Invariant Submanifolds Research Articles

Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces

In this work we study \(C^{k,\delta}, 2 \leq k \leq \infty, 0\leq \delta \leq 1,\) maps of a Banach space near a fixed point. We show the existence and uniqueness of a class of \(C^{k,\delta}\) local invariant sub-manifolds of the stable manifold which correspond to a spectral subspace satisfying a finite non-resonance condition of order \(L\leq k\) and an overriding condition of order \(L\leq k\) (condition (3) of Theorem 1). We study the dependence of these invariant manifolds on a parameter that lies in a Banach space. We also show that a \(C^{k,\delta}\) local weak-stable manifold that satisfies these two conditions is unique in the class of \(C^{k,\delta}\) maps. The uniqueness is due to the fact that our method does not require a cut-off function. An infinite dimensional Banach space does not always admit smooth cut-off functions.

Submanifolds of F‐structure manifold satisfying FK + (−)K+1F = 0

The purpose of this paper is to study invariant submanifolds of an n‐dimensional manifold M endowed with an F‐structure satisfying FK + (−)K+1F = 0 and FW + (−)W+1F ≠ 0 for 1 < W < K, where K is a fixed positive integer greater than 2. The case when K is odd (≥3) has been considered in this paper. We show that an invariant submanifold , embedded in an F‐structure manifold M in such a way that the complementary distribution Dm is never tangential to the invariant submanifold , is an almost complex manifold with the induced ‐structure. Some theorems regarding the integrability conditions of induced ‐structure are proved.

Perturbations of Lower Dimensional Tori in the Resonant Zone for Reversible Systems

Consider a nearly integrable system which is reversible with respect to an involution of (n,n) type. For a given l-resonant surface, 1<l<n, we show that there is a subset of it with positive measure, such that in each resonant torus corresponding to a point of this set, at least one dimensional invariant submanifold can survive any small perturbations.

Localization for Invariant Submanifolds

Localization for Invariant Submanifolds

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrodinger equation, and a hydrodynamic model.

Euler-Calogero-Moser model fromSU(2)Yang-Mills theory

The relation between SU(2) Yang-Mills mechanics, originated from the 4-dimensional SU(2) Yang-Mills theory under the supposition of spatial homogeneity of the gauge fields, and the Euler-Calogero-Moser model is discussed in the framework of Hamiltonian reduction. Two kinds of reductions of the degrees of freedom are considered: due to the gauge invariance and due to the discrete symmetry. In the former case, it is shown that after elimination of the gauge degrees of freedom from the SU(2) Yang-Mills mechanics the resulting unconstrained system represents the ID_3 Euler-Calogero-Moser model with an external fourth-order potential. Whereas in the latter, the IA_6 Euler-Calogero-Moser model embedded in an external potential is derived whose projection onto the invariant submanifold through the discrete symmetry coincides again with the SU(2) Yang-Mills mechanics. Based on this connection, the equations of motion of the SU(2) Yang-Mills mechanics in the limit of the zero coupling constant are presented in the Lax form.

Invariant manifolds for weak solutions to stochastic equations

Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C2 submanifold, is a strong solution. These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased interest in connection with a model for the stochastic evolution of forward rate curves.

Three counterexamples in the theory of inertial manifolds

An example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no finite-dimensionalC1 invariant submanifold of the phase space. The class of scalar reaction-diffusion equations in bounded domains Ω ⊂ ℝm without inertial manifolds\(\mathcal{M} \subset L^{\text{2}} (\Omega )\) with the property of absolute normal hyperbolicity on the setE of stationary points of the phase semiflow is described. Such equations may have inertial manifolds with the weaker property of normal hyperbolicity onE. Three-dimensional reaction-diffusion systems without inertial manifolds normally hyperbolic at stationary points are found.

Invariant jets of foliations along hyperbolic sets

It is shown that under certain conditions a hyperbolic set will possess an invariant structure in addition to the invariant submanifolds. This invariant structure is in the form of a ‘jet of a foliation’ along the leaves.

BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.

Hamiltonian structures of the first variation equations and symplectic connections

Necessary and sufficient conditions in terms of symplectic connections, ensuring that the first variation equation of a Hamiltonian system along a fixed invariant symplectic submanifold is also a Hamiltonian system with respect to some admissible symplectic structure are obtained. The class of admissible symplectic structures is distinguished by means of the natural condition of compatibility with the symplectic 2-form in the ambient space. Possible obstructions to the existence of a Hamiltonian structure on the first variation equation are investigated.

