Cusp bifurcation

Kinetic equation (6.35) will be rearranged to a simpler form using the substitution x = y + 1/3 and obtaining a standard (gradient) form of the cusp bifurcation... 

As /i increases within a resonant zone other periodic orbits with the same rotation number M/N may appear. In some cases, the boundary of the resonant zone can lose its smoothness at some points, like in the example shown in Fig. 11.7.4 here, the resonant zone consists of the union of two regions D and Z>2 corresponding to the existence of, respectively, one and two pairs of periodic orbits on the torus. The points C and C2 in Fig. 11.7.4 correspond to a cusp-bifurcation. At the point S corresponding to the existence of a pair of saddle-node periodic orbits the boundary of the resonant zone is non-smooth. 

These equations represent a parametric curve with parameter y. From a set of values of the parameter y it is possible to draw a curve in the xq — yo plane, so we obtain a bifurcation curve as a function of parameter y. This curve with a cusp point can be considered as the border that dividing the plane xq — yo into domains with one and three equilibrium states respectively. 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

It should be noted that if n is of the order 2, this is a cusp manifold (which is one which will show bifurcation behavior of the form discussed by Shore and Comins ). The analysis of this syst n presented by Ferrini et al. shows that this is indeed a system with multiple equilibrium states. Having this form for the potential, there exists a Fokker-Planck equation for the... 

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-... 

There is one last way to plot the results, which may appeal to you if you like to picture things in three dimensions. This method of presentation contains all of the others as cross sections or projections. If we plot the fixed points jt above the (r,/z) plane, we get the cusp catastrophe surface shown in Figure 3.6.5. The surface folds over on itself in certain places. The projection of these folds onto the (r,h) plane yields the bifurcation curves shown in Figure 3.6.2. A cross section at fixed h yields Figure 3.6.3, and a cross section at fixed r yields Figure 3.6.4. 

Figure 23. A calculated two-parameter bifurcation diagram for the formic acid model [Eq. (15)] showing the locations of the saddle-node (solid line), Hopf (dashed line), and saddle-loop bifurcations (dotted-dashed line). All three curves meet in a Takens-Bogdanov point close to the cusp. (Reprinted with permission from P. Strasser, M. Eiswirth and G. Ertl, J. Chem. Phys. 107, 991-1003, 1997. Copyright 1997 American Institute of Physics.)...

Fig. 19. Catastrophe surface M3, singularity set I3 and bifurcation set B3 of the cusp catastrophe (/t3).

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. 

Fig. 6.7. Bifurcation diagrams for stationary solutions. Detuning 5 and coupling strength 77 are plotted. The figures differ by the propagation phase parameter cp. Hopf bifurcations are denoted by dashed lines and saddle-nodes by solid lines. ZH — Zero-Hopf bifurcation CP — cusp GH — generalized Hopf bifurcation (where the Hopf bifurcation changes its criticality)...

Fig. 11.1 Two-dimensional bifurcation diagram calculated by continuation from the Citri-Epstein mechanism for the chlorite-iodide system. Plot of 2D space of constraints is chosen to be the ratio of input concentrations, [CIO ]o/[I ]o> versus the logarithm of the reciprocal resideuce time, logfco-Notation SSI, SS2, region of steady states with high [1 ] and low [I ], respectively Osc, region of periodic oscillations Exc, region of excitahihty snl, sn2, curves of saddle-node bifurcations of steady states hp, curve of Hopf bifiucatious sup, ciuve of saddle-node bifurcations of periodic orbits swt, swallow tail (a small area of tristability) C, cusp point TB, Takens-Bogdanov point (terminus of hp on snl). (From [5].)...

Actually there are two cusps which can interact in some cases (204) to form a higher order catastrophe called butterfly catastrophe because of the fancy shape of its bifurcation. 

The only variable in equation (8.11) is x, the electro-chemical potential difference. Equation (8.11) describes an MEA with ideal transport but reaction kinetic limitations. Let ro2 s) be monotone and rMe x) have timax local maxima, then the equation has at most timax + 1 solution branches rMe x). ro2 and rMe approximates for large and small ratios a an ideal ion source for the other - hence the IV urves for the extreme values of a are close to the electron reaction rate of rMe and, ro2 For intermediate values of a branches separated by cusps, complete the generic situation for one local maximum in one of the reaction rates (Figure 8.6). The study of the parameter dependence of the bifurcations is an additional source of information to verify kinetic models and related parameters. 

Following the method of Lorenz, one may obtain a quasi-one-dimensional map from successive local maxima Z (w = 1,2,...) of Z t), The result is a uni-modal map as shown in Fig. 7.17. Unlike the classical Lorenz chaos, the map seems to have a smooth maximum instead of a cusp structure. The route to chaos, if seen on the map, shows no difference from the usual period-doubling type except that the present map may not have a quadratic maximum the splitting of Ml into L/+1 and L/+i appears on the map as the bifurcation of 2-point cycles from a 2 -point cycle, and the mutual contact of L/4.1 and Z/+i at the saddle... 

Figure 1 The canonical cusp catastrophe function, Az = -f + 6x, at different values of the parameter a. The left panel illustrates the symmetric behavior for h = 0 the right panel for 6 = 1 illustrates the asymmetric behavior which occurs whenever b is different from zero. In the horizontal plane (a, x) we have drawn the locus of maxima as dotted lines, of minima as dashed lines and of inflections as dashed-dotted lines. The triply degenerate (or catastrophe) point occurs at a = 0 and 6 = 0 and bifurcation between single and double well modes occurs there for symmetric systems. For asymmetric systems, the bifurcation occurs on the fold line (at a = —1.9 for 6 = 1) where the cusp function has a doubly degenerate horizontal inflection point at x = 0.8.

The trajectories on the (x,y) plane do not exhibit equilibrium points for p<0, see top left of Fig. 7.4. For p = 0,sl cusp is observed in the proximity of the (0, 0) degenerate point, at which the bifurcation takes place. For p> 0 the typical phase portraits of a saddle and a vortex are observed. Homoclinic orbits, or trajectories, of the flow of this dynamical system join a saddle equilibrium point to itself. 

Fig. 7.4 On the left [right], supercritical [subcritical] saddle-centre bifurcation, Eq. (7.65) [Eq. (7.66)], for /t = —1,0, +1, from top to bottom. Regular orbits are blue. The homoclinic trajectories through saddle points and the trajectories joining at the cusp, i.e., at the degenerate critical point corresponding to the bifurcation, are red...

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