Asymptotics beyond All Orders for a Certain Type of Nonlinear Oscillators

This paper shows that a special class of smooth nonlinear oscillators, called bisuperlinear, has a family of adiabatically symmetric solutions. This was motivated by a problem studied in sloshing water waves. A potential application of the work is to compute the nontrivial leading order term of the adiabatic invariants for a certain type of nonlinear nearly periodic Hamiltonian systems.     

The <Emphasis Type="Italic">SU</Emphasis>(3) symmetry and macroscopic dynamics of magnets with spin <Emphasis Type="Italic">s</Emphasis> = 1

In the Hamiltonian approach, we derive nonlinear dynamic equations for magnetic media with spin s = 1. We introduce two types of magnetic exchange Hamiltonians corresponding to the Casimir invariants of the SU(3) group. We find the spectra of spin and quadrupole waves corresponding to the states with different symmetries under the time reversal transformation. We consider the effect of dissipative processes and find relaxation fluxes caused by the exchange symmetry of the magnetic Hamiltonian.     

Iterative calculation of strongly nonlinear self-vibrating viscoelastic mechanical systems

We consider self-excited vibrations of strongly nonlinear mechanical systems obeying the hereditary theory of viscoelasticity. using the Bubnov-Galerkin method, the problem is reduced to a system of ordinary nonlinear integrodifferential equations. The normal modes of vibration of nonlinear conservative elastic systems are chosen as the unperturbed solutions. Self-vibrating solutions are found by iteration to any degree of accuracy. The process converges for certain restrictions on the unperturbed functions and on the small parameter of the problem.Translated from Dinamicheskie Sistemy, No. 5, pp. 86–90, 1986.     

The Poincaré-Mel'nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. I

On the basis of the geometric ideas of Poincaré and Mel'nikov, we study sufficient criteria of the transversal splitting of heteroclinic separatrix manifolds of slowly perturbed nonlinear dynamical systems with a small parameter. An example of adiabatic invariance breakdown is considered for a system on a plane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1668–1681, December, 1993.     

Vassiliev invariants and finite-dimensional approximations of the euler equation in magnetohydrodynamics

We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen’s iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of n vortices on the plane. We study some dynamical properties of these systems.     

Torsion-free path geometries and integrable second order ODE systems

A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segré varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segré varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segré varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces.     

Shadowing Homoclinic Chains to a Symplectic Critical Manifold

We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system.The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of orderε| ln ε|. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.     

The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Received: 27 April 1998 / in final form: 8 August 1999     

Estimation for the region of attraction of nonlinear difference equations

Summary. We consider nonlinear systems of difference equations with and , where A is any matrix, is a continuous vector-function, and is a numeral parameter. The spectrum of A belongs to the unit circle . We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set M such that for solutions of the system with parameter the limit is true. Received April 7, 2000 / Revised version received January 8, 2001 / Published online November 15, 2001     

Applied aspects of the theory of adiabatic invariants

We consider the applicability of the theory of adiabatic invariants to determining the reaction of a linear oscillator that simulates the inverse problem of measurement technology taking account of the dynamics of a sensing element. An indication is given of the applicability of the ideas and methods developed in the papers of Ya. S. Podstrigach and his school to the solution of this circle of problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 163–167.     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.     

ADIABATIC INVARIANTS OF SLOWLY VARYING THREE－DIMENSIONAL SYSTEMS AND EXISTENCE OF INVARIANT TORI OF LOTKA－VOLTERRA EQUATION

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Optimization of a Class of Nonlinear Dynamic Systems: New Efficient Method without Lagrange Multipliers

This paper deals with optimization of a class of nonlinear dynamic systems with n states and m control inputs commanded to move between two fixed states in a prescribed time. Using conventional procedures with Lagrange multipliers, it is well known that the optimal trajectory is the solution of a two-point boundary-value problem. In this paper, a new procedure for dynamic optimization is presented which relies on tools of feedback linearization to transform nonlinear dynamic systems into linear systems. In this new form, the states and controls can be written as higher derivatives of a subset of the states. Using this new form, it is possible to change constrained dynamic optimization problems into unconstrained problems. The necessary conditions for optimality are then solved efficiently using weighted residual methods.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Canonical forms and invariants of some classes of control systems

We study the canonical forms and invariants of linear and bilinear control systems and the properties of orbits of these classes of systems under similarity transformations, that is, the action of the group G = GL _n (a change of coordinates in the state space) on the spaces of systems.     

Design of observers for nonlinear systems with H ∞ performance analysis

This paper investigates the problem of observer design for nonlinear systems. By using differential mean value theorem, which allows transforming a nonlinear error dynamics into a linear parameter varying system, and based on Lyapunov stability theory, an approach of observer design for a class of nonlinear systems with time‐delay is proposed. The sufficient conditions, which guarantee the estimation error to asymptotically converge to zero, are given. Furthermore, an adaptive observer design for a class of nonlinear system with unknown parameter is considered. A method of H _∞ adaptive observer design is presented for this class of nonlinear systems; the sufficient conditions that guarantee the convergence of estimation error and the computing method for observer gain matrix are given. Finally, an example is given to show the effectiveness of our proposed approaches. Copyright © 2013 John Wiley & Sons, Ltd.     

Local Symmetry of Harmonic Spaces as Determined by The Spectra of Small Geodesic Spheres

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm |?R|{{|\nabla{R}|}} of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants |R|² and |Ric|² of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.