Bifurcation of codimension-one

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. 

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. 

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. 

Surprisingly, even non-rough systems of codimension one may have infinitely many moduli. Of course, since the models of nonlinear dynamics are explicitly defined dynamical systems with a finite set of parameters, this creates a new obstacle which the classical bifurcation theory has not nm into. Although the case of homoclinic loops of codimension one does not introduce any principal problem, nevertheless codimensions two and higher are much less trivial as, for example, in the case of a homoclinic or heteroclinic cycle including a saddle-focus where the structure of the bifurcation diagrams is directly determined by the specific values of the corresponding moduli. 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

The cases where a bifurcation surface of codimension-one is imattainable from either or both sides are typical for multi-dimensional dynamical systems. 

This is the reason why the classification of principal bifurcations in multidimensional systems is not stated in terms of the degree of non-roughness, but it rather focuses on bifurcation sets of codimension-one. 

We can probe even deeper namely, it turns out that for any bifurcation of codimension s at least one of the values M is non-zero at i > s + 1. This implies that the part of the bifurcation set which corresponds to a bifurcation of codimension s is composed of surfaces (joined at /u = 0) of the form... 

Two-dimensional systems having a separatrix loop to a saddle with non-zero first saddle value ao form a bifurcation set of codimension one. Therefore, we can study such homoclinic bifurcations using one-parameter families. 

For the remainder of this section we consider only safe and dangerous stability boundaries of codimension one. This allows us to use only one bifurcation parameter. We therefore assume that at 5 = 0, the system... 

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. 

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. 

Because the defining conditions can be solved for the state variables and two parameters e.g. f and q), the above mentioned varieties are said to be of codimension-2. The dynamic model of the reactive flash contains several algebraic, but only one differential equation, when the holdup and pressure are fixed and the phase equilibrium is instantaneous. Such one-dimensional systems cannot exhibit Hopf bifurcations leading to oscillatory behavior. Therefore, dynamic classification is not necessary. 

Several papers have shown the direct relation between saddle-node bifurcations and voltage collapse problems, e.g., (Canizares and Alvarado, 1993 Canizares, 1995). Saddle-node bifurcations, also known as turning points, are generic codimension one local bifurcations of nonlinear dynamical systems of the form ... 

Suppose now that one is looking for bifurcations with codimension t. One variable only will be necessary to span the phase space. In other words,we shall first consider the interaction of steady states in phase space only in one dimension. In that case the solutions of the steady state equations will move along a line passing through the reference state Xi = 0, which can be projected out on any coordinate... 

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. 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

In Chap. 13 we will consider the bifurcations of a homoclinic loop to a saddle equilibrium state. We start with the two-dimensional case. First of all, we investigate the question of the stability of the separatrix loop in the generic case (non-zero saddle value), as well as in the case of a zero saddle value. Next, we elaborate on the cases of arbitrarily finite codimensions where the so-called Dulac sequence is constructed, which allows one to determine the stability of the loop via the sign of the first non-zero term in this sequence. 

As an example, let us consider the codimension-one bifurcation of three-dimensional systems with a homoclinic loop to a saddle-focus with the negative... 

If the saddle-node L is simple, then all neighboring systems having a saddle-node periodic orbit close to L constitute a codimension-one bifurcational surface. By construction (Sec. 12.2), the function /o depends continuously on the system on this bifurcational surface. Thus, if the conditions of Theorem 12.9 are satisfied by a certain system with a simple saddle-node, they are also satisfied by all nearby systems on the bifurcational surface. This implies that Theorem 12.9 is valid for any one-parameter family which intersects the surface transversely. In other words, our blue sky catastrophe occurs generically... 

This result gives us the last known principal (codimension one) stability boundary for periodic orbits. We will see below (Theorems 13.9 and 13.10) that the other cases of bifurcations of a homoclinic loop lead either to complex dynamics (infinitely many periodic orbits), or to the birth of a single saddle periodic orbit. 

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. 

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 computing algorithms of most of these bifurcations have been well developed and can therefore be implemented in software we mention here the packages designed to settle these bifurcation problems LOCBIF [76], AUTO [46] and CONTENT [83]. The exception is the blue sky catastrophe, Despite the fact that it is a codimension-one boundary, this bifurcation has not yet been found in applications of nonlinear dynamics although an explicit mathematical model does exist [53]. 

One can see that since the constant term is negative, it follows immediately from the Routh-Hurwitz criterion that the origin is an unstable equilibrium state. Furthermore, it may have no zero characteristic roots when a and b are positive. The codimension-2 point (a = b = 0) requires special considerations. We postpone its analysis to the last section, where we discuss the bifurcation of double zeros in systems with symmetry. 

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. 

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. 

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