# zbMATH — the first resource for mathematics

Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. (English) Zbl 1041.37032
Summary: We are concerned with perturbations of the Hamiltonian system $\ddot q-L(t)q+W_q(t,q)=0,\quad t\in \mathbb R, \tag{HS}$ where $$q = (q_1,\dots, q_N) \in \mathbb R^N$$, $$W \in C^1 (\mathbb R \times \mathbb R^N, \mathbb R)$$, and $$L(t) \in C (\mathbb R,\mathbb R^{N^2})$$ is a positive definite symmetric matrix. Variational arguments are used to prove the existence of homoclinic solutions for system (HS).

##### MSC:
 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
##### Keywords:
Homoclinic orbits; Duffing equations; Critical points
Full Text:
##### References:
 [1] Ambrosetti, A.; Bertotti, M., Homoclinic for second order conservative systems, (), 21-37 · Zbl 0804.34046 [2] Arioli, G.; Szulkin, A., () [3] Coti-Zelati, V.; Ekeland, I.; Sere, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [4] Coti-Zelati, V.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1992) · Zbl 0744.34045 [5] Rabinowitz, P.H., Homoclinic orbits for a class of Hamiltonian systems, (), 33-38 · Zbl 0705.34054 [6] Rabinowitz, P.H.; Tanaka, K., Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206, 473-499, (1991) · Zbl 0707.58022 [7] Ding, Y., Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear analysis, 25, 1095-1113, (1995) · Zbl 0840.34044 [8] Omana, W.; Willem, M., Homoclinic orbits for a class of Hamiltonian systems, Differential and integral equations, 5, 1155-1170, (1992) · Zbl 0759.58018 [9] Carriao, P.C.; Miyagaki, O.H., Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems, J. math. anal. appl., 230, 157-172, (1999) · Zbl 0919.34046 [10] Willem, M., Minimax theorems, (1996), Birkhäuser Boston, MA · Zbl 0856.49001 [11] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983), Springer New York · Zbl 0515.34001 [12] Alessio, F.; Caldiori, P.; Montecchiari, P., On the existence of homoclinic orbits for asymptotically periodic Duffing equation, Top. meth. nonl. analysis, 12, 275-292, (1998) · Zbl 0931.34028 [13] Spradlin, G., A Hamiltonian system with an even term, Top. meth. nonl. analysis, 10, 93-106, (1997) · Zbl 0988.37069 [14] Alves, C.O.; Carriao, P.C.; Miyagaki, O.H., Nonlinear perturbations of a periodic elliptic problem with critical growth, J. math. anal. appl., 260, 133-146, (2001) · Zbl 1139.35342 [15] Ding, Y.; Ni, W.Y., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. rat. mech. anal., 91, 283-308, (1986) · Zbl 0616.35029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.