Nonautonomous Hamiltonian Systems Research Articles

Research on Periodic Solutions of Non-autonomous Second-order Hamiltonian Systems

In the field of nonlinear science, Hamiltonian systems are crucial models extensively applied in engineering, physics, and biology. The existence of periodic solutions in non-autonomous second-order Hamiltonian systems, as a relatively specialized form, has been a focal and challenging research area. This paper employs variational methods to analyze the existence of periodic solutions in such systems, aiming to provide insights for related research endeavors.

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>

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>

Periodic Dynamics of a Class of Non-autonomous Contact Hamiltonian Systems

In this paper, we investigate the existence, number and stability of periodic orbits for the following contact Hamiltonian system H(p,q,s,t)=frac{p^{2}}{2m}+G(t,q,m)-mdq+cs(c>0). At the same time, unbounded conditions of each solution are also given. The contact Hamiltonian system actually represents a kind of physical phenomenon with non-conservation of energy, but the contact Hamiltonian system studied in this paper represents a one-dimensional damped oscillator system with constant variable sign damping coefficient under certain conditions. Therefore, it is of great physical significance to study the periodic dynamic properties of such system.

The Continuous-Time Singular LQR Problem and the Riddle of Nonautonomous Hamiltonian Systems: A Behavioral Solution

In this article, we deal with the continuous-time singular linear quadratic regulator (LQR) problems, which give rise to nonautonomous Hamiltonian systems. This case arises when the system’s transfer function matrix is not left-invertible. A special case of this problem can be solved using the constrained generalized continuous algebraic Riccati equation (CGCARE), when a certain condition on the input-cardinality of the Hamiltonian is satisfied. However, this condition is only a special case among many other possible cases. On the other hand, singular LQR problems with autonomous Hamiltonian systems have been well studied in the literature. In this article, we apply behavioral theoretic techniques to show that the general case of the singular LQR problem with nonautonomous Hamiltonian can be solved by a direct sum decomposition of the plant behavior, where one of the direct summands can be solved via CGCARE, while the other gives rise to an autonomous Hamiltonian system.

A time-dependent energy-momentum method

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of a class of non-autonomous Hamiltonian systems are described via foliated Lie systems, which opens a new field of application of such systems of differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden–Weinstein theorem and we provide conditions ensuring the stability of the projection of relative equilibrium points to the reduced space. As a byproduct, a geometrical extension of notions and results from Lyapunov stability theory on linear spaces to manifolds is provided. As an application, we study a class of mechanical systems, the hereafter called almost-rigid bodies, which covers rigid bodies as a particular instance.

Deforming lie algebras to frobenius integrable nonautonomous hamiltonian systems

Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems from [14].

Hamiltonian structure for a differential system from a modified Laguerre weight via the geometry of the modified third Painlevé equation

Recurrence coefficients of semi-classical orthogonal polynomials are often related to the solutions of special nonlinear second-order differential equations known as the Painlevé equations. Each Painlevé equation can be written in a standard form as a non-autonomous Hamiltonian system, so it is natural to ask whether differential systems satisfied by the recurrence coefficients also possess Hamiltonian structures. We consider recurrence coefficients for a modified Laguerre weight which satisfy a differential system known to be related to the modified third Painlevé equation and identify a Hamiltonian structure for it by constructing its space of initial conditions. We also discuss a transformation from this system to the modified third Painlevé equation which simultaneously identifies a discrete system for the recurrence coefficients with a discrete Painlevé equation.

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.

Periodic solutions and their stability for some perturbed Hamiltonian systems

We deal with non-autonomous Hamiltonian systems of one degree of freedom. For such differential systems, we compute analytically some of their periodic solutions, together with their type of stability. The tool for proving these results is the averaging theory of dynamical systems. We present some applications of these results.

A unified approach to periodic solutions for a class of non-autonomous second order Hamiltonian systems

This paper is dedicated to studying the periodic solutions of non-autonomous second order Hamiltonian systems. By virtue of control functions and the minimax methods in critical point theory, we propose a unified approach when the nonlinearity is unbounded while grows no more than |x| at infinity. We establish some new existence theorems on periodic solutions, which unify and generalize many previously known results in the literature.

Subharmonic solutions and minimal periodic solutions of first-order variant subquadratic Hamiltonian systems

Using the homological link theorem and iteration inequalities of Maslov-type index, we prove the multiplicity of subharmonic solutions for some variant subquadratic non-autonomous Hamiltonian systems. Moreover, the minimal period problem has also been considered for the variant subquadratic autonomous Hamiltonian systems.

