perturbations of linear oscillators (i.e. a i = b i = n 2 i for i = 1, 2) and not being applicable to the more general setting of (1.1).…”