Generalized Hopf Bifurcation Research Articles

Bifurcation at Infinity in Polynomial Vector Fields

We study here the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity: this bifurcation is a generalized Hopf bifurcation from the point at infinity. We start with the general theory and then specialize to the particular case of cubic polynomial systems for which we study the simultaneous bifurcation of limit cycles from the origin and from the equator. We finally discuss the transformation of a weak focus at infinity into a finite weak focus.

Hopf bifurcation withD 3 �D 3-symmetry

A generic Hopf bifurcation involving an eight dimensional center eigenspace is considered for systems possessing aD 3 ×D 3-symmetry. This kind of Hopf bifurcation can occur in systems of three interacting groups of oscillators, where each group itself is composed of three individual oscillators. The terminology “micro” and “Macro” is introduced here to denote symmetry operations acting on individual oscillators and on the whole groups, respectively. The normal form for the Hopf bifurcation admits 11 distinct periodic solutions with maximal isotropy subgroups. These are classified and their branching-types and stabilities are determined in terms of the cubic and relevant quintic coefficients of the normal form. The symmetry properties of these solutions when only certain Macro variables in the oscillator groups are observed are discussed in the context of the remaining symmetry. Furthermore, the relation of the normal form to the corresponding one for a singleD 3-symmetry is established by restricting the system to four dimensional fixed point subspaces associated with submaximal isotropy subgroups. Based on this information the possibility of quasiperiodic solutions and of a particular class of heteroclinic cycles is discussed.

Bifurcations in a short-pulse free-electron laser oscillator

With one-dimensional Maxwell-Lorentz equations, we have investigated the bifurcations in a short-pulse free-electron laser oscillator. A generic Hopf bifurcations and chaotic transition via period-doubling cascade have been described in the parametric space of three branching parameters: the cavity detuning parameter D, the slippage parameter S, and the superradiant parameter K. The present theory of the period of limit-cycle oscillation in the radiation power is well consistent with recent experimental results.

Sensitivity analysis of certain dynamic bifurcations

Control systems and other dynamic systems may lose stability at critical points from which bifurcations take place. However, due to the existence of small disturbances or imperfections, the dynamic behaviour of a system—in particular, the stability conditions—may change drastically. This paper is concerned with the imperfection sensitivity of dynamic bifurcations (degenerate or generalized Hopf bifurcations) arising in nonlinear autonomous systems. The analysis is carried out via the unification technique coupled with the intrinsic harmonic balancing procedure. It has been observed that the sensitivity properties associated with dynamic bifurcations are inherently different from those associated with static bifurcations. However, some similarities concerning the sensitivity curves do exist. A nonlinear feedback control system is analyzed to demonstrate the applicability of the analytic results.

Generic Hopf bifurcation in a class of integro-differential equations

Generic Hopf bifurcation in a class of integro-differential equations

Branch Inclusion in Generic Hopf Bifurcation: General Approach

AbstractBy using averaging transformations we determine approximations of higher order for the family of bifurcating periodic solutions in the generic case of Hopf bifurcation. These approximations permit the application of the Newton‐Kantorovič Theorem to derive inclusions of the amplitude and period.

Branch inclusion in generic Hopf bifurcation: The case of scaled van der Pol's equation

Asymptotic expansions of the amplitude and the primitive period to the family of periodic solutions in the case of generic Hopf bifurcation are well known. In this paper, by means of an averaging transformation we construct an approximation to the bifurcating family of periodic solutions in the case of scaled van der Pol's equation whose order of approximation permits the application of the Newton-Kantorovič theorem to get an error estimate to the corresponding asymptotic expansions of the amplitude and the primitive period.

Hopf bifurcation with non-semisimple 1:1 resonance

A generalised Hopf bifurcation, corresponding to non-semisimple double imaginary eigenvalues (case of 1:1 resonance), is analysed using a normal form approach. This bifurcation has linear codimension-3, and a centre subspace of dimension 4. The four-dimensional normal form is reduced to a three-dimensional system, which is normal to the group orbits of a phase-shift symmetry. There may exist 0, 1 or 2 small-amplitude periodic solutions. Invariant 2-tori of quasiperiodic solutions bifurcate from these periodic solutions. The authors locate one-dimensional varieties in the parameter space 1223 on which the system has four different codimension-2 singularities: a Bogdanov-Takens bifurcation a 1322 symmetric cusp, a Hopf/Hopf mode interaction without strong resonance, and a steady-state/Hopf mode interaction with eigenvalues (0, i,-i).

