Codimension-2 bifurcation

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. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

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). 

Subcritical Hopf transitions are found on the segments HM and GL of the Hopf curve and all other transitions are supercritical. The points H and G in figure 8 are located at (< ] = 0.019308, a2 = 0.030686) and ( i = 0.020668, a2 = 0.018330) respectively, and might be called metacritical. They are bifurcations of codimension two so that we expect only isolated metacritical points on the Hopf curve. These have to satisfy not only the conditions of (42), but also ... 

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 ... 

Figure 3.6 shows codimension-1 singularities in a parameter plane spanned by the activation energy of the anodic reaction yA and the activation energy of the electrolyte conductivity yK, when the total cell current I is used as the bifurcation parameter. Hysteresis and isola varieties are found. The nonlinear behavior becomes increasingly complicated for increasing values of yK. Steady-state multiplicities even exist for rather small values of yK, but vanish if the electrolyte s conductivity is temperature-independent, i.e., yK = 0. This confirms that the multiple steady states are caused not by the electrochemical reaction alone, but by the combination of reaction and varying electrical conductivity. 

Fig. 3.6. Codimension-1 singularities of a finite length system for the case of galvanostatic operation (/ is the bifurcation parameter) (a)-(f) point to the parameter values ofy and yA used in the one-parameter continuation of Fig. 3.7.

To summarize the results so far, we plot the bifurcation curves h = +hfr) in the (r,h) plane (Figure 3.6.2). Note that the two bifurcation curves meet tangentially at (r, /i) = (0,0) such a point is called a cusp point. VJe also label the regions that correspond to different numbers of fixed points. Saddle-node bifurcations occur all along the boundary of the regions, except at the cusp point, where we have a codimension-2 bifurcation. (This fancy terminology essentially means that we have had to tune two parameters, h and r, to achieve this type of bifurcation. Un-... 

Classification of catastrophes will be preceded by the centre manifold theorem which is a counterpart to the splitting lemma in elementary catastrophe theory. It will turn out that in the catastrophe theory of dynamical systems such notions of elementary catastrophe theory as the catastrophe manifold, bifurcation set, sensitive state, splitting lemma, codimension, universal unfolding and structural stability are retained. 

The examined physical system always satisfies stipulated additional conditions (an additional stationary point or suitable symmetry) and then the described catastrophe may appear in such systems in a structurally stable way. If the examined system does not fulfil these conditions, the description of the system is inadequate and a possibility of the description of a catastrophe of higher codimension should be considered. For example, such a structurally stable extension of the pitchfork bifurcation is the cusp catastrophe described in Section 5.5.3.2. 

A complete classification of the catastrophes in two state variables of codimension three has not yet been developed. Consequently, only the standard form of a catastrophe corresponding to the sensitive state of the Takens-Bogdanov bifurcation will be given. The generalized Takens--Bogdanov bifurcation has the corresponding standard form ... 

J. Guckenheimer, Multiple bifurcation problems of codimension two , SIAM J. Math. Anal., 15, 1 (1984). 

The stationary state (x2, y2, z2) will be stable when all the roots of equation (6.106) have negative real parts. We will investigate the conditions under which this stationary state loses stability, that is under which at least one solution with a positive real part appears. Next, in the region of control parameters corresponding to instability of the state (x2, y2, z2) we shall examine possible catastrophes of codimension 2. It follows from the classification given in Section 5.5 that the bifurcations of codimension one and two of a sensitive state corresponding to the requirement = 0 are theoretically possible the Hopf bifurcation for which a sensitive state is of... 

First, let us examine the possibility of the appearance of the bifurcations of codimension one and two associated with the sensitive state 2X = 0. Such a sensitive state is represented by equation (6.106a) in which the coefficient C, proportional to the product X1X2X3, is equal to zero. Since the parameter C, owing to inequality (6.106) cannot be zero, C > 0, catastrophes of codimension one and two, having the sensitive state Xt = 0, can be excluded. 

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