Showing papers on "Center manifold published in 1995"

The spectrum of an asymptotically hyperbolic Einstein manifold

Abstract: This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity It follows from work of R Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no embedded eigenvalues; however, in general there may be discrete eigenvalues below the continuous spectrum The main result of this paper is that, if the Yamabe invariant of the conformal structure on the boundary is non-negative, then there are no such eigenvalues This generalizes results of R Schoen, S-T Yau, and D Sullivan for the case of hyperbolic manifolds

Limit cycles, tori, and complex dynamics in a second-order differential equation with delayed negative feedback

Abstract: We analyze a second-order, nonlinear delay-differential equation with negative feedback. The characteristic equation for the linear stability of the equilibrium is completely solved, as a function of two parameters describing the strength of the feedback and the damping in the autonomous system. The bifurcations occurring as the linear stability is lost are investigated by the construction of a center manifold: The nature of Hopf bifurcations and more degenerate, higher-codimension bifurcations are explicitly determined.

Eigenvalues of Laplacians on a closed Riemannian manifold and its nets

Abstract: We show that the eigenvalues of the Laplacian of a closed manifold M is approximated in a certain sense by the eigenvalues of the Laplacian of the graph of a I-net in M as n -*oo . Our approximation needs no assumption on M except for dimension.

Center manifolds for quasilinear reaction-diffusion systems

Abstract: We consider strongly coupled quasilinear reaction-diffusion systems subject to nonlinear boundary conditions. Our aim is to develop a geometric theory for these types of equations. Such a theory is necessary in order to describe the dynamical behavior of solutions. In our main result we establish the existence and attractivity of center manifolds under suitable technical assumptions. The technical ingredients we need consist of the theory of strongly continuous analytic semigroups, maximal regularity, interpolation theory and evolution equations in extrapolation spaces.

Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback.

Abstract: A center manifold reduction and numerical calculations are used to demonstrate the presence of limit cycles, two‐tori, and multistability in the damped harmonic oscillator with delayed negative feedback. This model is the prototype of a mechanical system operating with delayed feedback. Complex dynamics are thus seen to arise in very plausible and commonly occurring mechanical and neuromechanical feedback systems.

Stochastic inertial manifold

Abstract: A nonlinear stochastic evolution equation in Hilbert space with generalized additive white noise is considered. A concept of stochastic mertial manifold is introduced, defined as a random manifold depending on time, which is finite dimensional, invariant for the dynamic, and attracts exponentially fast all the trajectories as t → ∞. Under the classical spectral gap condition of the deterministic theory, the existence of a stochastic inertial manifold is proved. It is obtained as the solution of a stochastic partial differential equation of degenerate parabolic type, studied by a variant of Bernstein method. A result of existence and uniqueness of a stationary inertial manifold is also proved; the stationary inertial manifold contains the random attractor, introduced in previous works.

Stability Results for Steady, Spatially--Periodic Planforms

Abstract: We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal lattice, and consider the bifurcation problem restricted to a finite-dimensional center manifold. For the square lattice we assume that the kernel of the linear operator, at the bifurcation point, consists of 4 complex Fourier modes, with wave vectors K_1=(a,b), K_2=(-b,a), K_3=(b,a), and K_4=(-a,b), where a>b>0 are integers. For the hexagonal lattice, we assume that the kernel of the linear operator consists of 6 complex Fourier modes, also parameterized by an integer pair (a,b). We derive normal forms for the bifurcation problems, which we use to compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. These solutions consist of rolls, squares, hexagons, a countable set of rhombs, and a countable set of planforms that are superpositions of all of the Fourier modes in the kernel. Since rolls and squares (hexagons) are common to all of the bifurcation problems posed on square (hexagonal) lattices, this framework can be used to determine their stability relative to a countable set of perturbations by varying a and b. For the hexagonal lattice, we analyze the degenerate bifurcation problem obtained by setting the coefficient of the quadratic term to zero. The unfolding of the degenerate bifurcation problem reveals a new class of secondary bifurcations on the hexagons and rhombs solution branches.

