Second-order Hamiltonian Systems Research Articles

Periodic solutions for second-order even and noneven Hamiltonian systems

AbstractIn this paper, we consider the second-order Hamiltonian system $$ \ddot{x}+V^{\prime}(x)=0,\quad x\in \mathbb{R}^{N}. $$ x ¨ + V ′ ( x ) = 0 , x ∈ R N . We use the monotonicity assumption introduced by Bartsch and Mederski (Arch. Ration. Mech. Anal. 215:283–306, 2015). When V is even, we can release the strict convexity hypothesis, which is used by Bartsch and Mederski combined with the monotonicity assumption. When V is noneven, we weaken the strict convexity assumption and introduce another hypothesis (see (V10)). Then in both cases, we can build the homomorphism between the Nehari manifold and the unit sphere of some suitable space. Using the Nehari manifold method introduced by Szulkin (J. Funct. Anal. 257:3802–3822 2009), we prove the existence of T-periodic solutions with minimal period T.

Periodic solutions of a class of non-autonomous second-order discrete Hamiltonian systems

<abstract><p>In this paper, in the view of control functions, the existence of periodic solutions of the following second-order discrete Hamiltonian system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \bigtriangleup ^{2} u(n-1) = \nabla F(n, u(n)), \ \ n\in \mathbb{Z} $\end{document} </tex-math></disp-formula></p> <p>with a generalized sublinear condition is further explored.</p></abstract>

High-order symmetric and energy-preserving collocation integrators for the second-order Hamiltonian system

High-order symmetric and energy-preserving collocation integrators for the second-order Hamiltonian system

A fourth-order energy-preserving and symmetric average vector field integrator with low regularity assumption

It is known that the average vector field (AVF) integrator is energy-preserving but only with second-order accuracy. Motivated by the construction of the classical AVF integrator, we devoted to constructing and analysing of high-order energy-preserving integrators for the second-order Hamiltonian systems in this work. Firstly, a novel improved average vector field (IAVF) integrator is derived by modifying the average vector field of the AVF integrator. Then the energy preservation and symmetry of the derived IAVF integrator are rigorously studied. Another significant investigation of the paper is that the IAVF integrator is shown to converge with order four under low regularity assumption q(t)∈C3([t0,T]). Numerical experiments are conducted to demonstrate the remarkable superiority in accuracy and efficiency over the traditional AVF integrator. Numerical results confirm the conclusions of our theoretical analysis.

Birational maps from polarization and the preservation of measure and integrals

The main result of this paper is the discretization of second-order Hamiltonian systems of the form , where K is a constant symmetric matrix and is a polynomial of degree in any number of variables n. The discretization uses the method of polarization and preserves both the energy and the invariant measure of the differential equation, as well as the dimension of the phase space. This generalizes earlier work for discretizations of first order systems with d = 3, and of second order systems with d = 4 and n = 1.

Existence of nontrivial rotating periodic solutions for second-order Hamiltonian systems

In this paper, we study existence of rotating periodic solutions for second-order Hamiltonian systems. We first define an index and give some properties of the index, and then build an index theory. By employing the index and the Leray–Schauder degree theory, the existence of nontrivial rotating periodic solutions is obtained.

A discrete second-order Hamiltonian system with asymptotically linear conditions

<abstract><p>This paper deals with a non-autonomous discrete second-order Hamiltonian system under asymptotically linear conditions. The existence of a periodic solution is obtained via the saddle point theorem.</p></abstract>

MULTIPLE PERIODIC SOLUTIONS FOR SUPERQUADRATIC AND SUBQUADRATIC SECOND-ORDER HAMILTONIAN SYSTEMS

In this paper, a class of second-order Hamiltonian systems is studied. Under the assumptions of superquadratic and subquadratic for the nonlinear term, the existence of six periodic solutions and nine periodic solutions is obtained by using the variational method and space decomposition. Finally, two examples are given to verify the feasibility of the new criteria.

Five periodic solutions for a class of subquadratic second-order even Hamiltonian systems

In this paper, we investigate a class of second-order subquadratic Hamiltonian systems with even potentials. The existence of at least five periodic solutions are obtained. Especially, two solutions possess the minimal period. The proof is based on the least action principle and the minimax methods in critical point theory.

