Showing papers on "Center manifold published in 2005"
TL;DR: This work surveys different approaches for computing a global stable or unstable manifold of a vector field, where it focuses on the case of a two-dimensional manifold.
Abstract: The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable manifold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example — the two-dimensional stable manifold of the origin in the Lorenz system.
236 citations
TL;DR: In this article, a delay-differential equation was used to model a bidirectional associative memory (BAM) neural network with three neurons and its dynamics were studied in terms of local analysis and Hopf bifurcation analysis.
217 citations
TL;DR: In this paper, the authors describe a method to establish existence and regularity of invariant manifolds and, at the same time, find simple maps which are conjugated to the dynamics on them.
196 citations
TL;DR: In this article, it was shown that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a poin.
Abstract: We consider dynamical systems possessing an attracting, invariant "slow manifold" that can be parameterized by a few observable variables. We present a procedure that, given a process for integrating the system step by step and a set of values of the observables, finds the values of the remaining system variables such that the state is close to the slow manifold to some desired accuracy. It should be noted that this is not equivalent to "integrating down to the manifold" since the latter process may significantly change the values of the observables. We consider problems whose solution has a singular perturbation expansion, although we do not know what it is nor have any way to compute it (because the system is not necessarily expressed in a singular perturbation form). We show in this paper that, under some conditions, computing the values of the remaining variables so that their (m + 1)st time derivatives are zero provides an estimate of the unknown variables that is an mth-order approximation to a poin...
188 citations
TL;DR: In this paper, the authors consider a delayed predator-prey system and derive explicit formulas to determine the stability and the direction of periodic solutions bifurcating from Hopf bifurlcations, using normal form theory and center manifold argument.
181 citations
TL;DR: The natural gradient method for neural networks is extended to the case where the weight vectors are constrained to the Stiefel manifold and a simpler updating rule is developed and one parameter family of its generalizations is developed.
175 citations
Abstract: The dynamics of a Nicholson's blowflies equation with a finite delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. Explicit algorithm for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived, using the theory of normal form and center manifold. Global existence of periodic solutions are established using a global Hopf bifurcation result of Wu (Trans. Amer. Math. Soc. 350 (1998) 4799), and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney (J. Differential Equations 106 (1994) 27).
129 citations
TL;DR: An overview of the contents of the papers collected in this special issue of Nonlinear Dynamics is given in this paper, along with some basic introductory ideas concerning dimension reduction and reduced order modelling.
Abstract: After presenting some basic introductory ideas concerning dimension reduction and reduced order modelling, an overview of the contents of the papers collected in this Special Issue of Nonlinear Dynamics is given.
106 citations
TL;DR: A model for a pendulum, which is free to rotate 360 degrees, attached to a cart, is described and it is shown that the system undergoes a supercritical Hopf bifurcation at the critical delay.
Abstract: We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos.
98 citations
TL;DR: In this paper, the authors applied the center manifold theory to a magnetohydrodynamic (MHD)-system and examined the influence of the Hall term on the width of the magnetic islands of the tearing-mode.
86 citations
TL;DR: Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens bifurcation for n-dimensional systems with arbitrary n ≥ 2.
Abstract: Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov–Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov–Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.
TL;DR: In this article, the authors demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, they can conveniently compute, using a legacy simulator, on a nearby manifold.
Abstract: If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of "slow" variables that can be used to parameterize the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results
TL;DR: This paper employs a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation and proves the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.
Abstract: This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi–Pasta–Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.
TL;DR: Applying the derived formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the criticalnormal form coefficients in the original basis.
Abstract: In this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues. These include all strong resonances, as well as the degenerate flip and Neimark--Sacker bifurcations. Applying our results to n-dimensional maps, one avoids the computation of the center manifold, but one can directly evaluate the critical normal form coefficients in the original basis. The formulas remain valid also for n=2 and allow one to avoid the transformation of the linear part of the map into Jordan form.
The developed techniques are tested on two examples: (1) a three-dimensional map appearing in adaptive control; (2) a periodically forced epidemic model. We compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models. To compute the required derivatives of the Poincare map for the epidemic model, the automatic differentiation package ADOL-C is used.
TL;DR: In this article, the authors studied the hopf bifurcation of an Internet congestion control algorithm, namely, Random Exponential Marking (REM) algorithm, with communication delay.
