90 research outputs found

    Infinitely many periodic solutions for second order Hamiltonian systems

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    In this paper, we study the existence of infinitely many periodic solutions for second order Hamiltonian systems u¨+uV(t,u)=0\ddot{u}+\nabla_u V(t,u)=0, where V(t,u)V(t, u) is either asymptotically quadratic or superquadratic as u|u|\to \infty.Comment: to appear in JDE(doi:10.1016/j.jde.2011.05.021

    Infinitely many fast homoclinic solutions for some second-order nonautonomous systems

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    We investigate the existence of infinitely many fast homoclinic solutions for a class of second-order nonautonomous systems. Our main tools are based on the variant fountain theorem. A criterion guaranteeing that the second-order system has infinitely many fast homoclinic solutions is obtained. Recent results from the literature are generalized and significantly improved.Досліджєно існування нескінченної кількості швидких гомоклінічних розв'язків для класу неавтономних систем другого порядку. Наш основний метод базується на модифікації теореми про фонтан. Отримано критерій, що гарантує наявність нескінченної кількості швидких гомоклінічних розв'язків системи другого порядку. Узагальнено та значно покращено нещодавно опубліковані результати

    Infinitely many homoclinic solutions for perturbed second-order Hamiltonian systems with subquadratic potentials

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    In this paper, we consider the following perturbed second-order Hamiltonian system u¨(t)+L(t)u=W(t,u(t))+G(t,u(t)), tR, -\ddot{u}(t)+L(t)u=\nabla W(t,u(t))+\nabla G(t,u(t)), \qquad \forall \ t\in \mathbb{R}, where W(t,u)W(t,u) is subquadratic near origin with respect to uu; the perturbation term G(t,u)G(t,u) is only locally defined near the origin and may not be even in uu. By using the variant Rabinowitz's perturbation method, we establish a new criterion for guaranteeing that this perturbed second-order Hamiltonian system has infinitely many homoclinic solutions under broken symmetry situations. Our result improves some related results in the literature

    Infinitely many homoclinic solutions for perturbed second-order Hamiltonian systems with subquadratic potentials

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    In this paper, we consider the following perturbed second-order Hamiltonian system −u¨(t) + L(t)u = ∇W(t, u(t)) + ∇G(t, u(t)), ∀ t ∈ R, where W(t, u) is subquadratic near origin with respect to u; the perturbation term G(t, u) is only locally defined near the origin and may not be even in u. By using the variant Rabinowitz’s perturbation method, we establish a new criterion for guaranteeing that this perturbed second-order Hamiltonian system has infinitely many homoclinic solutions under broken symmetry situations. Our result improves some related results in the literature

    New existence and multiplicity of homoclinic solutions for second order non-autonomous systems

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    In this paper, we study the second order non-autonomous system \begin{eqnarray*} \ddot{u}(t)+A\dot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \ \forall t\in\mathbb{R}, \end{eqnarray*} where AA is an antisymmetric N×NN\times N constant matrix, LC(R,RN×N)L\in C(\mathbb{R},\mathbb{R}^{N\times N}) may not be uniformly positive definite for all tRt\in\mathbb{R}, and W(t,u)W(t,u) is allowed to be sign-changing and local superquadratic. Under some simple assumptions on AA, LL and WW, we establish some existence criteria to guarantee that the above system has at least one homoclinic solution or infinitely many homoclinic solutions by using mountain pass theorem or fountain theorem, respectively. Recent results in the literature are generalized and significantly improved

    Existence and multiplicity of weak quasi-periodic solutions for second order Hamiltonian system with a forcing term

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    In this paper, we first obtain three inequalities and two of them, in some sense, generalize Sobolev's inequality and Wirtinger's inequality from periodic case to quasi-periodic case, respectively. Then by using the least action principle and the saddle point theorem, under subquadratic case, we obtain two existence results of weak quasi-periodic solutions for the second order Hamiltonian system: d[P(t)u˙(t)]dt=F(t,u(t))+e(t),\frac{d[P(t)\dot{u}(t)]}{dt}=\nabla F(t,u(t))+ e(t), which generalize and improve the corresponding results in recent literature [J. Kuang, Abstr. Appl. Anal. 2012, Art. ID 271616]. Moreover, when the assumptions F(t,x)=F(t,x)F(t,x)=F(t,-x) and e(t)0e(t)\equiv 0 are also made, we obtain two results on existence of infinitely many weak quasi-periodic solutions for the second order Hamiltonian system under the subquadratic case.

    Homoclinic Solutions for a Class of Second Order Nonautonomous Singular Hamiltonian Systems

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    We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems u¨+atWuu=0, (HS) where -∞<t<+∞, u=u1,u2, …,uN∈ℝNN≥3, a:ℝ→ℝ is a continuous bounded function, and the potential W:ℝN∖{ξ}→ℝ has a singularity at 0≠ξ∈ℝN, and Wuu is the gradient of W at u. The novelty of this paper is that, for the case that N≥3 and (HS) is nonautonomous (neither periodic nor almost periodic), we show that (HS) possesses at least one nontrivial homoclinic solution. Our main hypotheses are the strong force condition of Gordon and the uniqueness of a global maximum of W. Different from the cases that (HS) is autonomous at≡1 or (HS) is periodic or almost periodic, as far as we know, this is the first result concerning the case that (HS) is nonautonomous and N≥3. Besides the usual conditions on W, we need the assumption that a′t<0 for all t∈ℝ to guarantee the existence of homoclinic solution. Recent results in the literature are generalized and significantly improved

    Transport in Transitory Dynamical Systems

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    We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case. This requires knowing only the "action" of relevant heteroclinic orbits at the intersection of invariant manifolds of "forward" and "backward" hyperbolic orbits. These manifolds can be easily computed by leveraging the autonomous nature of the vector fields on either side of the time-dependent transition. As illustrative examples we consider a two-dimensional fluid flow in a rotating double-gyre configuration and a simple one-and-a-half degree of freedom model of a resonant particle accelerator. We compare our results to those obtained using finite-time Lyapunov exponents and to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure

    Bifurcation Theory of Dynamical Chaos

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    The purpose of the present chapter is once again to show on concrete new examples that chaos in one-dimensional unimodal mappings, dynamical chaos in systems of ordinary differential equations, diffusion chaos in systems of the equations with partial derivatives and chaos in Hamiltonian and conservative systems are generated by cascades of bifurcations under universal bifurcation Feigenbaum-Sharkovsky-Magnitskii (FShM) scenario. And all irregular attractors of all such dissipative systems born during realization of such scenario are exclusively singular attractors that are the nonperiodic limited trajectories in finite dimensional or infinitely dimensional phase space any neighborhood of which contains the infinite number of unstable periodic trajectories
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