Showing papers on "Center manifold published in 1990"

Homoclinic bifurcations with nonhyperbolic equilbria

Abstract: A general geometric approach is given for bifurcation problems with homoclinic orbits to nonhyperbolic equilibrium points of ordinary differential equations. It consists of a special normal form called admissible variables, exponential expansion, strong $\lambda $-lemma, and Lyapunov–Schmidt reduction for the Poincare maps under Sil’nikov variables. The method is based on the Center Manifold Theory, the contraction mapping principle, and the Implicit Function Theorem.

Normal hyperbolicity of center manifolds and Saint-Venant's principle

Abstract: The concept of normal hyperbolicity of center manifolds is generalized to infinite-dimensional differential equations, in particular, to elliptic problems in cylindrical domains. It is shown that all solutions u staying close to the center manifold for t ∈ (−l,l) satisfy an estimate of the form \(\left\| {u(t) - \tilde u(t)} \right\| \leqslant Ce^{ - \alpha (l - |t|)} \) where C and α are independent of l, and ũ is a solution on the center manifold. These results are applied to Saint-Venant's principle for the static deformation of nonlinearly elastic prismatic bodies. The use of the center manifold permits the effective treatment of the general case of non-zero resultant forces and moments acting on each cross-section.

Multiple coupling in chains of oscillators

Abstract: Chains of oscillators with coupling to more neighbors than the nearest ones are considered. The equations for the phase-locked solutions of an infinite chain of such type may be considered as a one-parameter family of $(2m - 1)$st-order discrete dynamical systems, whose independent variable is position along the chain, whose dependent variable is the phase between successive oscillators, and where m is the number of neighbors connected to each side. It is shown that for each value of the parameter in some range, the $(2m - 1)$st-order system has a one-dimensional hyperbolic global center manifold. This is done by using the theory of exponential dichotomies to show that the system “shadows” a simple one-dimensional system. The exponential dichotomy is constructed by exploiting an algebraic structure imposed by the geometry of the multiple coupling.For a finite chain, the dynamical system is constrained by manifolds of boundary conditions. It is shown that for open sets of such conditions, the solution to t...

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

Abstract: We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential

A generalized sideband stability theory via center manifold projection

Abstract: A center manifold theory is used to determine all modes that contribute significantly to the leading‐order sideband stability of finite‐amplitude monochromatic waves The classical multiscale theories based on the Ginzburg–Landau equation are extended away from near‐critical conditions and are shown to have omitted an important contribution from nonlinear interactions with low wave‐number modes Stability bounds on stable monochromatic waves are reported for dispersive systems that extend the classical Eckhaus bound for nondispersive systems and the Lange–Newell and Benjamin–Feir stability conditions for monochromatic waves with critical wave numbers These new stability bounds are verified numerically by computing the evolving spectrum of a model equation

Nonlinear study of the flow in a long rectangular cavity subjected to thermocapillary effect

Abstract: This paper is devoted to the theoretical study of motions of viscous fluids driven by a constant stress acting on an upper surface of a long rectangular cavity. This problem was originally addressed to the flow behavior of molten metals in open boats driven by thermocapillarity (Bridgman technique solidification). Computations of two‐dimensional Navier–Stokes equations show two totally different end circulations. Considering the spatial disturbances of the core flow (e.g., Couette flow), and, using the local theory of bifurcations (center manifold, normal forms), the appearance of two kinds of disturbances corresponding to these end circulations is explained. Moreover, on one hand a condition for the observability of the fully developed Couette flow in terms of aspect ratio (length/height) and Reynolds–Marangoni number is given, and on the other hand the analytical expression of the space periodic flow occurring in the direction of the cold wall is given.

