Bifurcation of codimension

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 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. 

Such a situation will henceforth be referred to as a a bifurcation of codimension A , and the surface 9Jl is called a bifurcation surface of codimension k (the codimension is equal to the number of the governing parameters). 

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... 

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. 

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. 

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]. 

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. 

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 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). 

Ringland, J. Rapid reconnaissance of a model of a chemical oscillator by numerical continuation of a bifurcation feature of codimension-2. J. Chem. Phys. 1991, 95, 555-562. 

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... 

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. 

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. 

The significance of higher degeneracies (starting from codimension three) in the linear part is that the effective normal forms become three-dimensional, and may, as a result, exhibit complex dynamics, the so-called instant chaos, even in the normal form itself. Such examples include the normal forms for a bifurcation of an equilibrium state with a triplet of zero characteristic exponents, and a complete or incomplete Jordan block, in which there may be a spiral strange attractor [18], or a Lorenz attractor [129], respectively (the latter case requires an additional symmetry). Since we will focus our considerations only on simple dynamics, we do not include these topics in this book. 

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. 

The case of zero saddle value was considered by E. A. Leontovich in 1951. Her main result is presented in Sec. 13.3, rephrased in somewhat different terms in the case of codimension n (i.e. when exactly the first (n — 1) terms in the Dulac sequence are zero) not more than n limit cycles can bifurcate from a separatrix loop on the plane moreover, this estimate is sharp. 

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