Numerical solution of Hopf bifurcation problems

We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ?. It has isolated solutions in the ?-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.     

Hopf bifurcation for non-densely defined Cauchy problems

In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille?CYosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.     

Symbolic software development for computing the normal form of double Hopf bifurcation

A method is presented for computing the normal form of double Hopf bifurcation associated with non-semi-simple 1:1 resonance. The method combines the normal form theory and the center manifold theory to deal with a general n-dimensional system. Explicit recursive formulas are derived for the coefficients of the normal form and the associated nonlinear transformations. User-friendly computer software is developed using a symbolic computation language Maple. This enables one to easily compute the normal form and nonlinear transformation for a given specific problem up to an arbitrary order. An illustrative example is given to show the applicability of the method and the convenience of the software.     

Stability and Hopf bifurcation analysis of a new system

In this paper, a new chaotic system is introduced. The system contains special cases as the modified Lorenz system and conjugate Chen system. Some subtle characteristics of stability and Hopf bifurcation of the new chaotic system are thoroughly investigated by rigorous mathematical analysis and symbolic computations. Meanwhile, some numerical simulations for justifying the theoretical analysis are also presented.     

A stochastic Hopf bifurcation

Summary Let {x _t :t0} be the solution of a stochastic differential equation (SDE) in ^d which fixes 0, and let denote the Lyapunov exponent for the linear SDE obtained by linearizing the original SDE at 0. It is known that, under appropriate conditions, the sign of controls the stability/instability of 0 and the transience/recurrence of {x _t :t0} on ^d \{0}. In particular if the coefficients in the SDE depend on some parameterz which is varied in such a way that the corresponding Lyapunov exponent ^z changes sign from negative to positive the (almost-surely) stable fixed point at 0 is replaced by an (almost-surely) unstable fixed point at 0 together with an attracting invariant probability measure ^z on ^d \{0}. In this paper we investigate the limiting behavior of ^z as ^z converges to 0 from above. The main result is that the rescaled measures (1/ ^z ) ^z converge (in an appropriate weak sense) to a non-trivial -finite measure on ^d \{0}.Research supported in part by Office of Naval Research contract N00014-91-J-1526     

Approximation of Hopf bifurcation

Summary We make several assumptions on a nonlinear evolution problem, ensuring the existence of a Hopf bifurcation. Under a fairly general approximation condition, we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions.     

An elementary approach to a generalized Hopf bifurcation

In the case of a generalized Hopf bifurcation several periodic solutions may branch off from the equilibrium. An elementary procedure is presented for establishing all those bifurcating solutions, as well as their stability behaviour, provided a certain non-degeneracy condition is satisfied.

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Zusammenfassung Im Falle einer verallgemeinerten Hopf-Verzweigung können mehrere periodische Lösungen von der Gleichgewichtslage abzweigen. Es wird ein elementares Verfahren vorgestellt, welches erlaubt, unter einer gewissen Nichtentartungsbedingung diese kleinen periodischen Lösungen sowie ihre Stabilität zu bestimmen.

     

An analytical method for computing Hopf bifurcation curves in neural field networks with space-dependent delays

We give an analytical parametrization of the curves of purely imaginary eigenvalues in the delay-parameter plane of the linearized neural field network equations with space-dependent delays. In order to determine if the rightmost eigenvalue is purely imaginary, we have to compute a finite number of such curves; the number of curves is bounded by a constant for which we give an expression. The Hopf bifurcation curve lies on these curves.     

Hopf bifurcation for functional equations

Using a “concrete” representation for the adjoint, the spectrum of the class of linear transformations T which are bounded on L^p(?∞, ∞), 1 < p < ∞ into itself and which satisfy the functional equation Tt(a) = m(a) t(a), ?∞ < a < ∞, a ≠ 0, where m(a) = 1 or (sgn a) and where (t(a)f)(x) = f(ax), is studied.     

A NEW DETECTING METHOD FOR CONDITIONS OF EXISTENCE OF HOPF BIFURCATION

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A Hopf bifurcation with spherical symmetry

Summary We provide a theoretical analysis of a Hopf bifurcation that can occur in systems with spherical geometry, based on the general theory of Hopf bifurcation in the presence of symmetry. In this particular bifurcation the imaginary eigenspace is a direct sum of two copies of the 5-dimensional irreducible representation of the groupSO(3). The same bifurcation has been studied by looss and Rossi (1988), using extensive computer-assisted calculations. Here we describe a simpler and more conceptual approach in which the representation ofSO(3) is realised as its conjugation action on the space of symmetric traceless 3 × 3 matrices. We prove the generic existence of five types of symmetry-breaking oscillation: two rotating waves and three standing waves. We analyse the stabilities of the bifurcating branches, describe the restrictions of the dynamics to various fixed-point spaces of subgroups ofSO(3), and discuss possible degeneracies in the stability conditions.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Hopf bifurcation in structural population models

We study a nonlinear PDE problem motivated by the peculiar patterns arising in myxobacteria, namely, counter‐migrating cell density waves. We rigorously prove the existence of Hopf bifurcations for some specific values of the parameters of the system. This shows the existence of periodic solutions for the systems under consideration. Copyright © 2016 John Wiley & Sons, Ltd.     

On the Hopf bifurcation for flows

Under fairly general hypotheses, we prove the existence of the families of periodic orbits obtained by Hopf bifurcation, with emphasis on their smoothness. A Banach version of a theorem of Lyapounov is obtained as a corollary. The proofs are complete, simple and original. To cite this article: M. Chaperon, S. López de Medrano, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Hopf bifurcation of the sunflower equation

In this paper, the problem of Hopf bifurcation of the sunflower equation is discussed on the basis of the newly proposed pseudo-oscillator analysis. The bifurcated periodic solution with different parameters is analyzed and compared with some results given in the literature by using the center manifold reduction and the Normal Form theory. It shows that the pseudo-oscillator analysis involves easier computation and yields better prediction on the bifurcated periodic solutions than the Norma Form theory.