Reaction-diffusion systems in nonconvex domains: Invariant manifold and reduced form

Abstract: A study is made of systems of weakly coupled, semilinear, parabolic equations, namely reaction-diffusion systems, subject to the homogeneous Neumann boundary conditions in parametrized nonconvex domains inR2. It is assumed that the domain approaches a union of two disjoint domains as the parameter varies. Under some conditions the long-time behavior of bounded solutions is discussed and the existence of a finite-dimensional invariant manifold is shown, together with its attractivity. This manifold is represented by a graph of some function defined in a possibly large bounded region of the phase space, and the original system is reduced to an ODE system on it. Since an explicit form of the reduced ODE system is given, its dynamics can be studied in detail, which in turn reveals the global dynamics of the original reaction-diffusion system. One can thereby prove, among other things, the existence of asymptotically stable equilibrium solutions of the original system having large spatial inhomogeneity. The existence and stability of a spatially inhomogeneous periodic solution of large amplitude are also discussed.

Linear stabilizability of planar nonlinear systems

Abstract: This paper considers scalar-input two-dimensional nonlinear systems for which the linearization, has a simple zero uncontrollable eigenvalue. The existence of linear stabilizing feedback laws is investigated, using center manifold techniques.

Feedback stabilization of nonlinear systems via center manifold reduction

Abstract: The center manifold theorem is applied to the local feedback stabilization of nonlinear systems in critical cases. The authors address two particular critical cases, for which the system linearization at the equilibrium point of interest is assumed to possess either a simple zero eigenvalue or a complex conjugate pair of simple, pure imaginary eigenvalues. In either case, the noncritical eigenvalues are taken to be stable. The results on stabilizability and stabilization are given explicitly in terms of the nonlinear model of interest in its original form, i.e. before reduction to the center manifold. Moreover, the formulation given uncovers connections between results obtained using the center manifold reduction and those of an alternative approach. >

Computation of Invariant Manifold Bifurcations

Abstract: Numerical methods for computation of homoclinic trajectories in ordinary differential equations depending upon parameters are presented. The situation is considered when the trajectory is formed by an intersection of the stable and the unstable manifolds of a saddle equilibrium point with only one positive eigenvalue. Explicit formulae are derived for quadratic expansions of the manifolds. Methods for computation of bifurcation parameter values for which a homoclinic trajectory exists are discussed. Procedures are presented for the computation of the orientability of the manifolds at the bifurcation parameter values. Some applications are given.

Quasiperiodicity involving twin oscillations in the complex Lorenz equations describing a detuned laser

Abstract: Quasiperiodical motion in the complex Lorenz equations describing a detuned laser is shown to consist of twin oscillations: the first oscillation originates from Hopf bifurcation and the second is a parastic oscillation of the first one. Equations for the twin asymptotic oscillations are analytically derived in the center manifold, showing explicitly the parastic property of the second oscillation: its frequency is proportional to the square of the amplitude of the first one. The phase of the second oscillation shows also certainanholonomy which is very similar to the characteristics of Berry's phase. Numerical results show further that the first oscillation follows the sequence of bifurcations from simple periodic through period-doubling to chaos, as one continuously increases the control parameter, whereas the frequency of the parastic oscillation does not change qualitatively during the bifurcation process.

Bifurcation Analysis: A Combined Numerical and Analytical Approach

Abstract: This paper proposes a generally applicable approach for the bifurcation analysis of nonlinear dissipative systems, described by smooth, ordinary and autonomous differential equations. The analysis is done in two stages. The first stage is the brute force calculation of Lyapunov exponents. The second stage is the more sophisticated partially analytical investigation of selected points of interest. There-fore we use a parametrized Taylor series approximation of the Poincare map, where the analytical derivations can be performed by using computer algebra. The approximation enables the determination of the type of bifurcation, the iterative calculation of bifurcation points and stability analysis at the critical value by application of center manifold theory. The proposed approach is demonstrated by application to the Duffing oscillator in the version of Ueda.

Methods for Simplifying Dynamical Systems

Abstract: When one thinks of simplifying dynamical systems, two approaches come to mind: one, reduce the dimensionality of the system and two, eliminate the nonlinearity. Two rigorous mathematical techniques that allow substantial progress along both lines of approach are center manifold theory and the method of normal forms. These techniques are the most important, generally applicable methods available in the local theory of dynamical systems, and they will form the foundation of our development of bifurcation theory in Chapter 3.