On the dynamics and stability of a codimension-3 autonomous system

Bifurcation and stability properties of a nonlinear, autonomous, codimension-3 system are studied. Attention is focused on the vicinity of a compound critical point at which the Jacobian has a double zero eigenvalue of index one and a pair of pure imaginary eigenvalues. A unification technique and harmonic balancing procedure yield simplified differential equations which govern the local dynamics. Thus, equilibrium solutions associated with generic bifurcations, Hopf bifurcations, secondary bifurcations, and bifurcations into quasi-periodic motions on invariant tori are discussed for the first time. Criticality conditions associated with various bifurcations as well as stability conditions are derived in explicit terms. An illustrative example is analyzed.

Degenerate Hopf Bifurcation and Nerve Impulse. Part II

The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...

About Generalized Hopf Bifurcation for Singularly Perturbed Systems

About Generalized Hopf Bifurcation for Singularly Perturbed Systems

Generalized Hopf Bifurcation and Its Dual Generalized Homoclinic Bifurcation

We show the duality between the generalized Hopf bifurcation (GHB) and the generalized homoclinic bifurcation (GHB*). This duality is twofold: (1) the Poincare normal forms at a weak focus and at a weak saddle, (2) the bifurcation diagrams of the GHB and the GHB*. Since the GHB is well known, the GHB* may be considered as the heart of the article.

Generalized Hopf bifurcations and applications to planar quadratic systems

Generalized Hopf bifurcations and applications to planar quadratic systems

Smoothness of bounded solutions of nonlinear evolution equations

It is shown that in many cases globally defined, bounded solutions of evolution equations are as smooth (in time) as the corresponding operator, even if a general solution of the initial-value problem is much less smooth; i.e., initial values for bounded solutions are selected in such a way that optimal smoothness is attained. In particular, solutions which bifurcate from certain steady states, such as periodic orbits, almost-periodic orbits and also homo- and heteroclinic orbits, have this property. As examples, a neutral functional differential equation, a slightly damped non-linear wave equation, and a heat equation are considered. In the latter case the space variable is included into the discussion of smoothness. Finally, generalized Hopf bifurcation in infinite dimensions is considered. Here smoothness of the bifurcation function is discussed and known results on the order of a focus are generalized.

On Generalized Hopf Bifurcations

Two distinct degenerate Hopf bifurcation phenomena associated with autonomous lumped-parameter systems are explored in great detail via the intrinsic harmonic balancing method. It is assumed that the Hopf’s transversality condition is violated and certain other conditions prevail. In one of the cases, the system exhibits a cusp shape bifurcation path which exists for either positive or negative values of the system parameter. On the other hand, the second case is concerned with a tangential bifurcation phenomenon which may not be exhibited unless an additional condition is satisfied. This existence condition is obtained in the course of analysis. The distinctive feature of the paper is that the results concerning the bifurcating paths and limit cycles are given in general, explicit forms which are expected to be very useful in a variety of applications.

Quasi-invariant manifolds, stability, and generalized Hopf bifurcation

Viene compiuta un'analisi completa del problema della biforcazione di Hopf relativa ad arbitrarie piccole perturbazioni del secondo membro di un'equazione differenziale in Rn, p=f0(p). Gli autovalori di f′0(O) soddisfano una condizione di non risonanza. I risultati sono forniti in termini delle proprietá di stabilità di un sistema dinamico piano convenientemente associato all'equazione imperturbata.

Attractivity properties of closed orbits in a case of generalized Hopf bifurcation

Attractivity properties of closed orbits in a case of generalized Hopf bifurcation

Nonresonant periodic perturbation of the hopf bifurcation

In this paper the following periodically perturbed autonomous system in which g is 2π/υ-periodic in t and is considered. It is assumed that tne autonomous system obtained by setting b = 0 undergoes a generic Hopf bifurcation from x =0 at η =0 and that the frequency, ω, is not in resonance with the frequency of the Hopf periodic solution. Under these conditions, we show that an integral manifold of solutions bifurcates from the unique 2π/ω-periodic solutionx∗, of (1) on one side of the neutral stability curve for x∗ in the (b,η) plane.

Generalized hopf bifurcation and h-asymptotic stability

Generalized hopf bifurcation and h-asymptotic stability

Generalized Hopf bifurcation in Hilbert space

AbstractWe consider a family of semilinear evolution equations in Hilbert space of the form equation image 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 equation image for some m ε N or equation image 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.