Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an (m−1)-dimensional invariant submanifold of a dynamical system in Rm is defined analogous to that in R. Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M₊ and M₋, after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.     

Hopf bifurcation control in the XCP for the Internet congestion control system

In this paper, we investigated the Hopf bifurcation of the eXplicit Control Protocol (XCP) for the Internet congestion control system. These bifurcation behaviors may cause heavy oscillation of average queue length and induce network instability. A time-delayed feedback control method was proposed for controlling Hopf bifurcation in the XCP system. Numerical simulation results are presented to show that the time-delayed feedback controller is efficient in controlling Hopf bifurcation.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

A global bifurcation result for a semilinear elliptic equation

We consider the problem

     

Hopf bifurcation for tearing instability in magnetohydrodynamics, theory and numerical results

Fusion plasmas inside a Tokamak may be represented using magnetohydrodynamic equations. An equilibrium solution consists in a stationary solution of these equations, generally depending only on some radial coordinate.

In Tokamak experiments, we try to obtain such equilibrium solutions; however, some instabilities may appear and destroy such an equilibrium configuration. For instance, the so-called tearing instability, which destroys the equilibrium magnetic configuration, is really observed in experiments. Above a critical value of some physical parameter, the instability appears first as a stationary solution; then, increasing again some physical parameter, oscillatory behaviour can be found, before more complicated states and exponential growth of the solutions.

These physically observed phenomena have also been numerically computed (new results using the DEMA code concern the so-called double-tearing instability).

The mathematical framework of bifurcation theory is well suited for such a study; we first recall the existence of a stationary branch of bifurcated solutions; then, using mainly a Center Manifold Theorem, we prove existence of Hopf bifurcation. Most of the results are obtained with given diffusion coefficients (viscosity, resistivity) but new results are also obtained when these coefficients depend nonlinearly of the unknowns.

We also give a new mathematical justification of the decomposition of the velocity and magnetic fields used in the previously cited DEMA code. Finally, we also recall some results concerning the existence of a global attractor (in dimension 2).     

Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.     

Concurrent homoclinic bifurcation and Hopf bifurcation for a class of planar Filippov systems

This paper investigates both homoclinic bifurcation and Hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of Filippov type. Firstly, based on a geometrical interpretation and a new analysis of the so-called successive function, sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems. Then, with the discussion about Poincaré map, bifurcation analyses of homoclinic orbit and parabolic–parabolic (PP) type pseudo-focus are presented. It is shown that two limit cycles can appear from the two different kinds of bifurcation in planar Filippov systems.     

A matroid invariant via the K-theory of the Grassmannian

Let G(d,n) denote the Grassmannian of d-planes in Cn and let T be the torus n(C^∗)/diag(C^∗) which acts on G(d,n). Let x be a point of G(d,n) and let be the closure of the T-orbit through x. Then the class of the structure sheaf of in the K-theory of G(d,n) depends only on which Plücker coordinates of x are nonzero - combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K^○(G(d,n)) to Z[t]. Letting gx(t) denote the image of , gx behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (2) (1993) 29-110], Hacking, Keel and Tevelev's very stable pairs [P. Hacking, S. Keel, E. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom. 15 (2006) 657-680] and the author's tropical linear spaces when they are realizable in characteristic zero [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527-1558]. Namely, in characteristic zero, a Lie complex or the underlying (d−1)-dimensional scheme of a very stable pair can have at most strata of dimensions n−i and d−i, respectively. This prove the author's f-vector conjecture, from [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527-1558], in the case of a tropical linear space realizable in characteristic 0.     

A NEW DETECTING METHOD FOR CONDITIONS OF EXISTENCE OF HOPF BIFURCATION

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Analysis of the Hopf bifurcation for the family of angiogenesis models

In this paper we study a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well-known models of tumour angiogenesis proposed by Hahnfeldt et al. and d?Onofrio-Gandolfi and is based on the Gompertz type of the tumour growth. As a consequence we start our analysis from the influence of delay onto the Gompertz model dynamics. The family of models considered in this paper depends on two time delays and a parameter α∈[0,1] which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. We focus on the analysis of the model in three cases: one of the delays is equal to 0 or both delays are equal, depending on the parameter α. We study the stability switches, the Hopf bifurcation and the stability of arising periodic orbits for different α∈[0,1], especially for α=1 and α=0 which reflects the Hahnfeldt et al. and the d?Onofrio-Gandolfi models. For comparison we use also the value α=1/2.     

Towards a bifurcation theory for nonautonomous difference equations

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. This article contains an approach to overcome this deficit in the context of nonautonomous difference equations. Based on special notions of attractivity and repulsivity, nonautonomous bifurcation phenomena are studied. We obtain generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation.     

A bifurcation theorem for the p-logistic equation

We consider a p-logistic equation with an equidiffusive reaction. Using variational methods and truncation techniques, we show that there is a critical parameter value λ^∗ > 0 such that for λ > λ^∗ the problem has a unique positive smooth solution, and for λ ∈ (0, λ^∗] the problem has no positive solution.     

A regularity result for critical points of conformally invariant functionals

Let be an open set inR ² andI be a conformally invariant functional defined onH ¹(,R ^d ). Letu be a critical point ofI. We show that, ifu is apriori assumed to be bounded, thenu is smooth in , up to (ifu _| is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.     

The integral bifurcation method and its application for solving a family of third-order dispersive PDEs

In this paper, an improved method named the integral bifurcation method is introduced. In order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations, a family of third-order dispersive partial differential equations which were given by A. Degasperis, D. Holm and A. Hone are studied. Many integral bifurcations are obtained for different parameter conditions. By using these integral bifurcations, many travelling wave solutions such as loop soliton solutions, solitary wave solutions, cusp soliton solutions and periodic wave solutions are obtained. In particular, under the conditions _c1<0,_c2=_c3=1$c_1<0,c_2=c_3=1$, a very peculiar periodic wave solution is obtained.     

Stability and bifurcation analysis for a discrete-time model of Lotka-Volterra type with delay

A discrete model of Lotka-Volterra type with delay is considered, and a bifurcation analysis is undertaken for the model. We derive the precise conditions ensuring the asymptotic stability of the positive equilibrium, with respect to two characteristic parameters of the system. It is shown that for certain values of these parameters, fold or Neimark-Sacker bifurcations occur, but codimension 2 (fold-Neimark-Sacker, double Neimark-Sacker and resonance 1:1) bifurcations may also be present. The direction and the stability of the Neimark-Sacker bifurcations are investigated by applying the center manifold theorem and the normal form theory.