First-order Hamiltonian System Research Articles

Homoclinic orbits for first-order Hamiltonian system with local super-quadratic growth condition

In this paper we study the first-order Hamiltonian system Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs.

Ground State Homoclinic Orbits for First-Order Hamiltonian System

In this paper, we study the following first-order Hamiltonian system $$\begin{aligned} \dot{z}=\mathscr {J}H_{z}(t,z). \end{aligned}$$Under some suitable conditions on the nonlinearity, we establish the existence of ground state homoclinic orbits by using variational methods. Moreover, we also explore some properties of these homoclinic orbits, such as compactness of set of and exponential decay of ground state homoclinic orbits.

First integrals and analytical solutions of some dynamical systems

This article investigates the first integrals and closed- form solutions of some nonlinear first-order dynamical systems from diverse areas of applied mathematics. We use the notion of artificial Hamiltonian, and we show that every first-order system of ordinary differential equations (ODEs) can be written in the form of an artificial Hamiltonian system [see Naz and Naeem (ZNA 73(4):323–330, 2018)]. One can also express the second-order ODE or system of second-order ODEs in the form of system of first-order artificial Hamiltonian system. Then the partial Hamiltonian approach is employed to compute the partial Hamiltonian operators and the corresponding first integrals. The first integrals are utilized to construct the closed-form solutions of two-stream model for tuberculosis and dengue fever, Duffing–van der Pol oscillator, nonlinear optical oscillators under parameter restrictions, nonlinear convection model and the two- dimensional galaxy model. We show that how one can apply the existing “partial Hamiltonian approach” for nonstandard Hamiltonian systems. This study provides a new way of solving the dynamical systems of first-order ODEs, second-order ODE and second-order systems of ODEs which are expressed into the artificial Hamiltonian system.

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

In this paper, we study the nonperiodic first-order Hamiltonian system \(\dot u = JL(t)u + JH'(t,u)\), where H ∈ C1(R × R2n). With some assumptions on L, the corresponding Hamiltonian operator has only discrete spectrum. By using the index theory for self-adjoint operator equation, we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.

HOMOCLINIC ORBITS FOR THE FIRST-ORDER HAMILTONIAN SYSTEM WITH SUPERQUADRATIC NONLINEARITY

In this paper, we consider the following first-order Hamiltonian system$$\dot{z} = \mathcal {J}H_{z}(t,z),$$where $H\in C^{1}(\mathbb{R}\times\mathbb{R}^{2N},\mathbb{R})$ isthe form $H(t,z)=\frac{1}{2}L(t)z\cdot z + R(t,z)$. By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we establish nontrivial and ground state solutions for the above system under conditions weaker than those in [39].

Homoclinic orbits of first-order superquadratic Hamiltonian systems

In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system\begin{equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}.\end{equation*}Under the Ambrosetti-Rabinowitz's superquadraticy condition, or no Ambrosetti-Rabinowitz's superquadracity condition, we present two results on the existence of infinitely many large energy homoclinic orbits when $H$ is even in $u$. We apply the generalized (variant) fountain theorems established recently by the author and Colin. Under no Ambrosetti-Rabinowitz's superquadracity condition, we also obtain the existence of a ground state homoclinic orbit by using the method of the generalized Nehari manifold for strongly indefinite functionals developed by Szulkin and Weth.

HOMOCLINIC ORBITS OF NONPERIODIC SUPERQUADRATIC HAMILTONIAN SYSTEM

In this paper, we study the following first-order nonperiodic Hamiltonian system $$\dot{z} = \mathcal {J}H_{z}(t,z),$$ where $H \in C^{1}(\mathbb{R} \times \mathbb{R}^{2N}, \mathbb{R})$ is the form $H(t,z) = \frac{1}{2} L(t)z \cdot z + R(t,z)$. Under weak superquadratic condition on the nonlinearitiy. By applying the generalized Nehari manifold method developed recently by Szulkin and Weth, we prove the existence of homoclinic orbits, which are ground state solutions for above system.

Nonexistence of Homoclinic Orbits for a Class of Hamiltonian Systems

We give several sufficient conditions under which the first-order nonlinear Hamiltonian systemx'(t)=α(t)x(t)+f(t,y(t)), y'(t)=-g(t,x(t))-α(t)y(t)has no solution(x(t),y(t))satisfying condition0<∫-∞+∞[|x(t)|ν+(1+β(t))|y(t)|μ]dt<+∞‍,whereμ,ν>1and(1/μ)+(1/ν)=1,0≤xf(t,x)≤β(t)|x|μ,xg(t,x)≤γ0(t)|x|ν,β(t),γ0(t)≥0, andα(t)are locally Lebesgue integrable real-valued functions defined onℝ.

Lyapunov-type inequalities for the first-order nonlinear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for the first-order nonlinear Hamiltonian system { x ′ ( t ) = α ( t ) x ( t ) + β ( t ) | y ( t ) | μ − 2 y ( t ) , y ′ ( t ) = − γ ( t ) | x ( t ) | ν − 2 x ( t ) − α ( t ) y ( t ) , which generalize or improve all related existing ones.

Homoclinic orbits of superlinear Hamiltonian systems

In this paper, we consider the first-order Hamiltonian system \[ J u ˙ ( t ) + ∇ H ( t , u ( t ) ) = 0 , t ∈ R . J\dot {u}(t)+\nabla H(t,u(t))=0,\quad t\in \mathbb {R}. \] Here the classical Ambrosetti-Rabinowitz superlinear condition is replaced by a general super-quadratic condition. We will study the homoclinic orbits for the system. The main idea here lies in an application of a variant generalized weak linking theorem for a strongly indefinite problem developed by Schechter and Zou.

Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems

Considering a decomposition R2N=A♁B of R2N , we prove in this work, the existence of at least (1+dimA) geometrically distinct periodic solutions for the first-order Hamiltonian system Jx'(t)+H'(t,x(t))+e(t)=0 when the Hamiltonian H(t,u+v) is periodic in (t,u) and its growth at infinity in v is at most like or faster than |v|a, 0≤a<1 , and e is a forcing term. For the proof, we use the Least Action Principle and a Generalized Saddle Point Theorem.

Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system

This paper deals with existence and exponential decay of homoclinic orbits in the first-order Hamiltonian system z ˙ = J H z ( t , z ) where the Hamiltonian function H ( t , z ) is nonperiodic in t ∈ R and superquadratic in z ∈ R 2 N . With certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems.