Showing papers on "Center manifold published in 2011"
TL;DR: This paper proposes a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance, and applies center manifold theory to show local exponential stability of the desired formation shape.
Abstract: This paper solves an n -agent formation shape control problem in the plane. The objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape. We consider two classes of directed, cyclic information architectures associated with so-called minimally persistent formations: leader-remote-follower and coleader. In our framework the formation shape is maintained by controlling certain interagent distances. Only one agent is responsible for maintaining each distance. We propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance. The resulting nonlinear closed-loop system has a manifold of equilibria, which implies that the linearized system is nonhyperbolic. We apply center manifold theory to show local exponential stability of the desired formation shape. The result circumvents the non-compactness of the equilibrium manifold. Choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero, and we show that this condition holds for all minimally persistent leader-remote-follower and coleader formations with generic agent positions. Simulations are provided.
145 citations
TL;DR: The dynamics of a delayed predator–prey system in the presence of harvesting in a modified version of the Leslie–Gower and Holling-type II scheme is analyzed in terms of local stability, global stability, influence of harvesting and bifurcation.
104 citations
TL;DR: In this article, a compartmental epidemic model, introduced by Gumel and Moghadas, is considered and conditions for the occurrence of backward bifurcation are derived from both the mathematical and epidemiological perspective.
Abstract: A compartmental epidemic model, introduced by Gumel and Moghadas [1], is considered. The model incorporates a nonlinear incidence rate and an imperfect preventive vaccine given to susceptible individuals. A bifurcation analysis is performed by applying the bifurcation method introduced in [2], which is based on the use of the center manifold theory. Conditions ensuring the occurrence of backward bifurcation are derived. The obtained results are numerically validated and then discussed from both the mathematical and the epidemiological perspective.
93 citations
TL;DR: In this paper, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the distribution of the eigenvalues.
Abstract: This paper is concerned with a delayed predator–prey system with diffusion effect. First, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the distribution of the eigenvalues. Next the direction and the stability of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.
89 citations
Book•
08 Sep 2011TL;DR: The first complete proof of what has come to be known as the Neimark-Sacker bifurcation theorem was given in this paper, where the reduction to normal form using weighted monomials which precludes using the Center Manifold theorem was presented.
Abstract: This article consists of Chapter 2 of the author's 1954 Dissertation bearing the same title and published as Courant Institute report IMM-NYU 333, available in full on the author's personal web site. Chapter 2 consists of the first complete proof of what has come to be known as the Neimark-Sacker bifurcation theorem. It includes the reduction to normal form using weighted monomials which precludes using the Center Manifold theorem which was not known to the author and was published the same year as the dissertation and called “a Reduction Principle” by V. Pliss and later named Center Manifold theorem by A. Kelley (see previous article for citations). After reduction to normal form, the resulting functional equations for the bifurcating invariant curve is solved in detail. Stand-alone decimals provide solutions of linear functional equations, a-priori estimates and interpolation inequalities between derivatives in the sup norm.
74 citations
Book•
28 Mar 2011
TL;DR: In this article, the authors discuss the controlability and observability of Affine Control Systems in topological spaces. But they focus on differentiable manifolds and differentiable algebra, Lie Group and Lie Algebra.
Abstract: Topological Space.- Differentiable Manifold.- Algebra, Lie Group and Lie Algebra.- Controllability and Observability.- Global Controllability of Affine Control Systems.- Stability and Stabilization.- Decoupling.- Input-Output Structure.- Linearization of Nonlinear Systems.- Design of Center Manifold.- Output Regulation.- Dissipative Systems.- L 2 -Gain Synthesis.- Switched Systems.- Discontinuous Dynamical Systems.
67 citations
TL;DR: In this paper, a three-species system with Holling type II functional response and feedback delays is introduced, and the authors derive explicit formulas to determine the direction of the Hopf bifurcation and the stability of periodic solution by using the normal-form method and center manifold theorem.
Abstract: A kind of three-species system with Holling type II functional response and feedback delays is introduced. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation are obtained. We derive explicit formulas to determine the direction of the Hopf bifurcation and the stability of periodic solution bifurcated out by using the normal-form method and center manifold theorem. Numerical simulations confirm our theoretical findings.
66 citations
TL;DR: In this article, a new three-dimensional chaotic system originates from the Lorenz system, which is different from the known Lorenz systems, Rossler system, Chen system, and includes Lu systems as its special case.
