Codimension two bifurcations

INTERACTION OF SHALLOW CELLS CELLULAR DYNAMICS Evolution of Shallow Cells The Role of Codimension Two Bifurcations. The importance of nonlinear interactions between spatially resonant structures is... 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

A periodically forced system may be considered as an open-loop control system. The intermediate and high amplitude forced responses can be used in model discrimination procedures (Bennett, 1981 Cutlip etal., 1983). Alternate choices of the forcing variable and observations of the relations and lags between various oscillating components of the response will yield information regarding intermediate steps in a reaction mechanism. Even some unstable phase plane components of the unforced system will become apparent through their role in observable effects (such as the codimension two bifurcations described above where they collide and annihilate stable, observable responses). 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

The three standard local codimensional-one bifurcations are the saddle-node, Hopf, and period doubling bifurcations and several have been continued numerically for this model and appear in figure 2. We have chosen not to show the curves of focus-node transitions because they do not represent any changes in stability, only changes in the approach to the steady behaviour. The saddle-node bifurcations that occur during phase locking of the torus at low amplitudes continue upward and either close upon themselves as in the case of the period 3 resonance horns or the terminate in some codimension-two bifurcation. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

Several codimension-two bifurcations have already been mentioned. Although they occur in restricted subspaces of parameter space and would therefore be difficult to locate experimentally, their usefulness lies in their role as centres for critical behaviour. Emanating from each local codimen-sion-two point will be two or more of the above codimension-one bifurcation curves. Their usefulness in studying dynamics is akin to that of the triple point in thermodynamic phase equilibria in which boundaries between three different phases come together at a point in a two-parameter diagram. Because some of these codimension-two points have been studied and classified analytically, finding one can provide clues about what other codimension-one bifurcation curves to expect near by and thus aids in the continuation of all of the bifurcation curves in the excitation diagram. 

Let us now come to bifurcations which can be formulated in a local way. An interesting class are the codimension two bifurcations, whereby two control parameters are varied. Consider, for instance, the most general form, also known as normal form, of equations involving two variables near a doubly degenerate critical eigenvalue of the linear stability operator3 ... 

The stationary bifurcation and the Hopf bifurcation typically occur as one parameter is varied and are therefore known as codimension-one bifurcations. They represent the generic ways in which a steady state of a two-variable system can become unstable. It is sometimes possible to make the stationary and Hopf instability threshold coalesce by varying two parameters. Such an instability, where T = A = 0, is known as a Takens-Bogdanov bifurcation or a double-zero bifurcation, since Ai = A.2 = 0 at such a point [175], This bifurcation is a codimension-two bifurcation, since it requires the fine-tuning of two system parameters. 

Greenberg, J. M. (1980) Spiral waves for A - w systems. SIAM J. Appl. Math. 39, 301 Guckenheimer, J. (1975) Isochrons and phaseless sets. J. Math. Biol. 1, 259 Guckenheimer, J. (1981) On codimension two bifurcations. In Dynamical Systems and Turbulence, Warwick 1980, ed. by D. A. Rand, L. S. Young, Lecture Notes in Math. (Springer, Berlin, Heidelberg, New York) p. 99... 

From the eigenvalue analysis of RW states, it has been possible to completely specify the codimension-two bifurcation to linear order. To gain a deeper understanding of the organizing center, one needs to understand the codimension-two bifurcation to higher order. For this one turns to a weakly nonlinear analysis, which is the subject of the next section. 

There is a problem, however, which makes this approach both difficult and interesting. The particular codimension-two bifurcation found for the spiral waves results from the interaction of Hopf and translational eigenmodes of a rotating wave, and existing bifurcation theory cannot be applied to such a case. The reason is that a theory has not been developed for noncompact symmetries such as translations. While translational symmetries are treated in many systems, this is done by requiring solutions to be spatially periodic and spiral waves are not (globally) periodic in space. Hence, it turns out that the codimension-two bifurcation at the heart of spiral meandering is of a fundamentally new type from the point of view of nonlinear dynamical systems. 

While a rigorous derivation of a normal form will have to await advances in symmetric bifurcation theory, I will nevertheless proceed by proposing a lowdimensional model which contains the same codimension-two bifurcation as was observed in the reaction-diffusion equations, and I will show that this model contains the essence of spiral meandering. 

The simplest low-order expansions for f,g, and h sufficient to give the desired codimension-two bifurcation, and which yield bounded trajectories for the model, are ... 

Throughout this chapter I have taken the point of view that the meandering of spiral waves in excitable media can and should be examined from the perspective of bifurcation theory. With this approach, it has been possible to show that the organizing center for spiral dynamics is a particular codimension-two bifurcation resulting from the interaction of a Hopf bifurcation from rotating waves with symmetries of the plane. From this observation has followed a simple ordinary-differential-equation model for spiral meandering. 

Section 13.6 discusses three main cases of codimension-two bifurcations of a homoclinic loop to a saddle. These cases were selected by Shilnikov in [138]... 

In this and the following sections we will review some codimension-two bifurcations of homoclinic loops and heteroclinic cycles which occur in various models. 

We will analyze the following three cases of codimension-two bifurcations of such homoclinic loops. 

Codimension-two bifurcations of homoclinic loops are essential for studying bifurcation phenomena in the Lorenz equation... 

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