Abstract: The purpose of this paper is to study bifurcation of an Internet congestion control algorithm, namely REM (Random Exponential Marking) algorithm, with communication delay. By choosing the delay constant as a bifurcation parameter, we prove that REM algorithm exhibits Hopf bifurcation. The formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, a numerical simulation is present to verify the theoretical results.
TL;DR: In this article, the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling is considered and an algorithm for determining the Hopf bifurcation properties is given.
TL;DR: A new algorithm for computing invariant manifolds that iteratively interpolates and constructs fat trajectories starting at the interpolation points, and allows errors to be controlled via the integration method used to find the fat trajectory and the width of the fat traje...
Abstract: We present a new algorithm for computing invariant manifolds. The algorithm uses two main theoretical results, presented in the appendices: formulae for the evolution of a second order local approximation of a bundle of trajectories (which we call a fat trajectory), and a proof of the existence and a constructive means of locating points where k (the dimension of the manifold) trajectories diverge.Invariant manifolds can be defined as the image under the flow of a (k-1)-dimensional manifold of starting points. The algorithm uses local approximations at points spaced along trajectories starting on the manifold of starting points to partially cover the invariant manifold. To finish the covering it then iteratively interpolates and constructs fat trajectories starting at the interpolation points. Unlike other methods, the resulting algorithm does not use an adaptively refined front, and it allows errors to be controlled via the integration method used to find the fat trajectory and the width of the fat traje...
TL;DR: In this paper, a delay differential equation (DDE) near a codimension 2 Hopf bifurcation point is studied, and analytical expressions for the double Hopf points are obtained.
Abstract: We study a well-known regenerative machine tool vibration model (a delay differential equation) near a codimension 2 Hopf bifurcation point. The method of multiple scales is used directly, bypassing a center manifold reduction. We use a nonstandard choice of expansion parameter that helps understand practically relevant aspects of the dynamics for not-too-small amplitudes. Analytical expressions are then obtained for the double Hopf points. Both sub- and supercritical bifurcations are predicted to occur near the reference point; and analytical conditions on the parameter variations for each type of bifurcation to occur are obtained as well. Analytical approximations are supported by numerics.
TL;DR: In this paper, the van der Pol equation with a time delay was considered, where the time delay is regarded as a parameter and it was found that Hopf bifurcation occurs when this delay passes through a sequence of critical values.
Abstract: In this paper, the van der Pol equation with a time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem.
TL;DR: In this article, the invariant manifolds of the spatial Hill's problem associated to the two liberation points are studied and a combination of analytical and numerical tools allow the normalization of the Hamiltonian and the computation of periodic and quasi-periodic orbits.
Abstract: The paper studies the invariant manifolds of the spatial Hill's problem associated to the two liberation points. A combination of analytical and numerical tools allow the normalization of the Hamiltonian and the computation of periodic and quasi-periodic (invariant tori) orbits. With these tools, it is possible to give a complete description of the center manifolds, association to the liberation points, for a large set of energy values. A systematic exploration of the homoclinic and heteroclinic connections between the center manifolds of the liberation points is also given.
TL;DR: In this paper, the basic problem of order reduction of nonlinear systems with time periodic coefficients is considered, where the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant.
TL;DR: For a class of reaction-convection-diffusion systems studied by Sattinger, notably including Majda's model of reacting flow, the authors rigorously characterize the transition from stability to time-periodic "galloping" instability of traveling-wave solutions as a relative Poincare-Hopf bifurca- tion arising in ODE with a group invariance.
Abstract: For a class of reaction-convection-diffusion systems studied by Sattinger, notably including Majda's model of reacting flow, we rigorously characterize the transition from stability to time-periodic "galloping" instability of traveling-wave solutions as a relative Poincare-Hopf bifurca- tion arising in ODE with a group invariance- in this case, translational symmetry. More generally, we show how to construct a finite-dimensional center manifold for a second-order parabolic evolu- tion equation inheriting an underlying group invariance of the PDE, by working with a canonical integro-differential equation induced on the quotient space. This reduces the questions of existence and stability of bounded solutions of the PDE to existence and stability of solutions of the reduced, finite-dimensional ODE on the center manifold, which may then be studied by more standard, finite- dimensional bifurcation techniques.
TL;DR: In this paper, the authors studied continuous nonlinear Schrodinger (NLS) equations, including the cubic NLS lattice with on-site interactions and the integrable Ablowitz-Ladik lattice, and showed that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equations near the zero-dispersion limit.
