Second-order Hamiltonian Systems Research Articles

Infinitely Many Rotating Periodic Solutions for Second-Order Hamiltonian Systems

In this paper, we consider a class of second-order Hamiltonian system in $\mathbb {R}^{N}$ with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., $x(t+T)=Qx(t)$ with $T>0$ and Q is an $N\times N$ orthogonal matrix. In the case $Q^{k}\neq I_{N}$ for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case $Q^{k}=I_{N}$ for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.

Multiplicity Solutions for a Class of Fractional Hamiltonian Systems With Concave–Convex Potentials

In this paper, we study the existence and nonuniqueness of nontrivial solutions for a class of fractional Hamiltonian systems with concave–convex potentials via the variational methods. A new twisted condition is introduced, which yields a new compact embedding theorem. Some results are new even for second-order Hamiltonian systems.

Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part

Existence of solutions for a class of second-order impulsive Hamiltonian system with indefinite linear part

Homoclinic orbits for a class of second-order Hamiltonian systems with concave-convex nonlinearities

Homoclinic orbits for a class of second-order Hamiltonian systems with concave-convex nonlinearities

Infinitely many periodic solutions for second-order discrete Hamiltonian systems

Infinitely many periodic solutions for second-order discrete Hamiltonian systems

Local super-quadratic conditions on homoclinic solutions for a second-order Hamiltonian system

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system u ̈ − L ( t ) u + ∇ W ( t , u ) = 0 , where t ∈ R , u ∈ R N , L : R → R N × N and W : R × R N → R . Applying the Mountain Pass Theorem, we prove the existence of nontrivial homoclinic solutions under new weaker conditions. In particular, we use a local super-quadratic condition lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ ( a , b ) for some − ∞ < a < b < + ∞ , instead of the common one lim | x | → ∞ W ( t , x ) | x | 2 = ∞ , uniformly in t ∈ R , which is essential to show the existence of nontrivial homoclinic solutions for the above system in all existing literature.

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

In this paper, we construct stochastic symplectic Runge--Kutta (SSRK) methods of high strong order for Hamiltonian systems with additive noise. By means of colored rooted tree theory, we combine conditions of mean-square order 1.5 and symplectic conditions to get totally derivative-free schemes. We also achieve mean-square order 2.0 symplectic schemes for a class of second-order Hamiltonian systems with additive noise by similar analysis. Finally, linear and non-linear systems are solved numerically, which verifies the theoretical analysis on convergence order. Especially for the stochastic harmonic oscillator with additive noise, the linear growth property can be preserved exactly over long-time simulation.

Notes on multiple periodic solutions for second-order discrete Hamiltonian system

ABSTRACTBy a new orthogonal direct sum decomposition EM = Y⊕Z, where Z is related to Δui(i = 1, 2, 3, …, M), and a new functional I(u), the method by Guo and Yu is improved to obtain new multiple periodic solutions with a negativity hypothesis on F for a second-order discrete Hamiltonian system. Moreover, we exhibit an instructive example to make our result more clear, which has not been solved by known results.

Multiple periodic solutions for second-order discrete Hamiltonian systems

Multiple periodic solutions for second-order discrete Hamiltonian systems

Homoclinic orbits for the second-order Hamiltonian systems

We deal with the following second-order Hamiltonian systems ü−L(t)u+∇W(t,u)=0, where L∈C(R,RN2) is a symmetric and positive define matrix for all t∈R, W∈C1(R×RN,R) and ∇W(t,u) is the gradient of W with respect to u. Under the superquadratic condition, we obtain the existence of ground state homoclinic orbits by means of the generalized Nehari manifold developed by Szulkin and Weth. Under the subquadratic condition, we employ variational techniques and the concentration-compactness principle to establish new criteria guaranteeing the multiplicity of classical homoclinic orbits. Recent results in literature are generalized and significantly improved.

Infinitely many periodic solutions to a class of perturbed second-order impulsive Hamiltonian systems

Infinitely many periodic solutions to a class of perturbed second-order impulsive Hamiltonian systems

A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems

In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages.

