Bifurcation of coisotropic invariant tori under locally Hamiltonian perturbations of integrable systems and nondegenerate deformation of symplectic structure

We study the bifurcation problem for a Cantor set of coisotropic invariant tori in the case where a Liouville-integrable Hamiltonian system undergoes locally Hamiltonian perturbations and, simultaneously, a deformation of the symplectic structure of the phase space. We consider a new case where the deformed symplectic structure generates a nondegenerate matrix of the Poisson brackets of action variables. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 221–232, April–June, 2006.     

Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

The Nosé–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nosé–Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé–Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.     

Bifurcation of periodic solutions and invariant tori for a four-dimensional system

In this paper, a four-dimensional system of autonomous ordinary differential equations depending on a small parameter is considered. Suppose that the unperturbed system is composed of two planar systems: one is a Hamiltonian system and another system has a focus. By using the Poincaré map and the integral manifold theory, sufficient conditions for the existence of periodic solutions and invariant tori of the four-dimensional system are obtained. An application of our results to a nonlinearly coupled Van der Pol–Duffing oscillator system is given.     

On invariant tori of weakly connected systems of differential equations

We consider a family of systems of differential equations depending on a sufficiently small parameter, whose zero value corresponds to a couple of independent systems. We use the method of Green-Samoilenko function for the construction of an invariant manifold of the perturbed system and present some examples of application. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 468–489, October–December, 2005.     

On the existence of infinite-dimensional invariant tori of nonlinear countable systems of difference-differential equations

In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for nonlinear countable systems of difference-differential equations defined on infinite-dimensional tori and containing an infinite set of constant deviations of a scalar argument.     

On the existence of invariant tori of countable systems of difference-differential equations defined on infinite-dimensional tori

In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of differential-difference equations defined on infinitedimensional tori and containing an infinite set of constant deviations of a scalar argument.     

Relationship between invariant sets of systems of differential equations and the corresponding difference equations

We study the relationship between invariant sets of systems of differential equations and the corresponding difference equations in terms of sign-constant Lyapunov functions. For systems of differential equations, we obtain a converse result concerning the existence of a positive-definite Lyapunov function whose zeros coincide with a given invariant manifold. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 280–285, April–June, 2006.     

Reducible Volterra and Levin–Nohel Retarded Equations with Infinite Delay

We consider two cases of reducible Volterra and Levin–Nohel retarded equations with infinite delay. In these cases reducibility arises from the use of a special type of memory functions with an exponential behavior. We address global questions like the existence of Liapunov functions and, consequently, of attractors for the nonlinear systems generated by these equations as well as the attractors for the reduced systems. For the reducible Volterra equations we exhibit cases of nontrivial Hamiltonian behaviour and for the reducible Levin–Nohel equation we identify Hopf and saddle connection bifurcations.     

Invariant Measures for Dissipative Systems and Generalised Banach Limits

Inspired by a theory due to Foias and coworkers (see, for example, Foias et al. Navier–Stokes equations and turbulence, Cambridge University Press, Cambridge, 2001) and recent work of Wang (Disc Cont Dyn Sys 23:521–540, 2009), we show that the generalised Banach limit can be used to construct invariant measures for continuous dynamical systems on metric spaces that have compact attracting sets, taking limits evaluated along individual trajectories. We also show that if the space is a reflexive separable Banach space, or if the dynamical system has a compact absorbing set, then rather than taking limits evaluated along individual trajectories, we can take an ensemble of initial conditions: the generalised Banach limit can be used to construct an invariant measure based on an arbitrary initial probability measure, and any invariant measure can be obtained in this way. We thus propose an alternative to the classical Krylov–Bogoliubov construction, which we show is also applicable in this situation.     

Invariant sets and comparison of dynamical systems

We propose a method for the construction and investigation of invariant sets of differential systems described by cone inequalities with the use of the operator of differentiation along the trajectories of the system. Well-known conditions for the positivity of linear and nonlinear differential systems with respect to typical classes of cones are generalized. A method for comparison and ordering is developed for a family of dynamical systems. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 163–176, April–June, 2007.     