TL;DR: In this article, a general model of nonlinear systems with distributed delays is derived from Chen's system with the weak kernel, and the local stability is analyzed by using the Routh-Hurwitz criterion, where the direction and the stability of the bifurcating periodic solutions are determined using the normal form theory and the center manifold theorem.
Abstract: In this paper, a general model of nonlinear systems with distributed delays is studied. Chen’s system can be derived from this model with the weak kernel. After the local stability is analyzed by using the Routh–Hurwitz criterion, Hopf bifurcation is studied, where the direction and the stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical analysis are also presented. Chaotic behavior of Chen’s system with the strong kernel is also found through numerical simulation, in which some waveform diagrams, phase portraits, and bifurcation plots are presented and analyzed.
TL;DR: In this paper, the authors focus on two particular reduction techniques, namely, computational singular perturbation (CSP) and zero-derivative principle (ZDP), for general nonlinear evolution equations involving multiple time scales.
Abstract: This article is concerned with general nonlinear evolution equations x ′ = g (x ) in RN involving multiple time scales, where fast dynamics take the orbits close to an invariant low-dimensional manifold and slow dynamics take over as the state approaches the manifold. Reduction techniques offer a systematic way to identify the slow manifold and reduce the original equation to an autonomous equation on the slow manifold. The focus in this article is on two particular reduction techniques, namely, computational singular perturbation (CSP) proposed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The University of Washington, Seattle, Washington, August 14–19, 1988 (The Combustion Institute, Pittsburgh, 1988), pp. 931–941] and the zero-derivative principle (ZDP) proposed recently by Gear and Kevrekidis [Constraint-defined manifolds: A legacy-code approach to low-dimensional computation, SIAM J. Sci. Comput., to appear]. It is shown that the tangent bundle to the state space offers a unifying framework for CSP and ZDP. Both techniques generate coordinate systems in the tangent bundle that are natural for the approximation of the slow manifold. Viewed from this more general perspective, both CSP and ZDP generate, at each iteration, approximate normal forms for the system under examination. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
TL;DR: In this article, a new fuzzy manifold K(E-infinity) was introduced and all the Betti numbers and other topological invariants of this manifold have been determined.
Abstract: The paper introduces a new Kahler-like fuzzy manifold K(E-infinity). All the Betti numbers and other topological invariants of this manifold have been determined. In particular it is found that the Euler characteristic is equal to 26 + k = 26.18033989 compared with 24 in the K3 Kahler case. On the other hand, the absolute value of the inverse signature was found to be equal to the Sommerfield electromagnetic fine structure constant lifted to 10 dimensions. This gives the manifold profound physical meaning.
TL;DR: In this article, the existence of travelling breathers in Klein-Gordon chains was studied in the case when the breather period and the inverse of its velocity are commensurate, and it was shown that the center manifold reduction method introduced by Iooss and Kirchgassner is still applicable when the problem is formulated in an appropriate way.
Abstract: We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrodinger equation (M Remoissenet, Phys Rev B33, n4, 2386 (1986), J Giannoulis and A Mielke, Nonlinearity17, p 551–565 (2004)) In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgassner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G Iooss, K Kirchgassner, Commun Math Phys211, 439–464 (2000)) However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate We show that the center manifold reduction method introduced by Iooss and Kirchgassner is still applicable when the problem is formulated in an appropriate way This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrodinger approximation
TL;DR: The study of a four-dimensional center manifold predicts a "Dimits shift" of the threshold for turbulence due to the excitation of zonal flows and establishes the exact value of that shift.
Abstract: The transition to collisionless ion-temperature-gradient-driven plasma turbulence is considered by applying dynamical systems theory to a model with 10 degrees of freedom. The study of a four-dimensional center manifold predicts a 'Dimits shift' of the threshold for turbulence due to the excitation of zonal flows and establishes (for the model) the exact value of that shift.
TL;DR: In this article, the cohomology and instantons number in E -infinity space were analyzed in fuzzy hyperbolic manifold and it was shown that the Betti numbers of this manifold lead to the conclusion that the Euler characteristic of such a manifold is χ ǫ −26 and that this is equal to the number of instantons.
Abstract: The paper deals with the cohomology and instantons number in E -infinity space. It is found that E -infinity may be described by a fuzzy hyperbolic manifold. The Betti numbers of this manifold lead to the conclusion that the Euler characteristic of this manifold is χ ≃ 26 and that this is equal to the number of instantons.
TL;DR: In this paper, the conformality of a Finsler manifold to a Berwald manifold is checked by a 1-form constructions on the underlying manifold by the help of integral formulas.