A stable/unstable “manifold” theorem for area preserving homeomorphisms of two manifolds

Abstract: The stable/unstable manifold theorem for hyperbolic diffeomorphisms has proven to be of extreme importance in differentiable dynamics. We prove a stable/unstable "manifold" theorem for area preserving homeomorphisms of orientable two manifolds having isolated fixed points of index less than 1. The proof relies uponl the concept of ftee mnodification which was first developed by Morton Brown for homeomorphisms of two manifolds and later extended by Pelikan and Slaminka for area preserving homeomorphisms of two manifolds.

Nonlinear oscillations and voltage collapse phenomenon in electrical power system

Abstract: The authors focus on bifurcation of modes associated with both the generator and load dynamics to show the relations between two different critical states. The authors observe that a system encounters both oscillatory-type instability and collapse-type instability using a sample power system model developed by I. Dobson et al. (1988). In addition to the collapse-type instability reported by Dobson et al., oscillatory-type instability is analyzed by using Hopf bifurcation theory. The authors analyze these two types of system instability in a sample power system via bifurcation theory. Collapse-type instability is studied through the center manifold reduction technique, which reduces the entire system dynamic model to a one dimensional manifold. >

Codimension 2 Bifurcation in Binary Convection with Square Symmetry

Abstract: Codimension two bifurcations with double zero eigenvalues (Takens-Bogdanov bifurcations) and square symmetries have recently been shown to exhibit chaotic behaviour [AGK]. Unlike chaotic solutions in other Codimension 2 situations [GH] the chaotic behaviour here is found in a “large” region in parameter space — a wedge with positive angle originating at the singularity. Hence Codimension 2 singularities with square symmetry should be particularly attractive from an experimental point of view. The intuitive reason for the appearance of these “large” chaotic regions is the existence of a nonintegrable Hamiltonian with two degrees of freedom in a scaling limit of the D 4 — symmetric Takens-Bogdanov normal form. This nonintegrable Hamiltonian system creates stochastic regions in phase space which, for a certain range of unfolding parameters are not quenched to the fixed points when one takes the dissipation into account. Section 2 gives a short overview on the D 4-symmetric Takens-Bogdanov normal form and its unfolding. We analyse the phase space for typical parameters and discuss chaotic and nonchaotic solutions. Section 3 deduces the physical meaning of these solutions in real space as opposed to phase space. In Section 4 we calculate the nonlinear terms in the normal form for the specific system of two sets of orthogonal convection rolls in a mixture of He 3/He 4 and analyse the resulting normal form. Section 5 deals with the implication of this theory for some experiments by Moses and Steinberg [MS] and Le Gal et al. [LeG].

An inertial manifold in the problem of nonlinear oscillations of an infinite panel

Abstract: For the problem of nonlinear oscillations of an infinite panel in supersonic gas flow, we prove the existence of a finite-dimensional invariant manifold, that exponentially attracts trajectories of the system and contains the maximal attractor.

Manifolds over plural numbers and semitangent structures

Abstract: If base manifolds of a multistep submersion are completed with jets of corresponding orders, then a new differentiable manifold having an integrable nilpotent structure of a general form from the standpoint of which this manifold serves as a real realization of a manifold over an algebra of plural numbers arises. We introduce lifts of tensor fields and connections with projectivity property relative to the given submersion. We study invariant objects of a holomorphic geodesic transformation of a complete lift connection and isolate holomorphic geodesic-flat manifolds.

