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)

The smallest bimolecular mass-action system with a vertical Andronov–Hopf bifurcation

We present a three-dimensional differential equation, which robustly displays a degenerate Andronov–Hopf bifurcation of infinite codimension, leading to a center, i.e., an invariant two-dimensional surface that is filled with periodic orbits surrounding an equilibrium. The system arises from a three-species bimolecular chemical reaction network consisting of four reactions. In fact, it is the only such mass-action system that admits a center via an Andronov–Hopf bifurcation

Interaction of Hopf and period doubling bifurcations of a vibro-impact system

International audienceAn inertial shaker as a vibratory system with impact is considered. By means of differential equations, periodicity and matching conditions, the theoretical solution of periodic n-1 impacting motion can be obtained and the Poincaré map is established. Dynamics of the system are studied with special attention to interaction of Hopf and period doubling bifurcations corresponding to a codimension-2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses -1 simultaneously for the Jacobi matrix. The four-dimensional map can be reduced to a three-dimensional normal form by the center manifold theorem and the theory of normal forms. The two-parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated. It is proved that there exist curve doubling bifurcation (a torus doubling bifurcation), Hopf bifurcation of 2–2 fixed points as well as period doubling bifurcation and Hopf bifurcation of 1–1 fixed points near the critical point. Numerical results indicate that the vibro-impact system presents complicated and interesting curve doubling bifurcation and Hopf bifurcation as the two controlling parameters vary

Dissipative periodic and chaotic patterns to the KdV--Burgers and Gardner equations

We investigate the KdV-Burgers and Gardner equations with dissipation and external perturbation terms by the approach of dynamical systems and Shil'nikov's analysis. The stability of the equilibrium point is considered, and Hopf bifurcations are investigated after a certain scaling that reduces the parameter space of a three-mode dynamical system which now depends only on two parameters. The Hopf curve divides the two-dimensional space into two regions. On the left region the equilibrium point is stable leading to dissapative periodic orbits. While changing the bifurcation parameter given by the velocity of the traveling waves, the equilibrium point becomes unstable and a unique stable limit cycle bifurcates from the origin. This limit cycle is the result of a supercritical Hopf bifurcation which is proved using the Lyapunov coefficient together with the Routh-Hurwitz criterion. On the right side of the Hopf curve, in the case of the KdV-Burgers, we find homoclinic chaos by using Shil'nikov's theorem which requires the construction of a homoclinic orbit, while for the Gardner equation the supercritical Hopf bifurcation leads only to a stable periodic orbit.Comment: 13 pages, 12 figure

Small aspect ratio Taylor-Couette flow: onset of a very-low-frequency three-torus state

The nonlinear dynamics of Taylor-Couette flow in a small aspect ratio annulus (where the length of the cylinders is half of the annular gap between them) is investigated by numerically solving the full three-dimensional Navier-Stokes equations. The system is invariant to arbitrary rotations about the annulus axis and to a reflection about the annulus half-height, so that the symmetry group is SO(2)×Z2. In this paper, we systematically investigate primary and subsequent bifurcations of the basic state, concentrating on a parameter regime where the basic state becomes unstable via Hopf bifurcations. We derive the four distinct cases for the symmetries of the bifurcated orbit, and numerically find two of these. In the parameter regime considered, we also locate the codimension-two double Hopf bifurcation where these two Hopf bifurcations coincide. Secondary Hopf bifurcations (Neimark-Sacker bifurcations), leading to modulated rotating waves, are subsequently found and a saddle-node-infinite-period bifurcation between a stable (node) and an unstable (saddle) modulated rotating wave is located, which gives rise to a very-low-frequency three-torus. This paper provides the computed example of such a state, along with a comprehensive bifurcation sequence leading to its onset.Postprint (published version

Hopf bifurcation analysis and control of three-dimensional Prescott neuron model

Neurons generate different firing patterns because of different bifurcations in the dynamical viewpoint. Various nerve diseases are relevant to the bifurcation of nervous system. Therefore, it is vital to control bifurcation since it may be potential ways of treating nerve diseases. This paper focuses on the critical Hopf bifurcation analysis and the problem of Hopf bifurcation control. We investigate the effects of key parameters on critical Hopf bifurcation and obtain the Hopf bifurcation occurrence region on parameter plane. With the theory of high-dimensional Hopf bifurcation, we analytically deduce the judgement criteria of Hopf bifurcation type for the three-dimensional models and judge the Hopf bifurcation type of Prescott model by using it. With application of the Washout filter, the subcritical Hopf bifurcation of Prescott model is controlled and converted to supercritical Hopf bifurcation. In addition, we make some discussions on Hopf bifurcation analysis of a coupled neural network. The results provided in this paper could bring new ways to controlling neurological diseases