Catastrophes of codimension one

The investigation of stability and catastrophes of specific dynamical systems may now proceed in the following way  

We will consider dynamical systems in which the vector field f depends on one control parameter 

The catastrophe of a change in the phase portrait nearby a stationary state requires (at least one) real zero eigenvalue or (at least one) purely imaginary conjugate pair of eigenvalues to be present among the eigenvalues of the stability matrix. 

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), 

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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. 

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 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. 

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... 

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. 

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... 

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 more codimension-one boundary of stability of periodic trajectories which corresponds to the blue sky catastrophe [152]. It may occur in n-dimensional systems where n > 3. 

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