Nonautonomous Hamiltonian Systems Research Articles

On the Constructive Algorithm for Stability Investigation of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in First-Order Resonance Case

We consider a non-autonomous Hamiltonian system with two degrees of freedom, whose Hamiltonian function is a 2π-periodic function of time and is analytic in the neighborhood of an equilibrium point. It is assumed that the system exhibits a first-order resonance, i.e., the linearized system in the neighborhood of the equilibrium point has a unit multiplier of multiplicity two. The case of the general position is considered when the monodromy matrix is not reduced to the diagonal form, and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system. In this paper, a constructive algorithm for the rigorous-stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed by Markeev. The sufficient conditions for the instability of the equilibrium position, as well as the conditions for its formal stability and stability in the third approximation, are expressed in terms of the coefficients of the normalized map. Explicit formulas are obtained that allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of the symplectic map. The developed algorithm is used to solve the problem of the stability of the resonant rotation of a symmetric satellite.

Existence of periodic solutions for second-order Hamiltonian systems with asymptotically linear conditions

We consider a class of asymptotically linear nonautonomous second-order Hamiltonian systems. Using the Saddle Point Theorem, we obtain the existence result, which extends some previously known results.

Solutions with minimal period for non-autonomous second-order Hamiltonian systems

In this paper, we study the existence of subharmonic solutions with prescribed minimal period for a class of second-order Hamiltonian systems with even potentials. By using the variational methods, we obtain some new existence theorems which improve some recent results in the literature.

Explicit symplectic geometric algorithms for quaternion kinematical differential equation

Solving quaternion kinematical differential equations U+0028 QKDE U+0029 is one of the most significant problems in the automation, navigation, aerospace and aeronautics literatures. Most existing approaches for this problem neither preserve the norm of quaternions nor avoid errors accumulated in the sense of long term time. We present explicit symplectic geometric algorithms to deal with the quaternion kinematical differential equation by modelling its time-invariant and time-varying versions with Hamiltonian systems and adopting a three-step strategy. Firstly, a generalized Euler U+02BC s formula and Cayley-Euler formula are proved and used to construct symplectic single-step transition operators via the centered implicit Euler scheme for autonomous Hamiltonian system. Secondly, the symplecticity, orthogonality and invertibility of the symplectic transition operators are proved rigorously. Finally, the explicit symplectic geometric algorithm for the time-varying quaternion kinematical differential equation, i.e., a non-autonomous and non-linear Hamiltonian system essentially, is designed with the theorems proved. Our novel algorithms have simple structures, linear time complexity and constant space complexity of computation. The correctness and efficiencies of the proposed algorithms are verified and validated via numerical simulations.

On the Stability of a Periodic Hamiltonian System with One Degree of Freedom in a Transcendental Case

The stability of an equilibrium of a nonautonomous Hamiltonian system with one degree of freedom whose Hamiltonian function depends 2π-periodically on time and is analytic near the equilibrium is considered. The multipliers of the system linearized around the equilibrium are assumed to be multiple and equal to 1 or–1. Sufficient conditions are found under which a transcendental case occurs, i.e., stability cannot be determined by analyzing the finite-power terms in the series expansion of the Hamiltonian about the equilibrium. The equilibrium is proved to be unstable in the transcendental case.

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.

Non-Atkinson Perturbations of Nonautonomous Linear Hamiltonian Systems: Exponential Dichotomy and Nonoscillation

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed family satisfies the properties of exponential dichotomy and global existence of $M^+$, then these two properties persist in a neighborhood of 0 which agrees either with the whole real line or with an open negative half-line; and in this last case, the ED fails at the right end value. The properties of ED and of global existence of $M^+$ are fundamental to guarantee the solvability of classical minimization problems given by linear-quadratic control processes.

New existence results on periodic solutions of non-autonomous second order Hamiltonian systems

In this paper, we are concerned with the existence of periodic solutions for the following non-autonomous second order Hamiltonian systems u ̈ ( t ) + ∇ F ( t , u ( t ) ) = 0 , a.e. t ∈ [ 0 , T ] , u ( 0 ) − u ( T ) = u ̇ ( 0 ) − u ̇ ( T ) = 0 , where F : R × R N → R is T -periodic ( T > 0 ) in its first variable for all x ∈ R N . When potential function F ( t , x ) is either locally in t asymptotically quadratic or locally in t superquadratic, we show that the above mentioned problem possesses at least one T -periodic solutions via the minimax methods in critical point theory, specially, a new saddle point theorem which is introduced in Schechter (1998).

