On the governing equations of motion of nonholonomic systems on Riemannian manifolds

We propose a geometric approach to formulate the governing equations of motion for a class of nonholonomic systems on Riemannian manifolds. We first present a coordinate-free geometric formulation of the D’Alembert–Lagrange equation. Then by explicating this geometric formulation with respect to an arbitrary frame, we obtain the governing equations of motion in generalized form. The governing equations so obtained directly eliminate the dependent variations without using undetermined multipliers. As examples, we apply the formulation to a rigid body and a system with general first-order nonholonomic constraints; we also demonstrate their equivalences to the known results.     

The invariant manifolds of systems with first integrals

Several additional possibilities of the Routh–Lyapunov method for isolating and analysing the stationarity sets of dynamical systems admitting of smooth first integrals are discussed. A procedure is proposed for isolating these sets together with the first integrals corresponding to the vector fields for these sets. This procedure is based on solving the stationarity equations of the family of first integrals of the problem in part of the variables and parameters occurring in this family. The effectiveness of this approach is demonstrated for two problems in the dynamics of a rigid body.     

Degenerate invariant manifolds of some completely integrable systems

We show that a large class of k degrees of freedom integrable Hamiltonian systems, the so-called Jacobi-Moser-Mumford systems, are Bott systems (this means that the regular critical points of the first integrals are nondegenerate). Thereby Fomenko's theory about classification of bifurcations of Liouville tori holds for such systems. Received March 16, 1998; in final form November 9, 1998     

On parameter dependence of bounded invariant manifolds of autonomous systems of differential equations

We consider bounded invariant manifolds of autonomous systems of differential equations and study the problem of their continuity and continuous differentiability with respect to a parameter.     

On the smoothness of bounded invariant manifolds of linear inhomogeneous extensions of dynamical systems

We establish sufficient conditions for the existence of smooth bounded invariant manifolds of dynamical systems.     

On discretizations of invariant foliations over inertial manifolds

Invariant foliations over inertial manifolds of partial differential equations under numerical discretizations are studied. It is proved that the numerical method considered as a discrete dynamical system has C¹-close invariant foliations. The rate of the C¹-convergence is estimated as well.     

Specific features of families of invariant manifolds of conservative systems

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.     

Global invariant manifolds

The main result of our article is an analog of the local manifold theorem for a hyperbolic point. We give a set of sufficient conditions that ensure the existence of the global invariant manifold.

We will prove the existence of invariant curves which lie in the quaternionic Julia set of the map f_c(X)=X²+c.     

Spiralling invariant manifolds

A definition of the concept of a multidimensional spiralling manifold is studied. Manifolds of this type are shown to occur as invariant manifolds of flows containing hyperbolic restpoints with a pair of complex conjugate eigenvalues. It is shown that spiralling can be a mechanism for producing intersections of invariant manifolds.     

Hamiltonization of non-holonomic systems in the neighborhood of invariant manifolds

The problem of Hamiltonization of non-holonomic systems, both integrable and non-integrable, is considered. This question is important in the qualitative analysis of such systems and it enables one to determine possible dynamical effects. The first part of the paper is devoted to representing integrable systems in a conformally Hamiltonian form. In the second part, the existence of a conformally Hamiltonian representation in a neighborhood of a periodic solution is proved for an arbitrary (including integrable) system preserving an invariant measure. Throughout the paper, general constructions are illustrated by examples in non-holonomic mechanics.     

Numerical Taylor expansions of invariant manifolds in large dynamical systems

Summary. In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14]. Received March 12, 1996 / Revised version reveiced August 8, 1997     

Center manifolds for smooth invariant manifolds

We study dynamics of flows generated by smooth vector fields in in the vicinity of an invariant and closed smooth manifold . By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of ) based on the information of the linearization along , which contains every locally bounded solution and is persistent under small perturbations.

     

On the Conley index in the invariant manifolds

Let ? be a flow on a manifold M and assume that N⊂M is an invariant manifold. The aim of this note is to compare the Conley indices of an isolated invariant set S⊂N with respect to the flow ? and the flow ? restricted to N.     

On the computation of invariant manifolds of fixed points

We present a method for the numerical computation of invariant manifoids of hyperbolic and pseudohyperbolic fixed points of diffeomorphisms. The derivation of this algorithm is based on well-known properties of (almost) invariant foliations. Numerical results illustrate the performance of our method.Supported by NWO grant 611-307-018.Supported by NWO grant 611-306-523.     

On a spin conformal invariant on manifolds with boundary

Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by:

Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.