Class Of Linear Hamiltonian Systems Research Articles

Renormalized oscillation theory for singular linear Hamiltonian systems

Working with a general class of linear Hamiltonian systems on intervals with at least one singular endpoint which can be limit-point, limit-circle, or limit-intermediate, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index associated with appropriately chosen paths of Lagrangian subspaces of C2n. In the first part of the analysis we associate our linear Hamiltonian systems with families of well-defined self-adjoint operators, and in the latter part we employ the renormalized oscillation approach to count the number of eigenvalues these operators have on fixed intervals (λ1,λ2) whose closures do not intersect the essential spectrum of the operators. We conclude the analysis with two illustrative examples, indicating how the theory can be implemented in practice. This extends previous work by the authors for regular linear Hamiltonian systems.

The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on $${\mathbb {R}}$$

Working with a general class of linear Hamiltonian systems specified on $${\mathbb {R}}$$ , we develop a framework for relating the Maslov index to the number of eigenvalues the systems have on intervals of the form $$[\lambda _1, \lambda _2)$$ and $$(-\infty , \lambda _2)$$ . We verify that our framework can be implemented for Sturm–Liouville systems, fourth-order potential systems, and a family of systems nonlinear in the spectral parameter. The analysis is primarily motivated by applications to the analysis of spectral stability for nonlinear waves, and aspects of such analyses are emphasized.

The efficiency of perturbations in the class of linear Hamiltonian systems

It is proved that the set of all limiting values of solutions’ arbitrary indicators under uniformly small perturbations of coefficients of a linear Hamiltonian system is the same as the similar set obtained by uniformly small Hamiltonian perturbations.

Oscillation of linear Hamiltonian systems

In this paper, sufficient conditions have been obtained for the oscillation of a class of linear Hamiltonian systems.

The HELP inequality for lim-p Hamiltonian systems

In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).

Probability Distributions and Hilbert Spaces: Quantum and Classical Systems

We use the fact that some linear Hamiltonian systems can be considered as “finite level” quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonian systems.

Ergodic Properties and Rotation Number for Linear Hamiltonian Systems

This paper is concerned with the dynamical behavior of the solutions of a class of linear Hamiltonian systems, including those to which Kotani's theory applies. We first present a symplecticL2Perron transformation which takes these systems into skew-symmetric form. This allows us to study the average of the trajectories and the Fourier coefficients of the solutions. In addition, from the construction of two invariant complex Lagrange planes, the differentiability of the rotation number is analyzed.

Orthosymplectic integration of linear Hamiltonian systems

The authors describe a continuous, orthogonal and symplectic factorization procedure for integrating unstable linear Hamiltonian systems. The method relies on the development of an orthogonal, symplectic change of variables to block triangular Hamiltonian form. Integration is thus carried out within the class of linear Hamiltonian systems. Use of an appropriate timestepping strategy ensures that the symplectic pairing of eigenvalues is automatically preserved. For long-term integrations, as are needed in the calculation of Lyapunov exponents, the favorable qualitative properties of such a symplectic framework can be expected to yield improved estimates. The method is illustrated and compared with other techniques in numerical experiments on the Henon-Heiles and spatially discretized Sine-Gordon equations.

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems under small average perturbations

Sharp bounds on mobility of the Lyapunov exponents of linear Hamiltonian systems are found for arbitrarily small average linear perturbations of the coefficients, with the help of the Millionshchikov turning method. The stabilizability and destabilizability of any solution by the indicated perturbations in the class of linear Hamiltonian systems are proved.