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Search results for: HOMOCLINIC POINT
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Dynamics near nonhyperbolic fixed points or nontransverse homoclinic points
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Homoclinics for singular strong force Lagrangian systems
PublicationWe study the existence of homoclinic solutions for a class of generalized Lagrangian systems in the plane, with a C1-smooth potential with a single well of infinite depth at a point ξ and a unique strict global maximum 0 at the origin.Under a strong force condition around the singular point ξ, via minimization of an action integral, we will prove the existence of at least two geometrically distinct homoclinic solutions.
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Homoclinic and Heteroclinic Orbits for a Class of Singular Planar Newtonian Systems
PublicationThe study of existence and multiplicity of solutions of differential equations possessing a variational nature is a problem of great meaning since most of them derives from mechanics and physics. In particular, this relates to Hamiltonian systems including Newtonian ones. During the past thirty years there has been a great deal of progress in the use of variational methods to find periodic, homoclinic and heteroclinic solutions...
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Two families of infinitely many homoclinics for singular strong force Hamiltonian systems
PublicationWe are concerned with a planar autonomous Hamiltonian system with a potential possessing a single well of infinite depth at a point X and a unique strict global maximum 0 at a point A. Under a strong force condition around the singularity X, via minimization of an action integral and using a shadowing chain lemma together with simple geometrical arguments, we prove the existence of infinitely many geometrically distinct homoclinic...
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Homoclinics for singular strong force Lagrangian systems in R^N
PublicationWe will be concerned with the existence of homoclinics for second order Hamiltonian systems in R^N (N>2) given by Hamiltonians of the form H(t,q,p)=Φ(p)+V(t,q), where Φ is a G-function in the sense of Trudinger, V is C^2-smooth, periodic in the time variable, has a single well of infinite depth at a point ξ and a unique strict global maximum 0 at the origin. Under a strong force type condition aroud the singular point ξ, we prove...
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On the existence of homoclinic type solutions of inhomogenous Lagrangian systems
PublicationWe study the existence of homoclinic type solutions for a class of inhomogenous Lagrangian systems with a potential satisfying the Ambrosetti-Rabinowitz superquadratic growth condition and a square integrable forcing term. A homoclinic type solution is obtained as a limit of periodic solutions of an approximative sequence of second order differential equations.
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On the Existence of Homoclinic Type Solutions of a Class of Inhomogenous Second Order Hamiltonian Systems
PublicationWe show the existence of homoclinic type solutions of a class of inhomogenous second order Hamiltonian systems, where a C1-smooth potential satisfies a relaxed superquadratic growth condition, its gradient is bounded in the time variable, and a forcing term is sufficiently small in the space of square integrable functions. The idea of our proof is to approximate the original system by time-periodic ones, with larger and larger...
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Homoclinic orbits for an almost periodically forced singular Newtonian system in R^3
Publication. This work uses a variational approach to establish the existence of at least two homoclinic solutions for a family of singular Newtonian systems in R^3 which are subjected to almost periodic forcing in time variable
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Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems
PublicationIn this work we will consider a class of second order perturbed Hamiltonian systems with a superquadratic growth condition on a time periodic potential and a small aperiodic forcing term. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system...
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A convergence result for mountain pass periodic solutions of perturbed Hamiltonian systems
PublicationIn this work, we study second-order Hamiltonian systems under small perturbations. We assume that the main term of the system has a mountain pass structure, but do not suppose any condition on the perturbation. We prove the existence of a periodic solution. Moreover, we show that periodic solutions of perturbed systems converge to periodic solutions of the unperturbed systems if the perturbation tends to zero. The assumption on...
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An approximative scheme of finding almost homoclinic solutions for a class of Newtonian systems
PublicationW niniejszej pracy badamy istnienie rozwiązań prawie homoklinicznych (ang. almost homoclinic solutions) dla pewnej klasy układów Newtona. Rozwiązanie prawie homokliniczne otrzymujemy jako granicę ciągu rozwiązań okresowych dla pewnego ciągu równań różniczkowych.
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A note on an approximative scheme of finding almost homoclinic solutions for Newtonian systems
PublicationIn this work we will be concerned with the existence of an almost homoclinic solution for a perturbed Newtonian system in a finite dimensional space. It is assumed that a potential is C^1 smooth and its gradient is bounded with respect to a time variable. Moreover, a forcing term is continuous, bounded and squere integrable. We will show that the appproximative scheme due to J. Janczewska for a time periodic potential extends to...
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Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3
PublicationWe consider a conservative second order Hamiltonian system \ddot{q}+ ∇V(q)=0 in R3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ∅ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.
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Subharmonic solutions for a class of Lagrangian systems
PublicationWe prove that second order Hamiltonian systems with a potential of class C1, periodic in time and superquadratic at infinity with respect to the space variable have subharmonic solutions. Our intention is to generalise a result on subharmonics for Hamiltonian systems with a potential satisfying the global Ambrosetti-Rabinowitz condition from [P. H. Rabinowitz, Proc. Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38]. Indeed, we weaken...
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Almost homoclinic solutions for the second order Hamiltonian systems
PublicationW niniejszej pracy badam istnienie rozwiązań prawie homoklinicznych (almost homoclinic) dla układu Hamiltona rzędu drugiego (układu Newtona): ü(t) + V_{u}(t,u) = f(t), gdzie t є R, u є R^{n}, V(t,u) = -K(t,u) + W(t,u), K,W: R x R^{n} → R są klasy C^{1}, K spełnia warunek ''pinching'', W_{u}(t,u)=o(|u|), gdy |u| → 0 jednostajnie względem t, f: R → R^{n} jest funkcją ciągłą, niezerową i odpowiednio małą w L^{2}(R,R^{n}). Przy tych...
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Topological Behaviour of Solutions of Vibro-Impact Systems in the Neighborhood of Grazing
PublicationThe grazing bifurcation is considered for the Newtonian model of vibro-impact systems. A brief review on the conditions, sufficient for the existence of a grazing family of periodic solutions, is given. The properties of these periodic solutions are discussed. A plenty of results on the topological structure of attractors of vibro-impact systems is known. However, since the considered system is strongly nonlinear, these attractors...