Symplecticness conditions of some low order partitioned methods for non-autonomous Hamiltonian systems

We consider the application of partitioned Runge-Kutta (PRK) methods to non-autonomous Hamiltonian systems. Necessary and sufficient conditions for the symplecticness of PRK methods are given, more particularly for two low order PRK methods: the partitioned (explicit-implicit) Euler method and the 2-stage Lobatto IIIA-B PRK method. Both methods are often the basis of composition schemes of higher order. In particular for irreducible PRK methods we show the necessity for the nodes of the two underlying PRK methods to be equal.

Generalized linking theorem with applications to nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold

Let [Formula: see text] be a Riemannian manifold with finite volume and [Formula: see text] be a linear topological space. We consider the strongly indefinite superlinear problem [Formula: see text] where [Formula: see text] is a self-adjoint linear operator, [Formula: see text] is a real Hilbert space with the compact embedding [Formula: see text] if [Formula: see text] for some [Formula: see text], and [Formula: see text]. We obtain the existence of two solutions provided that [Formula: see text] and [Formula: see text] for a certain choice of [Formula: see text], [Formula: see text]. Moreover, we prove that, if [Formula: see text] and [Formula: see text] small enough, there exist prescribed number of nontrivial solutions. As applications, the corresponding results hold true for nonautonomous Hamiltonian systems and Dirac equations on compact spin manifold.

On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem

The Yakubovich Frequency Theorem, in its periodic version and in its general nonautonomous extension, establishes conditions which are equivalent to the global solvability of a minimization problem of infinite horizon type, given by the integral in the positive half-line of a quadratic functional subject to a control system. It also provides the unique minimizing pair “solution, control” and the value of the minimum. In this paper, we establish less restrictive conditions under which the problem is partially solvable, characterize the set of initial data for which the minimum exists, and obtain its value as well a minimizing pair. The occurrence of exponential dichotomy and the null character of the rotation number for a nonautonomous linear Hamiltonian system defined from the minimization problem are fundamental in the analysis.

Multiplicity of Periodic Solutions for Second Order Hamiltonian Systems with Asymptotically Quadratic Conditions

A class of second order non-autonomous Hamiltonian systems with asymptotically quadratic conditions is considered in this paper. Using Fountain Theorem, one multiplicity result of periodic solutions is obtained, which improves some previous results.

Periodic and Subharmonic Solutions for a Class of Sublinear First-Order Hamiltonian Systems

By the aids of variational methods, we study the existence of periodic and subharmonic solutions, and the minimality of their periods, for the nonautonomous first order Hamiltonian system $$J{\dot{u}}(t)+\nabla H(t,u(t))+h(t)=0$$ under some kind of sublinear nonlinearity conditions. Our results are new and generalize and extend some results in the literature.

Nonautonomous fractional Hamiltonian system with critical exponential growth

In this paper, we study the following nonlocal nonautonomous Hamiltonian system on whole \({\mathbb {R}}\)$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^\frac{1}{2}~ u +u=Q(x) g(v)&{}\quad \text{ in } {\mathbb {R}},\\ (-\Delta )^\frac{1}{2}~ v+v = P(x)f(u)&{}\quad \text{ in } {\mathbb {R}}, \end{array}\right. \end{aligned}$$where \((-\Delta )^\frac{1}{2}\) is the square root Laplacian operator. We assume that the nonlinearities f, g have critical growth at \(+\,\infty \) in the sense of Trudinger–Moser inequality and the nonnegative weights P(x) and Q(x) vanish at \(+\infty \). Using suitable variational method combined with the generalized linking theorem, we obtain the existence of at least one positive solution for the above system.

Climate Vibrations and Chaotic Behavior of Earth’s Inclination in the Laskar’s Model of a Moonless Earth

The model of the moonless Earth, introduced by J. Laskar, has the form of a non-autonomous Hamiltonian system of differential equations for two variables: the cosine of the angle of inclination and the longitude of the axis of rotation of the Earth. The system describes the rotational dynamics of the Earth under the influence of the sun and planets. Earth perturbations from other planets of the solar system are considered periodic and are taken into account using the first four terms of the Fourier expansion of the corresponding part of the Hamilton function with known amplitudes and frequencies. The initial inclination of the Earth is considered as a parameter of the problem. The system was numerically integrated over a time period of 18 million years for various values of the initial inclination from 0 to 180 degrees. Three chaotic gaps of the initial inclination were found. During the bifurcation study, singular points were found and special segments of the non-autonomous system were obtained. A bifurcation diagram of the system is constructed by the initial inclination parameter. Poincare cartographic maps are constructed. The system is written in variations on the initial conditions for the Laskar system, and with its help the dependences of the problem parameter of the senior Lyapunov exponent and the averaged MEGNO indicator are calculated. The results confirm the presence of three chaotic and one regular region of variation of the bifurcation parameter of the problem.

Quasi-periodic solutions for a nonlinear non-autonomous Hamiltonian system

Quasi-periodic solutions for a nonlinear non-autonomous Hamiltonian system