Asymmetric Potentials and Motor Effect: A Large Deviation Approach

We provide a mathematical analysis for the appearance of concentrations (as Dirac masses) in the solutions to Fokker–Planck systems with asymmetric potentials. This problem has been proposed as a model to describe motor proteins moving along molecular filaments. The components of the system describe the densities of the different conformations of the proteins. Our results are based on the study of a Hamilton–Jacobi equation arising at the zero diffusion limit after an exponential transformation change of the phase function that yields a viscous Hamilton–Jacobi equation. We consider different classes of conformation transitions coefficients (bounded, unbounded and locally vanishing).     

Hamiltonian finite-dimensional oscillator-type reductions of Lax-integrable superconformal hierarchies

For bi-Hamiltonian superconformal hierarchies of nonlinear Benny-Kaup and Kaup-Broer dynamical systems, we develop a method for reduction to nonlocal finite-dimensional invariant subspaces of Neumann and Bargmann types, respectively. We prove that there exist even supersymplectic structures on these spaces and that the reduced commuting vector fields generated by the hierarchies are integrable in the Lax-Liouville sense. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 15–30, January–March, 2006.     

Quasi-Periodic Solutions with Prescribed Frequency in Reversible Systems

In this paper, we obtain a family of small-amplitude real analytic quasi-periodic solutions for a class of derivative nonlinear Schrödinger equations, subject to Dirichlet boundary conditions, which correspond to infinite-dimensional reversible systems with critical unbounded perturbations. We prove that the frequencies of the quasi-periodic solutions, accordingly, the tangential frequencies of the invariant tori for these reversible systems can be in a fixed direction.     

Existence of an invariant torus for one class of systems of differential equations

We consider the problem of the existence of an asymptotically stable toroidal set for a system of linear differential equations defined on an m-dimensional torus. We establish conditions under which a nonlinear system of differential equations has an invariant toroidal manifold. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 520–529, October–December, 2008.     

Strong Stability of KAM Tori: from the Point of View of Viscosity Solutions of H-J Equations

It is well-known that a KAM torus can be considered as a graph of smooth viscosity solution. Salamon and Zehnder (Comment Math Helv 64:84–132, 1989) have proved that there exist invariant tori having prescribed Diophantine frequencies for nearly integrable and positively definite Lagrangian systems with associated Hamiltonian H, whose Diophantine index is τ. If the invariant torus is represented as in the cotangent bundle , then we can show that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),

A Grobman–Hartman Theorem for Control Systems

We consider the problem of locally linearizing a control system via topological transformations. According to [2,3], there is no naive generalization of the classical Grobman–Hartman theorem for ODEs to control systems: a generic control system, when viewed as a set of under-determined differential equations parametrized by the control, cannot be linearized using pointwise transformations on the state and the control values. However, if we allow the transformations to depend on the control at a functional level (open loop transformations), we are able to prove a version of the Grobman–Hartman theorem for control systems.     

Backbone transitions and invariant tori in forced micromechanical oscillators with optical detection

Micromechanical oscillators often display rich dynamics due to nonlinearities in their response, actuation, and detection. This paper investigates the complicated response of a forced micromechanical oscillator. In particular, we investigate a thermally induced transition in the resonant response of a forced micromechanical oscillator with optical detection; and the branches of invariant tori formed at subsequent bifurcations that occur with increasing laser power. We use perturbation theory and continuation algorithms to investigate and compute these branches of invariant tori. The results of both methods are compared.     

Invariant Sets of Systems of Stochastic Differential Equations with Jumps

We introduce the notion of invariant surfaces for inhomogeneous stochastic differential equations with jumps. The results obtained enable one to determine invariant surfaces for stochastic differential equations of the type indicated. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 234–240, April–June, 2005.