Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case

The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered. We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with n degrees of freedom, in particular, this generalization includes those in [1].      

Oscillation of linear Hamiltonian systems

We establish new oscillation criteria for linear Hamiltonian systems using monotone functionals on a suitable matrix space. In doing so we develop new criteria for oscillation involving general monotone functionals instead of the usual largest eigenvalue. Our results are new even in the particular case of self-adjoint second order differential systems.

     

Construction of form factors of composite systems by a generalized Wigner-Eckart theorem for the Poincaré group

We generalize the previously developed relativistic approach for electroweak properties of two-particle composite systems to the case of nonzero spin. This approach is based on the instant form of relativistic Hamiltonian dynamics. We use a special mathematical technique to parameterize matrix elements of electroweak current operators in terms of form factors. The parameterization is a realization of the generalized Wigner-Eckart theorem for the Poincaré group, used when considering composite-system form factors as distributions corresponding to reduced matrix elements. The electroweak-current matrix element satisfies the relativistic covariance conditions and also automatically satisfies the conservation law in the case of an electromagnetic current.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 2, pp. 258–277, May, 2005.     

Limit circle invariance for two differential systems on time scales

In this paper we consider two linear differential systems on a time scale. Both systems depend linearly on a complex spectral parameter λ. We prove that if all solutions of these two systems are square integrable with respect to a given weight matrix for one value λ₀, then this property is preserved for all complex values λ. This result extends and improves the corresponding continuous time statement, which was derived by Walker (1975) for two non‐hermitian linear Hamiltonian systems, to appropriate differential systems on arbitrary time scales. The result is new even in the purely discrete case, or in the scalar time scale case, as well as when both time scale systems coincide. The latter case also generalizes a limit circle invariance criterion for symplectic systems on time scales, which was recently derived by the authors.     

Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems

We study some monotonicity and iteration inequality of the Maslov-type index i-1of linear Hamiltonian systems.As an application we prove the existence of symmetric periodic solutions with prescribed minimal period for first order nonlinear autonomous Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity.This result gives a positive answer to Rabinowitz’s minimal period conjecture in this case without strictly convex assumption.We also give a different proof of the existence of symmetric periodic solutions with prescribed minimal period for classical Hamiltonian systems which are semipositive,even,and superquadratic at zero and infinity which was proved by Fei,Kim and Wang in 2001.     

Restrictions of Quadratic Forms to Lagrangian Planes,Quadratic Matrix Equations,and Gyroscopic Stabilization

We discuss the symplectic geometry of linear Hamiltonian systems with nondegenerate Hamiltonians. These systems can be reduced to linear second-order differential equations characteristic of linear oscillation theory. This reduction is related to the problem on the signatures of restrictions of quadratic forms to Lagrangian planes. We study vortex symplectic planes invariant with respect to linear Hamiltonian systems. These planes are determined by the solutions of quadratic matrix equations of a special form. New conditions for gyroscopic stabilization are found.     

The monodromy problem and the tangential center problem

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center-focus problem asks for conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both of these questions for the case in which the Hamiltonian is hyperelliptic. As a by-product, we solve the corresponding problems for the 0-dimensional Abelian integrals defined by Gavrilov and Movasati.     

A note on the efficient implementation of Hamiltonian BVMs

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see Brugnano et al. [8] and references therein), also sketching their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.     

A note on the efficient implementation of Hamiltonian BVMs

We discuss the efficient implementation of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems (see Brugnano et al. [8] and references therein), also sketching their blended formulation. We also discuss the case of separable problems, for which the structure of the problem can be exploited to gain efficiency.     

Defining relations on the Hamiltonians of <Emphasis Type="Italic">XXX</Emphasis> and <Emphasis Type="Italic">XXZ R</Emphasis>-matrices and new integrable spin-orbital chains

We present several complete systems of integrability conditions on the density of the Hamiltonian of a spin chain matrix. The corresponding formulas for R-matrices are also given. The latter are expressed via the local Hamiltonian density in a form similar to spin one half XXX and XXZ models. The result is applied to the problem of integrability of SU(2) × SU(2)-and SU(2) × U(1)-invariant spin-orbital chains (the Kugel-Homskii-Inagaki model). Eight new integrable cases are found. One of these cases corresponds to the Temperley-Lieb algebra, three cases correspond to the algebra associated with the XXX model, one case corresponds to the algebra associated with the XXZ model, and one case corresponds to the algebra associated with the graded XXZ model. The remaining two R-matrices are also presented. Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 50–58.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.     

