Showing papers on "Center manifold published in 2008"
Book•
01 Jan 2008
TL;DR: The Fermi-Pasta-Ulam Problem and the Metastability Perspective in FPU was studied in this paper, where the authors provided an integral approximation for the Fermia-Pastora-Umm Lattice.
Abstract: to FPU.- Dynamics of Oscillator Chains.- Role of Chaos for the Validity of Statistical Mechanics Laws: Diffusion and Conduction.- The Fermi-Pasta-Ulam Problem and the Metastability Perspective.- Resonance, Metastability and Blow up in FPU.- Center Manifold Theory in the Context of Infinite One-Dimensional Lattices.- Numerical Methods and Results in the FPU Problem.- An Integrable Approximation for the Fermi-Pasta-Ulam Lattice.
202 citations
TL;DR: It is shown that the perturbation theory for dual semigroups (sun-star-calculus) is equally efficient for dealing with Volterra functional equations and both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem are obtained.
Abstract: We show that the perturbation theory for dual semigroups (sun-star-calculus) that has proved useful for analyzing delay-differential equations is equally efficient for dealing with Volterra functional equations. In particular, we obtain both the stability and instability parts of the principle of linearized stability and the Hopf bifurcation theorem. Our results apply to situations in which the instability part has not been proved before. In applications to general physiologically structured populations even the stability part is new.
150 citations
TL;DR: In this paper, the authors compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the normal form and the center-manifold theorem.
Abstract: We compare two approaches for determining the normal forms of Hopf bifurcations in retarded nonlinear dynamical systems; namely, the method of multiple scales and a combination of the method of normal forms and the center-manifold theorem. To describe and compare the methods without getting involved in the algebra, we consider three examples: a scalar equation, a single-degree-of-freedom system, and a three-neuron model. The method of multiple scales is directly applied to the retarded differential equations. In contrast, in the second approach, one needs to represent the retarded equations as operator differential equations, decompose the solution space of their linearized form into stable and center subspaces, determine the adjoint of the operator equations, calculate the center manifold, carry out details of the projection using the adjoint of the center subspace, and finally calculate the normal form on the center manifold. We refer to the second approach as center-manifold reduction. Finally, we consider a problem in which the retarded term appears as an acceleration and treat it using the method of multiple scales only.
134 citations
TL;DR: In this paper, the spatio-temporal dynamics of a generic integral-differential equation subject to additive random fluctuations are studied, and it is shown that the global fluctuations shift the Turing bifurcation threshold.
102 citations
TL;DR: In this paper, the existence of a true invariant manifold given an approximately invariant and hyperbolic manifold for an infinite-dimensional dynamical system was investigated. And they proved that if the given manifold is approximately invariance and approximately normal hyperbola, then the dynamical systems has a true-invariant manifold nearby, and applied this result to reveal the global dynamics of boundary spike states for the generalized Allen-Cahn equation.
Abstract: 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.
98 citations
TL;DR: In this paper, the authors considered a delayed Lotka-Volterra predator-prey system with a single delay and analyzed the characteristic equation of the linearized system of the original system at positive equilibrium.
Abstract: In this paper, we consider a delayed Lotka–Volterra predator–prey system with a single delay. By regarding the delay as the bifurcation parameter and analyzing the characteristic equation of the linearized system of the original system at the positive equilibrium, the linear stability of the system is investigated and Hopf bifurcations are demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.
73 citations
TL;DR: In this article, a time-delayed SIR model with a nonlinear incidence rate is considered and the existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values.
Abstract: In this paper, a time-delayed SIR model with a nonlinear incidence rate is considered. The existence of Hopf bifurcations at the endemic equilibrium is established by analyzing the distribution of the characteristic values. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. Numerical simulations to support the analytical conclusions are carried out.
64 citations
TL;DR: Wu et al. as discussed by the authors investigated a predator-prey system with two delays and showed that Hopf bifurcations can occur as τ crosses some critical values.
63 citations
TL;DR: This paper is concerned with a novel four-dimensional continuous autonomous hyperchaotic system, which is obtained by adding a simple dynamical state-feedback controller to a Lorenz-like three-dimensional autonomous chaotic system.
