Bifurcation set

We note that one can obtain two kinds of bifurcation sets resulting from little changes of the boundary conditions for spatial dissipative structures in continuous systems (1) the change of the parameter A that leads to the change of the form of the solution without change of the number of roots of Eq. (44), and (2) the change of the number of roots (or the number of positive roots) of Eq. (44). 

We have previously noted that the bifurcation set predicted by eqn (4.3) partitions a given control plane w 0 into two regions. At any point (u, v) contained in the region which is bounded by the hypocycloid-shaped cross-section of the bifurcation set, the function /(x, y ft) of eqn (4.3) exhibits two saddle points and another critical point, which is a local maximum if w < 0,... 

A set containing points in the control parameters space, obtained by projection of points of the set I on the control parameters space, will be referred to as the bifurcation set B. Hence, the set B is given by the following equation ... 

Fig. 17. Catastrophe surface M2, singularity set I2 and bifurcation set B2 of the fold catastrophe (.42).

The sets Z2 and B2 as well as the potential form in various regions of the bifurcation set are illustrated in Fig. 17. 

Fig. 19. Catastrophe surface M3, singularity set I3 and bifurcation set B3 of the cusp catastrophe (/t3).

The manifold M4 is a three-dimensional surface in a four-dimensional space and, consequently, it is difficult to describe. On the other hand, one may visualize the shape of the bifurcation set B4 which is a two-dimensional surface embedded in a three-dimensional space. 

To determine the form of the bifurcation set B4, the x-variable should be eliminated from equations (2.24), defining the set I4. This can be done in the following manner. The second of equations (2.24) is solved for x (a cubic equation has one or three real roots), which is then substituted to the first of equations (2.24). As a result, we obtain the equation... 

We shall now proceed to the examination of catastrophes modelled by Thom potential functions of two state variables. This will be done using the notions of the catastrophe manifold M, singularity set I and bifurcation set... 

Parametrization of the bifurcation set B follows from (2.50) as a projection on the parameter space ... 

The bifurcation set is shown in Fig. 26. An elliptic umbilic catastrophe is described by the potential function... 

Parametrization of the bifurcation set B is obtained from (2.57) by projection of I on the parametr subspace ... 

The catastrophes described by Thom potential functions in two variables may be described in the control parameters space. The catastrophe takes place when a trajectory of the system on the catastrophe surface M, projected on the control parameters space, intersects the bifurcation set B (i.e. the projection of the set Z). 

The bifurcation set B3 is given by the condition 4a3 + 21b2 = 0, which may be expressed in terms of pressure p and temperature T... 

It follows from (3.46) that when / has a degenerate critical point at x0, the denominator vanishes and the functions 4>(c), /(c) are divergent, that is, such a point x0 lies on the caustic. In this way, a very close relation between the short-wave optics and elementary catastrophe theory is obtained. More specifically, the caustic is set by the conditions V/= 0, det /y = 0, that is it corresponds to the singularity set Z. When the caustic is observed on a screen at the point c, which corresponds to a projection of the set Z on the control parameters space, it is the bifurcation set B which is then directly observed on the screen. 

Classification of catastrophes will be preceded by the centre manifold theorem which is a counterpart to the splitting lemma in elementary catastrophe theory. It will turn out that in the catastrophe theory of dynamical systems such notions of elementary catastrophe theory as the catastrophe manifold, bifurcation set, sensitive state, splitting lemma, codimension, universal unfolding and structural stability are retained. 

Each point of the bifurcation set for the two-component case corresponds to two types on the surface 0a and 0fl+A0a, 0/, and 0 +A0, where a and b indicates two different types of adsorbed molecules. Using the logarithmic form of the system (2.93)... 

Under certain assumptions the probability density P(x, y, t Wj) reduces to P H, t Ml). Deterministic and stochastic bifurcation sets coincide for particular/and g only. Depending on the special forms of the functions/and g, the bifurcation values presented by the deterministic approach shifts, or even new transitions are induced which are absent in the deterministic picture. 

Pogosyan, T. I. Construction of Bifurcation Sets in One Problem of Rigid-Body Dynamics. Mekhanika Tverdogo Tela, Kiev, issue 12 (1980), 9-16. 

Pogosyan, T. I., and Kharlamov, M. P. Bifurcation set and integral manifolds of the problem of the rigid-body motion in a linear field of forces. PriJd. Matem. i Mekh. 48 (1979), 419-428. 

Koper, M.T.M. (1995) Some simple bifurcation sets of an extended Van der Pol model and their relation to chemical oscillators. /. Chem. Phys., 102 (13), 5278-5287. 

The plane (c,a) is divided by the so called cusp-shaped curve, see Fig.13, which is a bifurcation set ... 

If one looks at how spiral states in the model system are organized as a function of control parameters, one finds a dynamics landscape, or flower garden as shown in Figure 2. In the nonlinear-dynamics literature, such a dynamics landscape is referred to either as a phase diagram or as a bifurcation set [18]. For the present discussion, the meaning of the two control parameters is not important. The relevant point is that the parameter plane is composed... 

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