Invariant tori of locally Hamiltonian systems close to conditionally integrable systems

We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure of the phase space is deformed. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007.     

The stability domains of hamiltonian systems

Linear Hamiltonian systems with an arbitrary number of degrees of freedom, which depend smoothly on a vector of real parameters, are investigated. All possible singularities of the boundary of the stability domain of Hamiltonian systems of general position are determined and described for the case of two and three parameters. In the first approximation, the geometry of these singularities (the orientation in the parameter space, angles, etc.) is determined on the basis of the first derivative of the matrix of the system with respect to the parameters, as are the eigenvectors and generalized eigenvectors evaluated at the singular point. A detailed investigation is made of gyroscopic systems as a special case of Hamiltonian systems. As mechanical examples, an account is given of the problem of the stability of the oscillations of a tube through which a fluid is flowing, and of the stability of the motion of a two-body system. The tangent cones to the stability domains of these systems at singular points of the “cusp” and “dihedral angle” type, which arise on the boundaries of these domains, are found.     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

Arnold Diffusion. I: Announcement of Results

We announce a proof of the existence of Arnold diffusion for a large class of small perturbations of integrable Hamiltonian systems with positive normal torsion in the case of time-periodic systems in two degrees of freedom and in the case of autonomous systems in three degrees of freedom.     

Symplectic invariants and covariants of matrices

This paper deals with symplectic algebraic invariants and covariants of matrices. We adapt the Aronhold symbolic method to characterize and compute the generators of algebras of symplectic invariants and covariants. New symplectic identities are obtained and applied to construct minimal generator systems of a matrix of order 4 and a Hamiltonian matrix of order 2n.     

Closed characteristics on non-degenerate star-shaped hypersurfaces in R<Subscript>2n</Subscript>

In this paper, firstly we establish the relation theorem between the Maslov-type index and the index defined by C. Viterbo for star-shaped Hamiltonian systems. Then we extend the iteration formula ofC. Viterbo for non-degenerate star-shaped Hamiltonian systems to the general case. Finally we prove that there exist at least two geometrically distinct closed characteristics on any non-degenerate star-shaped compact smooth hypersurface on R²ⁿ with n > 1. Here we call a hypersurface non-degenerate, if all the closed characteristics on the given hypersurface together with all of their iterations are non-degenerate as periodic solutions of the corresponding Hamiltonian system. We also study the ellipticity of closed characteristics whenn = 2     

Linear transformation and oscillation criteria for Hamiltonian systems

Using a linear transformation similar to the Kummer transformation, some new oscillation criteria for linear Hamiltonian systems are established. These results generalize and improve the oscillation criteria due to I.S. Kumari and S. Umanaheswaram [I. Sowjaya Kumari, S. Umanaheswaram, Oscillation criteria for linear matrix Hamiltonian systems, J. Differential Equations 165 (2000) 174-198], Q. Yang et al. [Q. Yang, R. Mathsen, S. Zhu, Oscillation theorems for self-adjoint matrix Hamiltonian systems, J. Differential Equations 190 (2003) 306-329], and S. Chen and Z. Zheng [Shaozhu Chen, Zhaowen Zheng, Oscillation criteria of Yan type for linear Hamiltonian systems, Comput. Math. Appl. 46 (2003) 855-862]. These criteria also unify many of known criteria in literature and simplify the proofs.     

Partial preservation of frequencies and floquet exponents in KAM theory

Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case). Original Russian Text ? M.B. Sevryuk, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 174–202.     

Dissipative and Hamiltonian Systems with Chaotic Behavior: An Analytic Approach

Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these systems is effectively reduced to finite-dimensional dynamics which can be unboundedly complex in a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks and some reaction–diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.