Abstract: This paper is concerned with a novel four-dimensional continuous autonomous hyperchaotic system, which is obtained by adding a simple dynamical state-feedback controller to a Lorenz-like three-dimensional autonomous chaotic system. This new system contains three parameters and each equation of the system has one quadratic cross-product term. Some basic properties of the system are studied first. Its complex dynamic behaviors are then analyzed by means of Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits and Poincare sections. It is shown that the system has several large hyperchaotic regions. When the system is evolving in a hyperchaotic region, the two positive LEs are both large, which can be larger than 1 if the system parameters are taken appropriately. The pitchfork bifurcation of the system is finally analyzed by using the center manifold theorem.
59 citations
TL;DR: In this article, the results on Takens-Bogdanov bifurcation obtained in [T. Faria et al., 1995] for scalar delay differential equations over to the case of delay differential systems with parameters were carried out.
55 citations
TL;DR: In this article, a van der Pol's equation with generally delayed feedback is considered and it is shown that there are Bogdanov-Takens bifurcation, triple zero and Hopf-zero singularities by analyzing the distribution of the roots of the associated characteristic equation.
TL;DR: In this paper, the Poincare-Hopf bifurcation of traveling-wave solutions of viscous conservation laws was studied and a nonstandard implicit function construction was carried out in the absence of a spectral gap between oscillatory modes and essential spectrum.
TL;DR: In this paper, the authors consider random dynamical systems with slow and fast variables driven by two independent metric dynamical system modeling stochastic noise and establish the existence of a random inertial manifold eliminating the fast variables.
Abstract: We consider random dynamical systems with slow and fast variables driven by two independent metric dynamical systems modeling stochastic noise. We establish the existence of a random inertial manifold eliminating the fast variables. If the scaling parameter tends to zero, the inertial manifold tends to another manifold which is called the slow manifold. We achieve our results by means of a fixed point technique based on a random graph transform. To apply this technique we need an asymptotic gap condition.
Posted Content•
TL;DR: In this paper, the complex Hessian equation over K\"ahler manifold was studied under the condition that the underline K''ahler manifolds have non-negative holomorphic bisectional curvature.
Abstract: In this paper, complex Hessian equation over K\"ahler manifold was studied. Under the condition that the underline K\"ahler manifold has non-negative holomorphic bisectional curvature, the existence and regularity of the solution was proved.
TL;DR: In this paper, a simple discrete two-neuron network model with three delays is considered, and the stability and direction of the three types of bifurcations are studied by applying the normal form theory and the center manifold theorem.
Abstract: In this paper, we consider a simple discrete two-neuron network model with three delays The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark–Sacker bifurcations The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem Our results are a very important generalization to the previous works in this field
11 Feb 2008
TL;DR: In this article, the authors employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map.
Abstract: In this paper we employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifurcation theory. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.
TL;DR: The local asymptotic stability of the positive equilibrium and the existence of local Hopf bifurcation of periodic solutions are investigated, and an explicit algorithm is given by applying the normal form theory and the center manifold reduction for functional differential equations.
TL;DR: In this paper, a neural network model with three neurons and a single delay is considered and the existence of local Hopf bifurcations is first considered and explicit formulas are derived by using the normal form method and center manifold theory.
Abstract: A neural network model with three neurons and a single delay is considered The existence of local Hopf bifurcations is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions A global Hopf bifurcation theorem due to Wu and a Bendixson's criterion for high-dimensional ODE due to Li and Muldowney are used to obtain a group of conditions for the system to have multiple periodic solutions when the delay is sufficiently large Finally, numerical simulations are carried out to support the theoretical analysis of the research
TL;DR: In this article, the authors studied the dynamics of Liu system using Routh-Hurwitz criteria, Center manifold theorem, Hopf bifurcation theorem and Hsu & Kazarinoff theorem.
Abstract: Dynamical behaviors of Liu system is studied using Routh–Hurwitz criteria, Center manifold theorem and Hopf bifurcation theorem. Periodic solutions and their stabilities about the equilibrium points are studied by using Hsu & Kazarinoff theorem. Linear feedback control techniques are used to stabilize and synchronize the chaotic Liu system.
TL;DR: In this paper, the authors investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation and derived linear stability criteria dependent on communication delay.
Abstract: In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.
TL;DR: A class of more general HIV infection models with time delay is proposed based on some important biological meanings and the explicit formulaes which determine the stability, the direction and the periodic of bifurcating period solutions are derived.
TL;DR: In this article, the relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied, based on Ljusternik-Schnirelmann category and Morse theory.