Feedback Stabilization via Center Manifold Reduction with Application to Tethered Satellites

Abstract: : Center manifold reduction has recently been introduced as a tool for design of stabilizing control laws for nonlinear systems in critical cases. In this dissertation, the center manifold approach is elaborated for general such nonlinear systems in several critical cases of interest, and the results are applied to the control of tethered satellite systems (TSS). In addition, to address stability questions for satellite deployment via TSS, we obtain new results in finite-time stability theory. The critical cases considered in the general feedback stabilization studies include the cases in which the system linearization possesses a simple zero eigenvalue (of multiplicity one or two), a pair of simple pure imaginary eigenvalues, one zero eigenvalues along with a pair of simple pure imaginary eigenvalues, and two pairs of simple pure imaginary eigenvalues. The calculations involve center manifold reduction, normal form transformations, and Liapunov function construction for critical systems. These calculations are explicit. The tethered satellite systems considered here consist of a satellite and subsatellite connected by a tether, in orbit around the Earth. The Lagrangian formulation of dynamics is used to obtain a nonlinear system of ordinary differential equations for TSS dynamics. For simplicity, a rigid, massless tether is assumed. Linear analysis reveals the presence of critical eigenvalues in the station-keeping mode of operation. This renders useful results on stabilization in critical cases to this application. The control variable assumed is tether tension feedback. Besides the design of stabilizing station-keeping controllers, stability of deployment and instability of retrieval are also shown for a constant angle deployment/retrieval scheme.

Subharmonics near an equilibrium point for hamitonian systems

Abstract: This work is devoted to the study of subharmonic solutions near an equilibrium for certain Hamiltonian systems. We impose a weak condition on the Floquet exponents of the linearized system, and a superquadratic condition on the higher order term. This last condition is reduced to the center manifold.

Transition to Turbulence via Spatiotemporal Intermittency: Modeling and Critical Properties

Abstract: When studying the transition to turbulence in closed flow systems, it has become traditional to make a distinction between them according to confinement effects. These effects can be measured by aspect-ratios, i.e. ratios of the lateral dimensions of the physical system to some internal length linked to the basic instability mechanism. Small aspect-ratio systems are strongly confined so that the spatial structure of the modes involved can be considered as frozen. As such, they experience a transition to turbulence according to scenarios understood in the framework of low dimensional dynamical systems theory. The general procedure in this field involves the reduction to center manifold dynamics by elimination of stable modes, the Poincare surface of section technique and the iteration of maps. Disorder is then interpreted rather as temporal chaos arising from the sensitivity of trajectories to initial conditions and small perturbations.

A sequence of Galerkin approximations to the center manifold for a certain class of nonlinear partial differential equational dynamical systems

Abstract: A sequence of Galerkin approximations to the center manifold at a bifurcation point is constructed for a certain important class of nonlinear p.d.e. (partial differential equation) dynamics that has a minimal property: the new modes adjoined in the kth approximation are $O( {x^k } )$ or higher, where x are coordinates on the center subspace $E^c $ that vanish at the equilibrium solution. The same holds for the unstable manifold and unstable subspace $E^u $, respectively, in the supercritical case. The general Galerkin approximation, usually constructed “ad hoc,” e.g., from good physical guesses as to the important modes to retain, is shown to have “gaps” in the sense that it leaves out some modes of low order while retaining higher order modes. The present theory is restricted by a “finite algebra” assumption, which should be relaxed in a future generalization to cover a wider class of physical bifurcation problems.

Symbolic computation and bifurcation methods

Abstract: From the numerical point of view, continuation methods constitute a widely used tool in the qualitative analysis of dynamical systems and, specifically, for the calculation of bifurcation points. However, analytical methods are frequently needed to characterize the bifurcation, being also a valuable guide for the numerical approach. As in the effective application of bifurcation methods the hand calculation of very long expressions is normally required, computer algebra implementations avoiding the intermediate expression swell problem are needed. Using the normal form approach and exploiting the possibilities of Lie transforms, several algorithms well suited to symbolic computation have been developed. In particular, algorithms implementing the center manifold reduction and the computation of Hopf bifurcation are reported. As an example, the above algorithms are applied to the Fitzhugh nerve equations, as studied by Golubitsky and Langford in a celebrated paper. A coefficient of its normal form never computed is explicitely shown, what implies an unfounded statement in the quoted work.