Invariant tori in nonlinear oscillations

The boundedness of all the solutions for semilinear Duffing equationx″ + ω² x + φ(x) =p(t), ω ∈ ℝ⁺ℕ is proved, wherep (t) is a smooth 2π-periodic function and the perturbation ⌽(x) is bounded.     

Invariant tori for commuting Hamiltonian PDEs

We generalize to some PDEs a theorem by Eliasson and Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with r integrals of motion and n degrees of freedom, r?n. The result we get ensures the persistence of an r-parameter family of r-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunov-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2-dimensional tori, while in the second case we construct 3-dimensional tori.     

Invariant tori for asymptotically linear impact oscillators

The existence of invariant tori and quasi-periodic solutions for asymptotically linear impact oscillators is proved by using the successor map and some generalized versions of the Moser's twist theorem.     

Invariant tori of differential equations in a Banach space

We establish conditions for the existence, uniqueness, and smoothness of the toroidal manifolds of differential equations with unbounded operator coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 13–21, January, 1998.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

Invariant tori for non conservative perturbations of integrable systems

We prove some results on the persistence of invariant tori for non hamiltonian perturbations of integrable systems. We also obtain persistence for the case where the unperturbed torus is resonant. In such a case the persistence of invariant tori is ensured for small, but outside of a neighbourhood of zero. Received June 1999 – Revised October 1999     

Existence of KAM tori in the phase-space of lattice vortex systems

Under very mild conditions on the circulations, and for arbitrary vortex configurations, the existence of quasi-periodic solutions for a lattice vortex model is shown.Control over the size of the perturbation in the KAM-theory is achieved by uniform scalings of the circulations, the vortex separations, and time. Thus, additional restrictions on the circulations and the ratios of vortex separations are not required; this makes the result physically meaningful.     

Invariant manifolds of singularly perturbed ordinary differential equations

For singularly perturbed systems of ODE's satisfying a certain stability assumption the existence of an asymptotically stable (unstable) invariant manifold is proved. This invariant manifold is -close to the so-called reduced manifold of such a system. As an illustrative example a 3-dimensional autonomous system describing a model in biochemistry is considered.

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Zusammenfassung Für singulär gestörte Differentialgleichungssysteme, die einer gewissen Stabilitätsbedingung genügen, wird die Existenz einer asymptotisch stabilen (instabilen) invarianten Mannigfaltigkeit nachgewiesen, die in einer -Umgebung der sogenannten reduzierten Mannigfaltigkeit eines solchen Systems liegt. Als Anschauungsbeispiel wird ein dreidimensionales autonomes System betrachtet, welches ein biochemisches Modell beschreibt.

     

Invariant tori of linear extensions which do not have Green's functions

We study the integral representation of invariant tori of linear extensions which do not have Green's functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 4, pp. 520–524, April, 1993.     

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.     

Linear chaotic resonance in vortex motion

For three-dimensional vortex motion, a linear mathematical model with random coefficients is considered, and formulas for the first two moment functions of solutions are derived. The conditions are found under which a linear chaotic resonance occurs; i.e., the mean angular velocity of the motion increases. The results show that the energy of the vortex increases because of the chaotic motions present in the flow.     

Invariant measures and regularity properties of perturbed Ornstein-Uhlenbeck semigroups

This paper deals with perturbations of the Ornstein-Uhlenbeck operator on L²-spaces with respect to a Gaussian measure μ. We perturb the generator of the Ornstein-Uhlenbeck semigroup by a certain unbounded, non-linear drift, and show various properties of the perturbed semigroup such as compactness and positivity. Strong Feller property, existence and uniqueness of an invariant measure are discussed as well.     

Invariant manifolds for singularly perturbed linear functional differential equations

The existence of “slow” and “fast” manifolds, and of invariant manifolds approaching the manifold of orbits of the degenerate system, is discussed for singularly perturbed systems of linear retarded functional differential equations (FDE). It is shown that these manifolds exist only in very degenerate situations and, consequently, the geometry of the flow of singularly perturbed ordinary differential equations does not generalize to FDEs.     

One case of vortex motion in fluid

This text presents an English translation of the significant paper [6] on vortex dynamics published by the outstanding Russian scientist S. A. Chaplygin, which seems to have almost escaped the attention of later investigators in this field. Although it was published more than a century ago, in our opinion it is still interesting and valuable. __________ Originally published in: Trudy otdeleniya fizicheskikh nauk imperatorskogo Moskovskogo obshchestva lyubitelei estestvoznaniya, antropologii i etnografii (Transactions of the Physical Section of Moscow Society of Friends of Natural Sciences, Anthropology and Ethnography), 1903, Vol. 11, No. 2, pp. 11–14. Translated from Russian by S. Ramodanov; edited by D. Blackmore; commented by V. V. Meleshko (Department of Theoretical and Applied Mechanics, Faculty of Machanics and Mathematics, Kiev National Taras Shevchenko University, 01030 Kiev, Ukraine. E-mail: meleshko@univ.kiev.ua) and G.J.F. van Heijst (Fluid Dynamics Laboratory, Faculty of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands. E-mail: g.j.f.v.heijst@tue.nl).