A Variant of Clark's Theorem and Its Applications for Nonsmooth Functionals without the Palais--Smale Condition

By introducing a new notion of the genus with respect to the weak topology in Banach spaces, we prove a variant of Clark's theorem for nonsmooth functionals without the Palais--Smale condition. In this new theorem, the Palais--Smale condition is replaced by a weaker assumption, and a sequence of critical points converging weakly to zero with nonpositive energy is obtained. As applications, we obtain infinitely many solutions for a quasi-linear elliptic equation which is very degenerate and lacks strict convexity, and we also prove the existence of infinitely many homoclinic orbits for a second-order Hamiltonian system for which the functional is not in $C^1$ and does not satisfy the Palais--Smale condition. These solutions cannot be obtained via existing abstract theory.

Existence of periodic solutions for nonautonomous second-order discrete Hamiltonian systems

In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.

Connecting orbits for a class of singular time-periodic second-order Hamiltonian systems

We show the existence of connecting orbits for a class of singular second-order Hamiltonian systemswhere, as opposed to most of the existing literature, we assume that the potential V has not one but any finite number of maxima of equal value. We use variational methods under the assumption that V (t, u) satisfies the so-called ‘strong-force’ condition at the singularity.

Infinitely many periodic solutions for a class of new superquadratic second-order Hamiltonian systems

In this paper, we establish the existence of infinitely many periodic solutions for a class of new superquadratic second-order Hamiltonian systems. Our technique is based on the Fountain Theorem due to Bartsch.

Infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems

In this paper, we mainly consider the existence of infinitely many homoclinic solutions for a class of subquadratic second-order Hamiltonian systems u¨−L(t)u+Wu(t,u)=0, where L(t) is not necessarily positive definite and the growth rate of potential function W can be in (1, 3/2). Using the variant fountain theorem, we obtain the existence of infinitely many homoclinic solutions for the second-order Hamiltonian systems.

Multiple Homoclinics for Nonperiodic Damped Systems with Superlinear Terms

We study the following second-order differential system $$\begin{aligned} -\ddot{u}(t)-M\dot{u}(t)+L(t)u(t)=H_u(t,u(t)),\quad t\in \mathbb {R}, \end{aligned}$$ (0.1) which can be regarded as a second-order Hamiltonian system with a damped term. Here, the nonlinearity H(t, u) is superquadratic as \(|u|\rightarrow \infty \). We do not need any periodic conditions, and we obtain infinitely many nontrivial homoclinic orbits of this system by variational methods. Our result improves and extends the corresponding results existed.

Infinitely many periodic solutions for a class of second-order Hamiltonian systems

In this paper we study the existence of infinitely many periodic solutions for second-order Hamiltonian systems $\left\{ {\begin{array}{*{20}c} {\ddot u(t) + A(t)u(t) + \nabla F(t,u(t)) = 0,} \\ {u(0) - u(T) = \dot u(0) - \dot u(T) = 0,} \\ \end{array} } \right. $ , where F(t, u) is even in u, and ∇F(t, u) is of sublinear growth at infinity and satisfies the Ahmad-Lazer-Paul condition.

Infinitely Many Homoclinic Solutions for a Class of Indefinite Perturbed Second-Order Hamiltonian Systems

In this paper, we study the existence of infinitely many homoclinic solutions of the perturbed second-order Hamiltonian system $$-\ddot{u}(t)+L(t)u=W_u(t,u(t))+G_u(t,u(t)),$$ where \({L(t)}\) and \({W(t,u)}\) are neither autonomous nor periodic in \({t}\). Under the assumptions that \({W(t,u)}\) is indefinite in sign and only locally superquadratic as \({|u|\to +\infty}\) and \({G(t,u)}\) is not even in \({u}\), we prove the existence of infinitely many homoclinic solutions in spite of the lack of the symmetry of this problem by Bolle’s perturbation method in critical point theory. Our results generalize some known results and are even new in the symmetric case.