Codimension one

If U = 5 the theorem is true. Let s assume U 5. Note that for every irreducible closed subset V of codimension one in 5, letting v,s be the generic points of V and of 5 respectively, we have ... 

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. 

In this section we will develop the phase-space structure for a broad class of n-DOF Hamiltonian systems that are appropriate for the study of reaction dynamics through a rank-one saddle. For this class of systems we will show that on the energy surface there is always a higher-dimensional version of a saddle (an NHIM [22]) with codimension one (i.e., with dimensionality one less than the energy surface) stable and unstable manifolds. Within a region bounded by the stable and unstable manifolds of the NHIM, we can construct the TS, which is a dynamical surface of no return for the trajectories. Our approach is algorithmic in nature in the sense that we provide a series of steps that can be carried out to locate the NHIM, its stable and unstable manifolds, and the TS, as well as describe all possible trajectories near it. 

The classification of sensitive states and the standard forms of dynamical systems describing a system in the vicinity of a specified sensitive state will be given. It is difficult to overestimate the efforts of mathematicians which have led them to classifying the standard forms of codimension one, two and (partly) three. Following the determination of the sensitive state of a given dynamical system, the nature of dynamics of catastrophes of this system may be inferred from knowledge of a suitable standard form. 

We will now discuss the shape of standard forms for the catastrophes of codimension one. These will be standard forms for which the function f(x c),... 

There are also catastrophes of codimension one not leading to a qualitative change of the stationary state. These are catastrophes of a global type. An example of the sensitive state corresponding to a global catastrophe is the state = A2. The phase trajectories for such sensitive states in linear systems were given in Section 5.2. 

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. 

Since Im < , Im have such large codimension, one certainly expects that for most g and (g - l)-dimensional principally polarized abelian varieties X, Y, P(Y) and <2>(X) would be disjoint. 

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. 

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 ... 

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. 

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]. 

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. 

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. 

All non-rough two-dimensional systems in a small neighborhood of a system with first-order of non-roughness are now known to form a surface of codimension-one. Moreover, due to Leontovich and Mayer, we know that all of them are identical in the sense that they have an identical topological... 

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

There are some other occurrences of moduli in structurally unstable three-dimensional systems of codimension-one with simple dynamics. For example, consider a three-dimensional system with a saddle-focus O and a saddle periodic orbit L. Let i 2 = p iu), and A3 be the characteristic roots at O such that /o < 0, cj > 0, A3 > 0, i.e. assume the saddle-focus has type (2,1) let i/ < 1 and I7I > 1 be the multipliers of the orbit L. Let one of the two sepa-ratrices P of O tend to L as t -> +00, i.e. T W[, as shown in Fig. 8.3.2. This condition gives the simplest structural instability. All nearby systems with similar trajectory behavior form a surface B of codimension-one. Belogui [28] had found that the value... 

The main idea of this approach is the following to a non-rough system X o some CO dimension k can be assigned. In the case of a finite degeneracy, the codimension k is identified with k equality-like conditions and a finite number of conditions of inequality type. Hence, Xeo is considered as a point on some Banach submanifold of codimension k in the space of dynamical systems. In other words, we have k smooth functionals defined in a neighborhood of Xeq whose zero levels intersect at B. In general, the inequality-like conditions secure the smoothness of B. In the case of codimension one Sotomayor [144, 145] had proved the smoothness of these functionals, and the smoothness of... 

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... 

Consider first the case where the first Lyapunov value I2 is non-zero. Following the scheme outlined in the preceding section, we first derive the equation of the boundary of the stability region near e = 0. Next we will find the conditions under which is a smooth surface of codimension one. Finally, we will select the governing parameter and investigate the transverse families. 

As explained in Sec. 11.2, this surface is composed of a number of smooth sheets of codimension one, corresponding to double roots. The line fio = =0... 

Jt is a -smooth surface of codimension one. Choose A( ) as the governing parameter pi and consider a one-parameter family transverse to 97J. On the center manifold 2/ = 0, this family assumes the form... 

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... 

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