Abstract: This paper formulates a new three-dimensional chaotic system that originates from the Lorenz system, which is different from the known Lorenz system, Rossler system, Chen system, and includes Lu systems as its special case. By using the center manifold theorem, the stability character of its non-hyperbolic equilibria is obtained. The Hopf bifurcation and the degenerate pitchfork bifurcation, the local character of stable manifold and unstable manifold, are also in detail shown when the parameters of this system vary in the space of parameters. Corresponding bifurcation cases are illustrated by numerical simulations, too. The existence or non-existence of homoclinic and heteroclinic orbits of this system is also studied by both rigorous theoretical analysis and numerical simulation.
62 citations
TL;DR: In this paper, a six-neuron BAM neural network model with discrete delays is considered, the linear stability of the model is investigated, and Hopf bifurcation is demonstrated.
59 citations
TL;DR: In this article, a reaction diffusion system with general time-delayed growth rate and kernel functions is considered and the existence and stability of the positive spatially nonhomogeneous steady-state solution is obtained.
58 citations
TL;DR: In this paper, a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds is presented, where primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories.
Abstract: This paper presents a method to construct optimal transfers between unstable periodic orbits of differing energies using invariant manifolds. The transfers constructed in this method asymptotically depart the initial orbit on a trajectory contained within the unstable manifold of the initial orbit and later, asymptotically arrive at the final orbit on a trajectory contained within the stable manifold of the final orbit. Primer vector theory is applied to a transfer to determine the optimal maneuvers required to create the bridging trajectory that connects the unstable and stable manifold trajectories. Transfers are constructed between unstable periodic orbits in the Sun–Earth, Earth–Moon, and Jupiter-Europa three-body systems. Multiple solutions are found between the same initial and final orbits, where certain solutions retrace interior portions of the trajectory. All transfers created satisfy the conditions for optimality. The costs of transfers constructed using manifolds are compared to the costs of transfers constructed without the use of manifolds. In all cases, the total cost of the transfer is significantly lower when invariant manifolds are used in the transfer construction. In many cases, the transfers that employ invariant manifolds are three times more efficient, in terms of fuel expenditure, than the transfer that do not. The decrease in transfer cost is accompanied by an increase in transfer time of flight.
TL;DR: Stability and Hopf bifurcation analysis for a two-species Lotka-Volterra predator-prey model with two delays and the linear stability of the positive equilibrium is investigated and hopf bIfurcation is demonstrated.
Abstract: In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.
TL;DR: In this article, the effect of the degree of habitat complexity and gestation delay on the stability of a predator-prey model was studied. But the authors did not consider the relationship between the delay and the abundance of the population.
TL;DR: In this paper, an SEIV epidemic model with vaccination and nonlinear incidence rate is formulated, and the analysis of the model is presented in terms of the basic reproduction number R ≥ 0.
Abstract: In this paper, an SEIV epidemic model with vaccination and nonlinear incidence rate is formulated. The analysis of the model is presented in terms of the basic reproduction number R
0. It is shown that the model has multiple equilibria and using the center manifold theory, the model exhibits the phenomenon of backward bifurcation where a stable disease-free equilibrium coexists with a stable endemic equilibrium for a certain defined range of R
0. We also discuss the global stability of the endemic equilibrium by using a generalization of the Poincare–Bendixson criterion. Numerical simulations are presented to illustrate the results.
TL;DR: A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces and it is shown that there exists a finite-dimensional center manifold on a space of generalized functions.
Abstract: A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is sin θ, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not sin θ, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.
TL;DR: In this article, an eco-epidemiological model with a stage structure is considered, and the asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated.
Abstract: In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.
TL;DR: In this article, a three-species Lotka-Volterra food chain system with two discrete delays is considered and the asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcations are investigated.
TL;DR: In this article, a bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces, and it is shown that there exists a finite-dimensional center manifold on a space of generalized functions.
Abstract: A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin \theta $, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin \theta $, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.
TL;DR: In this paper, a delayed predator-prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered, and the local asymptotic stability of the positive equilibrium of the system is investigated.
Abstract: In this paper, a delayed predator–prey model with Holling type II functional response incorporating a constant prey refuge and diffusion is considered. By analyzing the characteristic equation of linearized system corresponding to the model, we study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Further, an example is presented to illustrate our main results. Finally, recurring to the numerical method, the influences of impulsive perturbations on the dynamics of the system are also investigated.
TL;DR: It is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions, corresponding to the diffusive N-waves.
Abstract: The large-time behavior of solutions to the Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted $L^2$ space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this “metastable” manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.
TL;DR: Fiedler et al. as discussed by the authors showed that the Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system.
TL;DR: In this paper, the authors investigated the evolution of scalar fields with arbitrary potential trapped in a Randall-Sundrum's Braneworld of type II and derived sufficient conditions for the asymptotic stability of the de Sitter solution.