A convergence result for mountain pass periodic solutions of perturbed Hamiltonian systems

In 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 the potential that guarantees the mountain pass geometry of the corresponding action functional is of independent interest as it is more general than those by Rabinowitz [Homoclinic orbits for a class of Hamiltonian systems, Proc. R. Soc. Edinburgh A 114 (1990) 33–38] and the authors [M. Izydorek and J. Janczewska, Homoclinic solutions for a class of the second-order Hamiltonian systems, J. Differ. Equ. 219 (2005) 375–389].

On nonsmooth Hamiltonian systems

In this paper, and by using a new variant of the standard theorem ([8], Theorem 1.1), we prove the existence of infinitely many homoclinic solutions for a new class of partial differential equations known as the nonsmooth second-order Hamiltonian systems.

Multiple minimal periodic solutions for subquadratic second-order Hamiltonian systems

By using a generalized version of the Weierstrass theorem and a new space decomposition, we obtain the existence of two minimal periodic solutions for a class of subquadratic even Hamiltonian systems. To our best knowledge, this is the first result of the existence of multiple minimal periodic solutions for Hamiltonian systems with subquadratic potentials.

Existence of Three Solutions for Nonlinear Operator Equations and Applications to Second-Order Differential Equations

The existence of three solutions for nonlinear operator equations is established via index theory for linear self-adjoint operator equations, critical point reduction method, and three critical points theorems obtained by Brezis-Nirenberg, Ricceri, and Averna-Bonanno. Applying the results to second-order Hamiltonian systems satisfying generalized periodic boundary conditions or Sturm-Liouville boundary conditions and elliptic partial differential equations satisfying Dirichlet boundary value conditions, we obtain some new theorems concerning the existence of three solutions.

Existence and Multiplicity of Periodic Solutions for Nonautonomous Second-Order Discrete Hamiltonian Systems

In this paper, we consider the periodic solutions of the following non-autonomous second order discrete Hamiltonian system $$\Delta^{2}u(n-1)=\nabla F(n,u(n)), \quad n\in\mathbb{Z}.$$ When the nonlinear function $F(n,x)$ is like-quadratic for $x$, we obtain some existence and multiplicity results under twisting conditions by using the least action principle and a multiple critical point theorem. The methods and main ideas using in this paper are variational method and critical point theory. The twisting conditions in our results are different from that in the lituratures.

Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems

In this paper, we consider a class of second-order Hamiltonian systems of the form \begin{document}$ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $\end{document} where \begin{document}$ L:R\rightarrow R^{N^2} $\end{document} and \begin{document}$ W \in C^1(R\times R^N, R) $\end{document} are asymptotically periodic in \begin{document}$ t $\end{document} at infinity. Under the reformative perturbation conditions and weaker superquadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is established. The main tools employed here are the local mountain pass theorem and the concentration-compactness principle.

Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

In this paper, we investigate infinitely many solutions for the generalized periodic boundary value problem −x″−B0tx+B1tx=λ∇xVt,xa.e.t∈0,1,x1=Mx0,x′1=Nx′0 under the potential function Vt,x without the evenness assumption and obtain two new existence results by the multiple critical point theorem. Meanwhile, we give two corollaries for the periodic solutions of second-order Hamiltonian systems and an example that illustrates our results.

Notes on Nontrivial Multiple Periodic Solutions for Second-Order Discrete Hamiltonian System

We obtain new existence of multiple nontrivial M-periodic solutions for a class of second-order discrete Hamiltonian system with the force F(t, x) being neither supquadratic nor subquadratic growth in x. Moreover, we exhibit four instructive examples to illustrate our results.

Existence of Homoclinic Solutions for a Class of Second-Order Hamiltonian Systems with Locally Subquadratic Potentials

In this paper, we mainly consider the existence of homoclinic orbits for the following second-order Hamiltonian systems $$\begin{aligned} \ddot{u}(t)-L(t)u(t)+\nabla W\bigl (t,u(t)\bigr )=f(t), \end{aligned}$$where L(t) is a positive definite and symmetric matrix for all $$t\in \mathbb {R}$$ and the potential function W(t, u) is locally subquadratic. Here, the coefficient of the upper bound for W is a positive constant, whereas in the previous literature the corresponding coefficient need to be some integrable functions a(t) on $$\mathbb {R}$$.

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

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

Two solutions to superlinear Hamiltonian systems with impulsive effects

As continuing research of the paper (Liu et al., 2019), in this article the existence of two solutions to superlinear second-order Hamiltonian systems that have damped and impulsive effects is proved. The main tools used are variational methods and two recent two-critical-points theorems. An example is presented to prove the value of the obtained theorems.