Order Hamiltonian Systems Research Articles

Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

In this paper, we consider the second order Hamiltonian system $$\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= −d 2/dt 2 −A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.

A note on the minimal periodic solutions of nonconvex superlinear Hamiltonian system

In this paper, we consider the multiplicity result of minimal periodic solutions for the first order Hamiltonian system x˙(t)=J∇H(x(t)). We point out a mistake in the literatures [M. Girardi, M. Matzeu, Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem, Nonlinear Anal. T.M.A. 10 (4) (1986) 371–382] and [Tianqing An, On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems, J. Math. Anal. Appl. 329 (2007) 1273–1284]. When the potential is even, under some stronger restricted conditions, we give the correct proof and obtain more distinct nonconstant periodic solutions with minimal period T.

Stability analysis for standing pulse solutions to FitzHugh–Nagumo equations

This paper studies standing pulse solutions to the FitzHugh–Nagumo equations. Since the reaction terms are coupled in a skew-gradient structure, a standing pulse solution is a homoclinic orbit of a second order Hamiltonian system. In this work, an index theory for the Hamiltonian system is employed to study the stability of standing pulses for the FitzHugh–Nagumo equations. Related results for more general skew-gradient systems are also obtained.

New existence results on periodic solutions of nonautonomous second order Hamiltonian systems with $(q,p)$-Laplacian

Some new existence theorems are obtained for periodic solutions of nonautonomous second order Hamiltonian systems with $(q,p)$-Laplacian by using the least action principle and the minimax methods.

New periodic solutions for a class of singular Hamiltonian systems

In this paper, we apply a variant of the famous Mountain Pass Lemmas of Ambrosetti-Rabinowitz and Ambrosetti-Coti Zelati with (PSC)c type condition of Palais-Smale-Cerami to study the existence of new periodic solutions with a prescribed energy for symmetrical singular second order Hamiltonian conservative systems with weak force type potentials.

Homoclinic solutions for second order Hamiltonian systems with general potentials near the origin

In this paper, we study the existence of innitely many homoclinic solutions for a class of second order Hamiltonian systems with general potentials near the origin. Recent results from the literature are generalized and signicantly improved.

Homoclinic solutions for second order Hamiltonian systems with small forcing terms

Homoclinic solutions for second order Hamiltonian systems with small forcing terms

Homoclinic orbits for a class of singular second order Hamiltonian systems in ℝ3

Abstract We consider a conservative second order Hamiltonian system $$\ddot q + \nabla V(q) = 0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ {0} = ϑ 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.

Periodic solutions for second order Hamiltonian systems

By using the least action principle and minimax methods in critical point theory, some existence theorems for periodic solutions of second order Hamiltonian systems are obtained.

Infinitely many periodic solutions for discrete second order Hamiltonian systems with oscillating potential

In this article we obtain two sequence of infinitely many periodic solutions for discrete second order Hamiltonian systems with an oscillating potential. One sequence of solutions are local minimizers of the functional corresponding to the system, the other sequence are minimax type critical points of the functional.

Non-constant periodic solutions for second order Hamiltonian system with a p-Laplacian

Abstract In this paper, some existence theorems are obtained for nonconstant periodic solutions of second order Hamiltonian system with a p-Laplacian by using the Linking Theorem.

Odd Homoclinic Orbits for a Second Order Hamiltonian System

Abstract The aim of this paper is to find an odd homoclinic orbit for a class of reversible Hamiltonian systems. The proof is variational and it employs a version of the concentration compactness principle of P. L. Lions in a lemma due to Struwe.

Homoclinic solutions for a class of second order Hamiltonian systems with locally defined potentials

In this paper, we study the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems ü−L(t)u+Wu(t,u)=0, ∀t∈R, where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is only locally defined near the origin with respect to u and not assumed to be periodic with respect to t.

Periodic Solution of Some Non-Autonomous Second Order Hamiltonian Systems

By making use of the least action principle and the minimax methods,we study the existence of periodic solutions for some non-autonomous second order Hamiltonian systems in the cases of the sublinear growth and linear growth.Some new solvability conditions are obtained.

Periodic solutions for second order Hamiltonian systems on an arbitrary energy surface

Two theorems about the existence of periodic solutions with prescribed energy for second order Hamiltonian systems are obtained. One gives existence for almost all energies under very natural conditions. The other yields existence for all energies under a

Homoclinic solutions for a class of second order Hamiltonian systems

(Communicated by M. Paˇ sic) Abstract. The existence of homoclinic solutions is obtained by the Mountain Pass theorem for a class of the second order Hamiltonian systems ¨ q(t)+∇V(t,q(t)) = 0, where V(t,x )= −K(t,x )+ W(t,x) ∈ C 1 (R ×R N ,R), K(t,x) is not a quadratic form in x and W(t,x) is su- perquadratic in x.

Direct modeling method of generalized Hamiltonian system and simulation simplified

Generalized Hamiltonian model can give dynamics mechanism of inner parameters connected, however, it will meet many difficult while an actual physical system is converted to Generalized Hamiltonian model. And due to complexity of the structure matrix and damping matrix of high order Hamiltonian system, analysis and application to inner parameters connected mechanism is not ease and convenience. In this paper one kind of Hamiltonian model, four order and single input affine nonlinear system, is given basic form, it is a formulation formulize form, that is generalize Hamiltonian model can be directly derived by formulate calculation. The hydro turbine Hamiltonian model is taken as case to introduce this modeling method. At last, a simulation simplified method based-physical background is proposed, and used to simplify the hydro turbine Hamiltonian model.

The Existence of Homoclinic Solutions for Second Order Hamiltonian System

The research of homoclinic orbits for Hamiltonian system is a classical problem, it has valuable applications in celestial mechanics, plasma physis, and biological engineering. For example, homoclinic orbits rupture can yield chaos lead to more complex dynamics behaviour. This paper studies the existence of homoclinic solutions for a class of second order Hamiltonian system, we will prove this system exists at least one nontrivial homoclinic solution.

Variational Approach to a Class of Second Order Hamiltonian Systems on Time Scales

In this paper, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for the second order Hamiltonian system on time scale $\mathbb{T}$ $$\left\{\begin{array}{[email protected]{\quad}l}u^{\Delta^{2}}(t)+A(\sigma(t))u(\sigma(t))+\nabla F(\sigma(t),u(\sigma(t)))=0,& \hbox{\ $\Delta$-a.e. $t\in [0,T]_{_{\mathbb{T}}}^{\kappa}$,} \\u(0)-u(T)=0,\qquad u^{\Delta}(0)-u^{\Delta}(T)=0,& \hbox{}\end{array}\right.$$ where u Δ(t) denotes the delta (or Hilger) derivative of u at t, $u^{\Delta^{2}}(t)=(u^{\Delta})^{\Delta}(t)$ , � is the forward jump operator, T is a positive constant, A(t)=[d ij (t)] is a symmetric N�N matrix-valued function defined on $[0,T]_{\mathbb{T}}$ with $d_{ij}\in L^{\infty}([0,T]_{\mathbb{T}},\mathbb{R})$ for all i,j=1,2,�,N, and $F:[0,T]_{\mathbb{T}}\times \mathbb{R}^{N}\rightarrow\mathbb{R}$ . By establishing a proper variational setting, two existence results and two multiplicity results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.

Periodic solutions for nonautonomous second order Hamiltonian systems with sublinear nonlinearity

Some existence and multiplicity of periodic solutions are obtained for nonautonomous second order Hamiltonian systems with sublinear nonlinearity by using the least action principle and minimax methods in critical point theory.Mathematics Subject Classification (2000): 34C25, 37J45, 58E50.