Catastrophe Hopf bifurcations

The context of Uppal, Ray, and Poore s work was the revival of interest in Hopf bifurcation at the end of the 1960s and the soon-to-be-transmogrified catastrophe theory of Zeeman. Ray has followed up this type of analysis in studies of polymerization and other important processes. 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

When the condition (1.9) is not met in (1.6), we deal with dynamical catastrophes. In some cases, for example for the so-called Hopf bifurcation, dynamical catastrophes may be examined by static methods of elementary catastrophe theory or singularity theory (Chapter 5). General dynamical catastrophes, taking place in autonomous systems, are dealt with by generalized catastrophe theory and bifurcation theory (having numerous common points). Some information on general dynamical catastrophes will be provided in Chapter 5. 

Subsequently, we will give the method enabling a demonstrative representation of certain dynamical catastrophes, such as the Hopf bifurcation, in the spirit of elementary catastrophe theory. The Hopf bifurcation will be shown to be in principle an elementary catastrophe (with potential). Finally, we shall discuss an application of the above mentioned methods to the... 

In Section 5.6 we will discuss a catastrophe occurring in the reduced (van der Pol) system (5.41) when the parameter 5 changes sign. A catastrophe of this type the appearance of a limit cycle, associated with a loss of stability by the stationary point (0, 0), i.e. the Hopf bifurcation, also occurs in the original system (5.40). The resulting limit cycle is in this case localized on the surface z = x2, see Fig. 75. 

A catastrophe of this type is called the Hopf bifurcation. It may be represented by a bifurcation diagram in which the position of a stationary state and a limit cycle are plotted as a function of the parameter c (Figs. 83, 84). 

The Hopf bifurcation is a dynamical catastrophe, since a stable stationary state bifurcates to a limit cycle hence, state variables change with time. Interestingly, the Hopf bifurcation may be visualized by elementary catastrophe theory despite the fact that the Hopf bifurcation may not appear in a gradient system and, furthermore, it is a dynamical catastrophe. 

Such an analysis of the Hopf bifurcation (and some other catastrophes) will be presented in Section 5.6. 

The saddle node catastrophe and the Hopf bifurcation may be shown to be structurally stable. Certain additional conditions (see Sections 5.5.2.2, 5.5.2.3) are imposed on the transcritical bifurcation and the pitchfork bifurcation. The system is structurally stable under perturbations not disturbing these additional conditions on the other hand, when arbitrary... 

Hence, the stationary point (0,0) for small (as regards the absolute value) e is a stable focus and for small positive e is an unstable focus. When the parameter e changes sign, a catastrophe — a change in the nature of trajectories, takes place in the system. In addition, At 2(0) = +i hence, the state of the system corresponding to e = 0 is a sensitive state typical for the Hopf bifurcation. 

Evidently, for e > 0 a stable limit cycle appears in the system. A catastrophe of this type is called the Hopf bifurcation. Interestingly, the system can be assigned a certain energy or, more specifically, the change in energy per revolution, AH. The states of the system approach the closed trajectory for which AH = 0. Hence, we may introduce a potential function V, dependent on the state parameter x, x = R2, such that the state which is approached by all trajectories corresponds to the condition of minimum of this potential... 

The way of presentation of the material discussed in Chapter 5 is based on papers of Guckenheimer and on the ideas contained in papers by Stewart. A paper by Nicolis and an article by Othmer in a book published by Field and Burger constitute a very good supplement to these papers. A book by Arnol d (1983), although rather difficult, provides much additional material. The elementary method of analysis of some dynamical catastrophes presented in Section 5.6 is patterned after ArnoPd s approach to the Hopf bifurcation in the van der Pol system described in his book (1975). A book by Gilmore provides basic information on catastrophes in dynamical systems. A paper by Stewart contains another proof (compared to Section 5.5) that Hopf bifurcation is an elementary catastrophy. 

The second sensitive state is that corresponding to Re(2j 2) = 0, that is a = a0 = (y — l)-1. For a0 < a < a2 = [(v/y + l)/(y — 1)] the stationary state is an unstable focus, hence the sensitive state corresponds to the Hopf bifurcation. This catastrophe will be examined in more detail. The state a = a2 is a third sensitive state for this a value we have kt = k2> 0 (the discriminant of equation (6.71) vanishes) and for a2 < a the stationary state is an unstable node. This catastrophe has a global character and is analogous to the catastrophe occurring for the value a = av... 

On the basis of the above analysis we arrive at a conclusion that in the system represented by the equations not accounting for diffusion (6.82) the Hopf bifurcation may appear. We shall examine the catastrophe taking place in more detail using the method described in Section 5.6. 

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... 

On the other hand, the abundance of experimental material stimulates an evolution of the theories explaining non-linear phenomena. For example, as shown above, the transition in a chemical reaction from the stationary state to the state of periodical oscillations, the so-called Hopf bifurcation, is a certain elementary catastrophe. The transition in a chemical reaction to the chaotic state may be explained in terms of catastrophes associated with a loss of stability of a certain iterative process or by using the notion of a strange attractor (anyway, it turns out that both the systems are closely related). The equations of a chemical reaction with diffusion have been extensively studied lately. Based on the progress being made in this area, further interesting achievements in theory may be anticipated, particularly for the phenomena associated with catastrophes — the loss of stability by a non-linear system. 

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