The dynamics of resonant capture

Abstract: Resonant capture describes the behavior of a weakly coupled multi-degree-of-freedom system when two or more of its uncoupled frequencies become locked in resonance. Flow on the region of phase space near the resonance (the resonance manifold) involves a region bounded by a separatrix in the uncoupled (e = 0) system. Capture corresponds to motions which appear to cross into the interior of the separated region for e > 0.

Quasiperiodic oscillations in robot dynamics

Abstract: Delayed robot systems, even of low degree of freedom, can produce phenomena which are well understood in the theory of nonlinear dynamical systems, but hardly ever occur in simple mechanical models. To illustrate this, we analyze the delayed positioning of a single degree of freedom robot arm which leads to an infinite dimensional dynamical system. Restricting the dynamics to a four dimensional center manifold, we show that the system undergoes a codimension two Hopf bifurcation for an infinite set of parameter values. This provides a mechanism for the creation of two-tori in the phase space and gives a theoretical explantion for self-excited quasiperiodic oscillations of force controlled robots. We also compare our results with experimental data.

Non-linear overstability in the thermal convection of viscoelastic fluids

Abstract: The existence and stability of elastic overstability in the presence of non-negligible inertia for an Oldroyd-B fluid are examined for the Rayleigh-Bernard thermal convection problem. The study is based on the four-dimensional non-linear dynamical system presented by Khayat (J. Non-Newtonian Fluid Mech., 53 (1994) 227) which constitutes a generalization of the classical Lorenz system for a Newtonian fluid. It is shown that elastic overstability can only set in once the Deborah number exceeds a critical value which depends on the Prandtl number and fluid retardation. Fluid elasticity is found to precipitate the onset of overstability while retardation tends to delay it. The conditions of existence of the corresponding Hopf bifurcation are examined as functions of fluid elasticity, retardation and thermal conductivity. The stability of the periodic orbit (in phase space) is investigated using center manifold theory. It is found that the orbit is asymptotically stable to perturbations about the conductive state, with the initial period of oscillation decreasing with Deborah number, reaching a minimum, and increasing asymptotically toward a constant value.

Internal equilibrium control of a bicycle

Abstract: Internal equilibrium control is applied to the problem of path-tracking with balance for the bicycle using steering and rear-wheel torque as inputs. From the internal dynamics of the bicycle an internal equilibrium manifold, a submanifold of the state-space, is constructed. The internal equilibrium controller makes a neighborhood of the manifold attractive and invariant. This results in approximate tracking of time-parameterized paths in the plane while retaining balance.

Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations

Abstract: In this study a new procedure for analysis of nonlinear dynamical systems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (L-F) transformation. The L-F transformation is obtained by computing the state transition matrix associated with the linear part of the problem. The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in time t which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quasi-linear equations with periodic coefficients, a dynamically similar system is obtained whose linear part is time-invariant and the nonlinear part consists of coefficients which are periodic. Due to this property of the L-F transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the L-F transformation, is conducted by employing time-dependent center manifold reduction and time-dependent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method. Furthermore, the technique is applicable also to those systems where the periodic term of the linear part does not contain a small parameter which is certainly not the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed that analysis of this nature has been reported for the first time for this class of systems.

Monopoles on asymptotically flat manifolds

Abstract: We consider the relation between the theory of monopoles on ℝ 3 and the analogous problem on a manifold M obtained from ℝ3 by forming the connected sum with a compact manifold M. The main result is that the moduli space of monopoles whose action is concentrated away from M is a smooth manifold which can be described in terms of the homology of M and the part, we determine the asymptotics of these functions. As a result, we determine the «end» of the moduli space in some special cases.

Dynamics of a minimal power system: invariant tori and quasi-periodic motions

Abstract: The paper provides an extensive analysis of local and global bifurcation phenomena in the voltage-angle dynamic interactions of a minimal power system model. Using nonlinear analysis and normal form theory, it is proved that this system will experience quasi-periodic motions near certain degenerate local bifurcations which are explicitly characterized. The results in the paper provide strong analytical evidence for the possible occurrence of complicated behavior in the power system from the interactions of voltage and angle instability mechanisms. Computational methods for the detection of invariant 2-tori in higher dimensional systems using tools from center manifold theory and normal form theory are introduced briefly and these techniques are illustrated on a fourth order power system model.

