Bifurcation of codimension-two

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 ... 

We study some homoclinic bifurcations of codimension two in Secs. 13.6. In Sec. 13.7, we review the results on the bifurcations of a homoclinic-8, and on the simplest heteroclinic cycles. 

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. 

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. 

A straightforward generalization of two-dimensional bifurcations was developed soon after. So were some natural modifications such as, for instance, the bifurcation of a two-dimensional invariant torus from a periodic orbit. Also it became evident that the bifurcation of a homoclinic loop in high-dimensional space does not always lead to the birth of only a periodic orbit. A question which remained open for a long time was could there be other codimension-one bifurcations of periodic orbits Only one new bifurcation has so far been discovered recently in connection with the so-called blue-sky catastrophe as found in [152]. All these high-dimensional bifurcations are presented in detail in Part II of this book. 

One must bear in mind, however, that a truncated normal form does not always guarantee a complete reconstruction of the dynamics of the original system. For instance, when the truncated normal forms possess additional symmetries, these symmetries are, in principle, broken if the omitted higher-order terms are taken back into account, and this can even lead to an onset of chaos in some regions of the parameter space. These regions are extremely narrow near a bifurcation point of codimension two but their size may expand rapidly as we move away from the bifurcation point over a finite distance. 

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]... 

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

Fig. 11.1.2. The bifurcation surface XfV of codimension two is a curve in a three-parameter family.

We can now discuss the bifurcation of an equilibrium state with two zero characteristic exponents. This bifurcation is worth being distinguished because its analysis includes nearly all bifurcations of codimension one. 

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

In general, the bifurcation of a homoclinic butterfly is of codimension two. However, the Lorenz equation is symmetric with respect to the transformation (x y z) <-)> (—X, —y z). In such systems the existence of one homoclinic loop automatically implies the existence of another loop which is a symmetrical image of the other one. Therefore, the homoclinic butterfly is a codimension-one phenomenon for the systems with symmetry. 

Case (a) corresponds to a codimension-three bifurcation, while Cases (b) and (c) are of codimension four. However, if the system exhibits some symmetry, then all of the above three bifurcations reduce to codimension two. It was established in [126, 127, 129] that a symmetric homoclinic butterfly with either a = 0 or A = 0 appears in the so-called extended Lorenz model, and in the Shimizu-Morioka system, as well as in some cases of local bifurcations of codimension three in the presence of certain discrete symmetries [129]. 

This bifurcation has codimension two the governing parameters (/ii,/X2) are chosen here to be the coordinates of the point of intersection of the onedimensional unstable separatrix of 0 with some cross-section transverse to the one-dimensional stable separatrix of the other saddle O2. Since the... 

The point NS. This point is of codimension two as <7 = 0 here. The behavior of trajectories near the homoclinic-8, as well as the structure of the bifurcation set near such a point depends on the separatrix value A (see formula (13.3.8)). Moreover, they do not depend only on whether A is positive (the loops are orientable) or negative (the loops are twisted), but it counts also whether A is smaller or larger than 1. If A < 1, the homoclinic-8 is stable , and unstable otherwise. To find out which case is ours, one can choose an initial point close sufficiently to the homoclinic-8 and follow numerically the trajectory that originates from it. If the figure-eight repels it (and this is the case in Chua s circuit), then A > 1. Observe that a curve of double cycles with multiplier 4-1 must originate from the point NS by virtue of Theorem 13.5. 

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. 

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. 

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