On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-order Resonance Case

This paper is concerned with a nonautonomous Hamiltonian system with two degrees of freedom whose Hamiltonian is a 2π-periodic function of time and analytic in a neighborhood of an equilibrium point. It is assumed that the system exhibits a secondorder resonance, i. e., the system linearized in a neighborhood of the equilibrium point has a double multiplier equal to −1. The case of general position is considered when the monodromy matrix is not reduced to diagonal form and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system. In this paper, a constructive algorithm for a rigorous stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed in [1]. The main idea of this method is to construct and normalize a symplectic map generated by the phase flow of a Hamiltonian system. It is shown that the normal form of the Hamiltonian function and the generating function of the corresponding symplectic map contain no third-degree terms. Explicit formulae are obtained which allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of a symplectic map. The developed algorithm is applied to solve the problem of stability of resonant rotations of a symmetric satellite.

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.

Subharmonic Pl-solutions of first order Hamiltonian systems

In this paper, for any symplectic matrix P, the existence of subharmonic Pl-solutions of the first order non-autonomous superquadratic Hamiltonian systems is considered. Under the convex condition, the existence of infinitely many geometrically distinct Pl-solutions is proved.

О периодических движениях неавтономной гамильтоновой системы в одном случае кратного параметрического резонанса

О периодических движениях неавтономной гамильтоновой системы в одном случае кратного параметрического резонанса

Multiple periodic solutions of Hamiltonian systems confined in a box

We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.

Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth

Using a homologically link theorem in variational theory and iteration inequalities of Maslov-type index, we show the existence of a sequence of subharmonic solutions of non-autonomous Hamiltonian systems with the Hamiltonian functions satisfying some anisotropic growth conditions, i.e., the Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Moreover, we also consider the minimal period problem of some autonomous Hamiltonian systems with anisotropic growth.

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.

Existence and multiplicity of periodic solutions for nonautonomous second order Hamiltonian systems

Using the least action principle, the existence of periodic solutions for some nonautonomous second order Hamiltonian systems is obtained. Using minimax methods, the multiplicity of periodic solutions is obtained. Our results extend some previous results.

Existence of periodic solutions for a class of second order discrete Hamiltonian systems

By using the variational minimizing method and the saddle point theorem, the periodic solutions for non-autonomous second-order discrete Hamiltonian systems are considered. The results obtained in this paper complete and extend previous results.

Some Collision Solutions of the Rectilinear Periodically Forced Kepler Problem

Abstract In [5], Ortega has analyzed “generalized” collision solutions of the periodically forced rectilinear Kepler problem. In this note, we explain a different approach to study these solutions by embedding the non-autonomous Hamiltonian system into the zero-energy level of an autonomous Hamiltonian system and by employing the Levi-Civita regularization to regularize the double collisions. In addition to this, under a certain smoothness hypothesis of the periodic force term, we show that there exists a set of positive measure of generalized quasi-periodic solutions in the extended phase space, each of them accumulated by generalized periodic solutions of the system. The energy of these quasi-periodic solutions can have an arbitrary large absolute value.

On periodic solutions of nonautonomous second order Hamiltonian systems with ( q , p ) -Laplacian

On periodic solutions of nonautonomous second order Hamiltonian systems with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo form="postfix">)</mml:mo> </mml:mrow> </mml:math>-Laplacian

Multi-pulse Orbits and Homoclinic Trees in a Non-autonomous Resonant Hamiltonian System

In this study, we develop the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems. Our generalized energy-phase method applies to integrable, two-degree-of freedom non-autonomous resonant Hamiltonian systems. As an example, we investigate the multi-pulse orbits and homoclinic trees for a parametrically excited, simply supported rectangular thin plate of two-mode approximation. In both the Hamiltonian and dissipative case we find homoclinic trees, which describe the repeated bifurcations of multi-pulse solutions, and we present visualizations of these complicated structures.