On the stabilization of a class of nonholonomic systems using invariant manifold technique

Abstract: This paper presents an asymptotically stabilizing discontinuous feedback controller for a class of nonholonomic systems. The controller consists of two parts: the first part yields an invariant manifold on which all trajectories of the closed-loop system tend to the origin, and the latter part renders the invariant manifold attractive, while avoiding a discontinuity surface. The controller yields exponential stability so that the convergence can be chosen arbitrarily fast.

Topology and closed characteristics of K-contact manifolds.

Abstract: We prove that the characteristic ow of a K-contact form has at least n+1 closed leaves on a closed 2n+1-dimensional manifold. We also show that the rst Betti number of a closed sasakian manifold with nitely many closed characteristics is zero.

Parametric manifolds. I. Extrinsic approach

Abstract: A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) one‐form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, all based on a nonstandard action of vector fields on functions. There is a new geometric object, called the deficiency, which behaves much like torsion, and which measures whether a parametric manifold can be viewed as a one‐parameter family of orthogonal hypersurfaces.

Preliminary results on asymptotic stabilization of Hamiltonian systems with nonholonomic constraints

Abstract: This paper presents some preliminary results on asymptotic stabilization of nonholonomic mechanical systems using the Hamiltonian formulation proposed by van der Schaft and Maschke (1994). Our work seeks to establish a general formulation for designing time-varying controllers for some mechanical system described in the generalized coordinates (position and momentum). The paper gives the change of coordinates that transforms the Hamiltonian system to the form needed to apply the center manifold theorem. We also present a worked example for which stability is analyzed.

Weakly nonlinear behavior of periodic disturbances in two-layer plane channel flow of upper-convected Maxwell liquids

Abstract: The stability of a plane Couette-Poiseuille flow consisting of two layers of different fluids is analyzed. The fluids have different viscosities, densities and relaxation times and there is surface tension at the interface. The center manifold reduction method is used to derive the final amplitude evolution equation. The nonlinear calculations are carried out with two alternative approaches. One approach is to keep the combined volume flux fixed, and the other is to keep the pressure gradient in the horizontal direction fixed. Numerical results are presented for the one-fluid Poiseuille flow, and some two-layer Poiseuille flows with reference to co-extrusion through a slit die. Situations with subcritical and supercritical bifurcations are described.

Homoclinic bifurcation in blasius boundary-layer flow

Abstract: In an attempt to elucidate the laminar/turbulent transition mechanism in a Blasius boundary‐layer flow, a nonsemisimple resonance of phase‐locked secondary instability modes is investigated. The local nonlinear behavior is described by means of a center manifold reduction. The numerically computed normal form is of the symmetric Takens–Bogdanov type and predicts a homoclinic orbit which is possibly related to a physical bursting process. A global continuation procedure for equilibrated three‐dimensional (3‐D) waves in the full Navier–Stokes system validates some of the local predictions and very closely outlines the experimentally observed skin friction domain including subcritical transition.

Bifurcations from periodic solution in a simplified model of two‐dimensional magnetoconvection

Abstract: A two‐dimensional Boussinesq fluid with nonlinear interaction between Rayleigh–Benard convection and an external magnetic field is investigated numerically and analytically. A simplified model consisting of a fifth‐order system of nonlinear ordinary differential equations with five parameters is introduced and integrated numerically in certain parameter regions. Various types of bifurcations from periodic solutions are found numerically: period‐doubling bifurcation, heteroclinic bifurcation, intermittency, and saddle‐node bifurcation. A normal form equation is also derived from the fifth‐order system, and center manifold theory is applied to it. An expression for the renormalized Holmes–Melnikov boundary for the evaluation of the numerical results is given. It is shown from the normal form equation that each property of the two phase portraits described by the Duffing equation and the van der Pol equation emanates from one common attractor in the five‐dimensional space of the fifth‐order system.

Co2-laser dynamics with feedback

Abstract: The validity of the four-level model is controlled in an experiment with a ${\mathrm{CO}}_{2}$ laser with electronic feedback. A global application of center manifold theory allows one to reduce the number of varieties from six to four. Qualitative differences are found with the case of loss modulations.