Linear Hamiltonian Systems Research Articles

A note on eigenvalues of a class of singular continuous and discrete linear Hamiltonian systems

In this paper, we show that the analytic and geometric multiplicities of an eigenvalue of a class of singular linear Hamiltonian systems are equal, where both endpoints are in the limit circle cases. The proof is fundamental and is given for both continuous and discrete Hamiltonian systems. The method used in this paper also works for both endpoints are regular, or one endpoint is regular and the other is in the limit circle case.

The Maslov index and the spectral flow\u2014revisited

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of self-adjoint first-order operators.We particularly pay attention to the continuity of the latter path of operators, where we consider the gap-metric on the set of all closed operators on a Hilbert space. Finally, we obtain from Cappell, Lee and Miller’s theorem a spectral flow formula for linear Hamiltonian systems which generalises a recent result of Hu and Portaluri.

A necessary and sufficient condition for the stability of linear Hamiltonian systems with periodic coefficients

Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with time-dependent coefficients can be parameterized by an envelope matrix w(t), which has a clear physical meaning and satisfies a nonlinear envelope matrix equation. It is proved that a linear Hamiltonian system with periodic coefficients is stable if and only if the envelope matrix equation admits a solution with periodic w†w and a suitable initial condition. The mathematical devices utilized in this theoretical development with significant physical implications are time-dependent canonical transformations, normal forms for stable symplectic matrices, and horizontal polar decomposition of symplectic matrices. These tools systematically decompose the dynamics of linear Hamiltonian systems with time-dependent coefficients and are expected to be effective in other studies as well, such as those on quantum algorithms for classical Hamiltonian systems.

Krylov projection methods for linear Hamiltonian systems

We study geometric properties of Krylov projection methods for large and sparse linear Hamiltonian systems. We consider in particular energy preservation. We discuss the connection to structure preserving model reduction. We illustrate the performance of the methods by applying them to Hamiltonian PDEs.

Multiple solutions for asymptotically linear 2p-order Hamiltonian systems with impulsive effects

Multiple solutions for asymptotically linear 2p-order Hamiltonian systems with impulsive effects

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems

In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the origin and at infinity, has at least $|i_\infty-i_0|$ periodic solutions, and an additional one if $i_0$ is even. Our argument combines the Poincar\'e--Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.

Riccati equations for linear Hamiltonian systems without controllability condition

In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.

Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems

In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.

On the Reducibility of a Class of Linear Almost Periodic Hamiltonian Systems

In this paper, we study the reducibility problem for a class of analytic almost periodic linear Hamiltonian systems $$\begin{aligned} \frac{dx}{dt} = J[{A}+\varepsilon {Q}(t)]x \end{aligned}$$ where A is a symmetric matrix, J is an anti-symmetric symplectic matrix, Q(t) is an analytic almost periodic symmetric matrix with respect to t, and $$\varepsilon $$ is a sufficiently small parameter. It is also assumed that JA has possible multiple eigenvalues and the basic frequencies of Q satisfy the non-resonance conditions. It is shown that, under some non-resonant conditions, some non-degeneracy conditions and for most sufficiently small $$\varepsilon $$ , the Hamiltonian system can be reduced to a constant coefficients Hamiltonian system by means of an almost periodic symplectic change of variables with the same basic frequencies as Q(t).

On First Integrals of Linear Hamiltonian Systems

In the paper, Lax pairs for linear Hamiltonian systems of differential equations are found. Besides, first integrals of the system which are obtained from the Lax pairs are investigated.

Solution of Eigenvalue Problems for Linear Hamiltonian Systems with a Nonlinear Dependence on the Spectral Parameter

A method for solving self-adjoint eigenproblems for linear Hamiltonian systems with equation, coefficient, and boundary conditions nonlinearly dependent on the spectral parameter is presented. The suggested approach is based on the iterative Newton procedure with spectral correction. The fast convergence of the method is demonstrated, and two-sided estimates of the eigenvalue sought are obtained. The results of the test application of the outlined algorithm are presented for the problem of the transverse natural oscillations of nonhomogeneous rods with a density defect, using the Euler–Bernoulli, Rayleigh, and Timoshenko models.

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.

On First Integrals of Linear Hamiltonian Systems

On First Integrals of Linear Hamiltonian Systems

Existence of Solutions for Second-Order Hamiltonian Systems with Resonance

In 2014, J. Pipan and M. Schechter discussed periodic solutions for second-order Hamiltonian systems with resonance. In this paper, we will generalize their results. To this end, we will first establish an index theory for second-order linear Hamiltonian systems with coefficient matrix in $$L^1$$. Then we propose generalized assumptions. We also investigate multiple solutions for symmetric second-order Hamiltonian systems.

Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and the comparative index

We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval.

On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian system x˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.

Reducibility for wave equations of finitely smooth potential with periodic boundary conditions

In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation utt−uxx+Mu+ε(V0(ωt)uxx+V(ωt,x)u)=0,x∈R/2πZ can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in (t,x), quasi-periodic in time t with Diophantine frequency ω∈Rn, and V0 is finitely smooth and quasi-periodic in time t with Diophantine frequency ω∈Rn. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.

Strongly Exponentially Separated Linear Systems

In the study of linear differential systems, an important concept is that of exponential separation. It is closely related to the concept of exponential dichotomy and has played a key role in the theory of Lyapunov exponents. Usually in the study of exponential separation, it is assumed that the coefficient matrix A(t) is bounded in norm. Our first aim here is to develop a theory of exponential separation which applies to unbounded systems. It turns that in order to have a reasonable theory it is necessary to add the assumption that the angle between the two separated subspaces is bounded below (note this follows automatically for bounded systems). Our second aim is to show that if a bounded linear Hamiltonian system is exponentially separated into two subspaces of the same dimension, then it must have an exponential dichotomy.

Proof of Simultaneous Conditional Exponential Stabilization and Destabilization of Linear Hamiltonian Systems

It is proved that any linear Hamiltonian system is simultaneously conditionally (with respect to a subspace of half dimension) exponentially stabilizable and destabilizable by uniformly small Hamiltonian perturbations.

Properties of canonical transformations of linear Hamiltonian systems

In this paper we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems have an important role in fluid mechanics and in statistical mechanics. Firstly, we prove some properties of canonical transformations of linear Hamiltonian systems. Further, we get the new method to find the generating function of a canonical transformation. We get a system of matrix equations for finding the generating function of a canonical transformation. In various cases we obtain the solution of the system. Then we use the system for normalization the Hamiltonian matrix. We apply the system of matrix equations to transform the Hamiltonian matrix from one normal form to another. Further, we get the solution of the matrix Riccati equation. This nonlinear equation has an important role in optimal control problems, multivariable and large-scale systems, scattering theory, estimation, detection and transportation. Finally, we transform the Hamiltonian from the complex form to the real form. An illustrative example for the proposed method is given. Therefore, we obtain the new method of normalization of the quadratic Hamiltonian. With this method we can investigate the stability of the solution of the Hamiltonian systems.