Abstract: In this paper we investigate, from the dynamical systems perspective, the evolution of an scalar field with arbitrary potential trapped in a Randall-Sundrum's Braneworld of type II. We consider an homogeneous and isotropic Friedmann-Robertson-Walker (FRW) brane filled also with a perfect fluid. Center Manifold Theory is employed to obtain sufficient conditions for the asymptotic stability of de Sitter solution. We obtain conditions on the potential for the stability of scaling solutions as well for the stability of the scalar-field dominated solution. We prove the there are not late time attractors with 5D-modifications (they are saddle-like). This fact correlates with a transient primordial inflation. In the particular case of a scalar field with potential $V=V_{0}e^{-\chi\phi}+\Lambda$ we prove that for $\chi 0$ the de Sitter solution is unstable (of saddle type).
TL;DR: The weakly nonlinear stability of a stationary pulse undergoing a Hopf bifurcation in a neural field model with an excitatory or Mexican hat synaptic weight function and Heaviside firing rate nonlinearity is analyzed.
Abstract: We analyze the weakly nonlinear stability of a stationary pulse undergoing a Hopf bifurcation in a neural field model with an excitatory or Mexican hat synaptic weight function and Heaviside firing rate nonlinearity. The presence of a spatially localized input inhomogeneity $I(x)$ precludes the 0 eigenvalue related to translation invariance of the pulse. Consequently, in the spectral analysis of the linearization about the stationary pulse $\EuScript{U}(x)$, there are two spatial modes, either of which can undergo a Hopf bifurcation in the Mexican hat network to produce a periodic orbit that either expands/contracts (breather) or moves side-to-side (slosher). We derive the normal form for each mode becoming critical in the Hopf bifurcation by (i) the method of amplitude equations and (ii) center manifold reduction, which are shown to agree. Importantly, the critical third order coefficient of the normal form is found to be in strong agreement with numerical simulations of the full model, particularly when...
TL;DR: In this paper, a new logistic competition model is proposed, and the invariant manifolds, including the important centre manifolds and their bifurcation is investigated via numerical simulations.
Abstract: In this paper, we study a new logistic competition model. We will investigate stability and bifurcation of the model. In particular, we compute the invariant manifolds, including the important centre manifolds, and study their bifurcation. Saddle-node and period-doubling bifurcation route to chaos are exhibited via numerical simulations.
TL;DR: In this article, a methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented, which is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations.
01 Jan 2011
TL;DR: In this article, the Hopf bifurcation formula is used to determine the stability of the periodic motion in DDE and the Two Variable Expansion Method (also known as Multiple Scales) is employed to investigate the transient behavior involved in the approach to periodic motion, and Center Manifold Analysis is applied to reduce the DDE from an infinite dimensional evolution equation on a function space to a two dimensional ODE on the center manifold.
Abstract: Periodic motions in DDE (Differential-Delay Equations) are typically created in Hopf bifurcations. In this chapter we examine this process from several points of view. Firstly we use Lindstedt’s perturbation method to derive the Hopf Bifurcation Formula, which determines the stability of the periodic motion. Then we use the Two Variable Expansion Method (also known as Multiple Scales) to investigate the transient behavior involved in the approach to the periodic motion. Next we use Center Manifold Analysis to reduce the DDE from an infinite dimensional evolution equation on a function space to a two dimensional ODE (Ordinary Differential Equation) on the center manifold, the latter being a surface tangent to the eigenspace associated with the Hopf bifurcation. Finally we provide an application to gene copying in which the delay is due to an observed time lag in the transcription process.
TL;DR: This work classifies the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere, and focuses on the case where the system has an additional sign-change symmetry.
Abstract: Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics of degrees $\ell$ and $\ell+1$. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each $\ell$. For small values of $\ell$, the center manifold equations are constructed, and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions or at tertiary bifurcations. The results are consistent with numerical simulations of ...
TL;DR: In this article, the existence of centers on local center manifolds at equilibria of the Lu system of differential equations on R 3 was shown to be a conjecture of Mello and Coelho.
TL;DR: This work systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks and demonstrates that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking.
Abstract: Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG) activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.
TL;DR: In this article, the Hopf bifurcation behavior of a TCP fluid model of Internet congestion control system is investigated and the parameter condition that the hopf biffurcation occurs is deduced.
Abstract: In this paper the Hopf bifurcation behavior of a TCP fluid model of Internet congestion control system is investigated. The parameter condition that the Hopf bifurcation occurs is deduced. The stability and direction of the bifurcating periodic solutions are analyzed by applying the normal form theory and the center manifold theorem. Numerical simulations demonstrate the complex behavior of the system and verify the theoretical analysis.