Persistence of noncompact normally hyperbolic invariant manifolds in bounded geometry

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in Riemannian manifolds of bounded geometry. The bounded geometry of the ambient manifold is a crucial assumption in order to control the uniformity of all estimates throughout the proof.     

Approximately invariant manifolds and global dynamics of spike states

We investigate the existence of a true invariant manifold given an approximately invariant manifold for an infinite-dimensional dynamical system. We prove that if the given manifold is approximately invariant and approximately normally hyperbolic, then the dynamical system has a true invariant manifold nearby. We apply this result to reveal the global dynamics of boundary spike states for the generalized Allen–Cahn equation.     

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.     

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.

     

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.     

Domain perturbation and invariant manifolds

We consider a reaction-diffusion equation defined on a sequence of bounded open sets ${(\Omega_n)_n \in \mathbb{N}}$ , converging to ${\Omega}$ in the sense of Mosco, and we prove stability of invariant manifolds of the flux with respect to domain perturbation.     

Differential geometry on almost tangent manifolds

Summary We start from a tensor field Q of type (1, 1) defined in a2n-dimensional manifold M which satisfies Q ²⁼⁰ and has rank n. The tensor field Q defines an almost tangent structure in M. We then introduce another tensor field P of the same type and having properties similar to those of Q. We then define and study the tensors H=PQ, V=QP, J=P−Q, K=P+Q, L=PQ−QP, (J, K, L) defining an almost quaternion structure of the second kind on M. We study the differential geometry on almost tangent manifolds in terms of these tensors. To ProfessorBeniamino Segre on his seventieth birthday Entrata in Redazione il 7 giugno 1973.     

Orbits heteroclinic to invariant manifolds

Using the theory of invariant manifolds, we give local expressions of the stable and unstable manifolds for normally hyperbolic invariant tori, and study the existence of transverse orbits heteroclinic to hyperbolic invariant tori. These extend and improve the corresponding results obtained in [3–5].Supported by the National Natural Science Foundation of China and Shanghai Natural Science Foundation.     

Stabilization of DAEs and invariant manifolds

Summary. Many methods have been proposed for the stabilization of higher index differential-algebraic equations (DAEs). Such methods often involve constraint differentiation and problem stabilization, thus obtaining a stabilized index reduction. A popular method is Baumgarte stabilization, but the choice of parameters to make it robust is unclear in practice. Here we explain why the Baumgarte method may run into trouble. We then show how to improve it. We further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature. Our approach is to (i) consider stabilization of ODEs with invariants, (ii) discretize the stabilizing term in a simple way, generally different from the ODE discretization, and (iii) use orthogonal projections whenever possible. The best methods thus obtained are related to methods of coordinate projection. We discuss them and make concrete algorithmic suggestions. Received September 1992/Revised version received May 13, 1993     

An invariant of multiply connected manifolds

The conditions under which a multiply connected open manifold has the homotopic type of a finite complex are studied. Examples are analyzed.Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 155–164, August, 1968.     

Complex geometry of moment-angle manifolds

Moment-angle manifolds provide a wide class of examples of non-Kähler compact complex manifolds. A complex moment-angle manifold \(\mathcal {Z}\) is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold \(\mathcal {Z}\) is the total space of a holomorphic bundle over a toric variety with fibres compact complex tori. In general, a complex moment-angle manifold \(\mathcal {Z}\) is equipped with a canonical holomorphic foliation \({\mathcal {F}}\) which is equivariant with respect to the \(({\mathbb {C}}^\times )^m\)-action. Examples of moment-angle manifolds include Hopf manifolds of Vaisman type, Calabi–Eckmann manifolds, and their deformations. We construct transversely Kähler metrics on moment-angle manifolds, under some restriction on the combinatorial data. We prove that any Kähler submanifold (or, more generally, a Fujiki class \(\mathcal {C}\) subvariety) in such a moment-angle manifold is contained in a leaf of the foliation \({\mathcal {F}}\). For a generic moment-angle manifold \(\mathcal {Z}\) in its combinatorial class, we prove that all subvarieties are moment-angle manifolds of smaller dimension and there are only finitely many of them. This implies, in particular, that the algebraic dimension of \(\mathcal {Z}\) is zero.     

Differential geometry of f-structure manifolds

We give the necessary concepts for an investigation of f-structure manifolds. We mention the basic geometric facts obtained in the investigation of framed manifolds. We give a proof of a theorem on a symmetric affine connection.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 15, pp. 95–125, 1983.     

Supersymmetry on noncompact manifolds and complex geometry

Special properties of realizations of supersymmetry on noncompact manifolds are discussed. On the basis of the supersymmetric scattering theory and the supersymmetric trace formulas, the absolute or relative Euler characteristic of a barrier inR ^N can be obtained from the scattering data for the Laplace operator on forms with absolute or relative boundary conditions. An analog of the Chern-Gauss-Bonnet theorem for noncompact manifolds is also obtained. The map from the stationary curve of an antiholomorphic involution on a compact Riemann surface to the real circle on the Riemann sphere, generated by a real meromorphic function is considered. An analytic expression for its topological index is obtained by using supersymmetric quantum mechanics with meromorphic superpotential on the Klein surface. Bibliography: 27 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 77–99. Translated by B. M. Bekker.     

Numerical Taylor expansions for invariant manifolds

Summary. We consider numerical computation of Taylor expansions of invariant manifolds around equilibria of maps and flows. These expansions are obtained by writing the corresponding functional equation in a number of points, setting up a nonlinear system of equations and solving this system using a simplified Newtons method. This approach will avoid symbolic or explicit numerical differentiation. The linear algebra issues of solving the resulting Sylvester equations are studied in detail.Mathematics Subject Classification (1991): 65Q05, 65P, 37M, 65P30, 65F20, 15A69Dedicated to Gerhard Wanner on the occasion of his 60th birthdayAcknowledgments. The authors like to thank Olavi Nevanlinna for discussions and his suggestion to use complex evaluation points.     

An existence theorem for invariant manifolds

For systems of the form =Ax +F ₁ (x, y, z), =By +F ₂ (x, y,z), =Cz +F ₃ (x, y, z) possessingP = {(0,0,z)} as invariant manifold we present sufficient conditions for the extension ofP to an invariant manifold of the form (x, s (x, z), z). Hereby we assume that the spectrum _A ofA is located to the left and the spectrum _b ofB to the right of a vertical straight linel in . In the case where the spectrum _C ofC lies to the left ofl too such an extension ofP is rather simple. We consider the situation where _{A C} cannot be separated from _B by a vertical line in .

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Zusammenfassung Für Systeme der Form =Ax +F ₁ (x, y, z), =By +F ₂ (x, y,z), =Cz +F ₃ (x, y, z) mitP = {(0,0,z)} als invarianter Mannigfaltigkeit geben wir Bedingungen an, unter welchen sichP zu einer invarianten Mannigfaltigkeit der Form (x, s (x, z), z) fortsetzen läßt. Wir gehen stets davon aus, daß das Spektrum _A vonA links und das Spektrum _B vonB rechts einer vertikalen Geradenl in liegen. Eine derartige Fortsetzung vonP ist einfach, falls das Spektrum _C vonC ebenfalls links vonl liegt. Wir untersuchen den Fall, in welchem _{A C} sich nicht durch eine vertikale Gerade von _B trennen läßt.

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Dedicated to H. W. Knobloch on the occasion of his sixtieth birthday

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Supported by the Volkswagenwerk foundation