SUBMANIFOLDS OF PRODUCT RIEMANNIAN MANIFOLD

This paper discusses submanifolds of product Riemannian manifold, and proves that an invariant submanifold of product Riemannian manifold can be written as a production.

Almost invariant submanifolds for compact group actions

Abstract. We define a C1 distance between submanifolds of a riemannian manifold M and show that, if a compact submanifold N is not moved too much under the isometric action of a compact group G, there is a G-invariant submanifold C1-close to N. The proof involves a procedure of averaging nearby submanifolds of riemannian manifolds in a symmetric way. The procedure combines averaging techniques of Cartan, Grove/Karcher, and de la Harpe/Karoubi with Whitney's idea of realizing submanifolds as zeros of sections of extended normal bundles.

QR -Submanifolds of ( p - 1) QR -Dimension in a Quaternionic Projective Space QP (n+p)/4

The purpose of this paper is to study n-dimensional QR-submanifolds of (p - 1) QR-dimension isometrically immersed in a quaternionic projective space QP(n+p)/4 and to give sufficient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.

Hyperbolic manifolds with the strongly shadowing property

Let f be a C1 diffeomorphism of a compact smooth manifold M and λ ⊂ M a C1 compact invariant submanifold with a hyperbolic structure as a subset of M. We show that the diffeomorphism | λ is Anosov if and only if λ has the strongly shadowing property, and find hyperbolic sets which have the strongly shadowing property.

Optimization based nonlinear observers revisited

Abstract In this paper, we present some new theoretical results concerning Optimization Based Observers. Despite the title we use, our observer is not based on the search for a global minimum of a cost function. It is based on a descent-like approach without the use of the Hessian matrix. A nice feature is that the resulting observer takes a classical form in the sense that it is defined by a set of ordinary differential equations. Beside the theoretical results, some implementation issues are discussed. In particular, we propose to use the post stabilization approach, well known when dealing with differential systems with invariant submanifolds. This enables to increase the sampling period while keeping a very good precision. An example is presented to illustrate the efficiency of the proposed method.

Optimization based non-linear observers revisited

Optimization based non-linear observers are those where the integrated output prediction error is used to define the dynamic of the observer's state. Their main advantage is genericity and the natural assumptions that are needed for the related convergence results to hold. Here, we present some new theoretical results concerning these observers. Despite the title we use, our observer is not based on the search for a global minimum of a cost function. It is based on a descent-like approach without the use of the Hessian matrix. A nice feature is that the resulting observer takes a classical form of in the sense that it is defined by a set of ordinary differential equations. Besides the theoretical results, some implementation issues are discussed. In particular, we propose to use the post stabilization approach, well known when dealing with differential systems with invariant submanifolds. This enables to increase the sampling period while keeping very good precision. Two illustrative examples are presented to show the efficiency of the proposed method.

On Invariant Submanifolds of C- and S-Manifolds

We characterize totally geodesic invariant submanifolds of C- and S-manifolds. We prove that the second fundamental form is parallel if and only if the submanifold of an S-manifold is totally geodesic. We show that this is not true for the C-manifolds.

Invariant sub-manifolds and modes of nonlinear autonomous systems

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

Synchronization, intermittency and critical curves in a duopoly game

The phenomenon of synchronization of a two-dimensional discrete dynamical system is studied for the model of an economic duopoly game, whose time evolution is obtained by the iteration of a noninvertible map of the plane. In the case of identical players the map has a symmetry property that implies the invariance of the diagonal x 1= x 2, so that synchronized dynamics is possible. The basic question is whether an attractor of the one-dimensional restriction of the map to the diagonal is also an attractor for the two-dimensional map, and in which sense. In this paper, a particular dynamic duopoly game is considered for which the local study of the transverse stability, in a neighborhood of the invariant submanifold in which synchronized dynamics takes place, is combined with a study of the global behavior of the map. When measure theoretic, but not topological, attractors are present on the invariant diagonal, intermittency phenomena are observed. The global behavior of the noninvertible map is investigated by studying of the critical manifolds of the map, by which a two-dimensional region is defined that gives an upper bound to the amplitude of intermittent trajectories. Global bifurcations of the basins of attraction are evidenced through contacts between critical curves and basin boundaries.