TL;DR: In this paper, a delayed cooperation diffusion system with Dirichlet boundary conditions is considered and the asymptotic stability of positive equilibrium and Hopf bifurcation is investigated.
Abstract: This paper is concerned with a delayed cooperation diffusion system with Dirichlet boundary conditions. By applying the implicit function theorem, the normal form theory and the center manifold reduction, the asymptotic stability of positive equilibrium and Hopf bifurcation are investigated. It is shown that an increase in delay will destabilize the positive equilibrium and lead to the occurrence of a supercritical Hopf bifurcation when the delay crosses through a sequence of critical values. Based on the normal form theory and the center manifold reduction for partial functional differential equations (PFDEs), we find that the bifurcating periodic solution occurring from the first Hopf bifurcation point is stable on the center manifold and those occurring from the other bifurcation points are unstable. Finally, some numerical simulations are given to illustrate our results.
TL;DR: In this paper, the authors studied the behavior of perturbations of small nonlinear Dirac standing waves in the presence of a symmetry of the Dirac operator and a resonance condition.
Abstract: We study the behavior of perturbations of small nonlinear Dirac standing waves. We assume that the linear Dirac operator of reference $H=D_m+V$ has only two double eigenvalues and that degeneracies are due to a symmetry of H (theorem of Kramers). In this case, we can build a small four-dimensional manifold of stationary solutions tangent to the first eigenspace of H. Then we assume that a resonance condition holds, and we build a center manifold of real codimension 8 around each stationary solution. Inside this center manifold any $H^s$ perturbation of stationary solutions, with $s>2$, stabilizes towards a standing wave. We also build center-stable and center-unstable manifolds, each one of real codimension 4. Inside each of these manifolds, we obtain stabilization towards the center manifold in one direction of time, while in the other, we have instability. Eventually, outside all of these manifolds, we have instability in the two directions of time. For localized perturbations inside the center manifold...
TL;DR: In this article, the authors investigated a congestion control algorithm with communication delay and showed that Hopf bifurcation would occur when the delay exceeds a critical value, and derived necessary and sufficient conditions for Hopfbifurcation to occur for this model.
Abstract: In this paper, we investigate a novel congestion control algorithm, ie, exponential RED algorithm, with communication delay We derive some necessary and sufficient conditions ensuring Hopf bifurcation to occur for this model By choosing the delay as a bifurcation parameter, we demonstrated that Hopf bifurcation would occur when the delay exceeds a critical value A formula for determining the bifurcation direction and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem Some numerical simulations for justifying the theoretical results are also provided
TL;DR: In this article, the authors uncover the geometric structure responsible for moving spikes on a non-hyperbolic critical manifold in a nonautonomous dynamical system, and illustrate their results on analytical and numerical examples of off-wall fluid flow separation.
TL;DR: In this article, the authors derive a linearization theorem in the framework of dynamic equations on time scales and investigate the behavior of the topological conjugacy under parameter variation, which is tailor-made for future applications in analytical discretization theory, i.e., to study the relationship between ODEs and numerical schemes applied to them.
TL;DR: In this paper, the authors used Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure, and they used this theorem to show that every manifold in the SnapPea cusped census has the same structure.
Abstract: The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximations of SNAP, a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating the manifold, identifying consistency and completeness equations with respect to this triangulation, and then trying to solve the system of equations using Newton's Method. This produces an approximate, not actual solution. The method developed here uses Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure.
TL;DR: In this article, the authors studied quasilinear systems of parabolic partial differential equa- tions with fully nonlinear boundary conditions on bounded or exterior domains and derived the asymptotic behavior of the solutions in the vicin-ity of an equilibrium.
Abstract: We study quasilinear systems of parabolic partial differential equa- tions with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicin- ity of an equilibrium. The local center, center-stable, and center-unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.
TL;DR: In this article, a delay differential mathematical model that described HIV infection of CD4+ T-cells is analyzed, and the stability of the non-negative equilibria and the existence of Hopf bifurcation are investigated.
Abstract: A delay differential mathematical model that described HIV infection of CD4+ T-cells is analyzed. The stability of the non-negative equilibria and the existence of Hopf bifurcation are investigated. A stability switch in the system due to variation of delay parameter has been observed, so is the phenomena of Hopf bifurcation and stable limit cycle. The estimation of the length of delay to preserve stability has been calculated. Using the normal form theory and center manifold argument, the explicit formulaes which determine the stability, the direction and the periodic of bifurcating period solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.