The centre manifold theorem

When we deal with a dynamical system in which a sensitive state occurs (assumptions of the Grobman-Hartman theorem are not fulfilled), it may turn out that the sensitive state is associated only with a part of state variables. The variables related to the sensitive state may then be separated and the catastrophes occurring in a system dependent on a smaller number of state variables examined. 

A counterpart to the splitting lemma of elementary catastrophe theory is the centre manifold theorem (also called a neutral manifold), generalizing the Grobman-Hartman theorem to the case of occurrence of sensitive states (5.28). The centre manifold theorem allows us to establish an equivalence (nonequivalence) of two autonomous systems. In this sense it is also a generalization to the case of autonomous systems of the equivalence relationship introduced in Chapter 2 for potential functions. 

Let for c = c0 the functions f, gj vanish together with their first derivatives at (x, y) = (0, 0) (the origin of the coordinate system), that is, not contain linear terms. Then the system (5.29) has the stationary point (0, 0). Linearization of the system (5.29) in the neighbourhood of the point (0, 0) yields the linear system 

Assume now that the eigenvalues of the matrices Bik and satisfying the equations 

Let us briefly examine properties of the system (5.29). Firstly, the stationary state (0, 0) is a sensitive state in view of equations (5.34a). Secondly, to simplify notation we assumed that there are no unstable solutions for which Re (2) 0. Furthermore, it is apparent that the sensitive state is associated only with the variable x. 

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The Tikhonov theorem has an important generalization, called the centre manifold theorem, which will be discussed in Sections 5.4.5-5.4.7. In classification of catastrophes occurring in dynamical systems and represented by systems of autonomous equations, the centre manifold theorem plays the role of the splitting lemma (see Section 2.3.4). 

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. 

When conditions (1)—(3) are not fulfilled, the Grobman-Hartman theorem is not valid. As will be shown later, then we have to deal with the sensitive state of a dynamical system (this corresponds to a degenerate critical point in elementary catastrophe theory). A generalization of the Grobman-Hartman theorem, the centre manifold theorem which may be regarded as a counterpart to the splitting lemma of elementary catastrophe theory, has been found to be very convenient in that case. 

The centre manifold theorem deals with the modification of this pattern in the non-linear system (5.29). It follows from the centre manifold... 

The geometrical meaning of the centre manifold theorem is expressed by the graphs of phase trajectories, Fig. 74 (for the non-linear system (5.29) and Fig. 73 (for the linearized system (5.30)). 

Relation between the Tikhonov theorem and the centre manifold theorem... 

The centre manifold theorem may be regarded as a generalization of the Tikhonov theorem to a case when the time hierarchy does not explicitly occur in the examined system, i.e. when the equations cannot be divided into... 

We shall now show an application of the centre manifold theorem to the system without marked time hierarchy (this is the Duffing equation) ... 

A paper by Feinn and Ortoleva deals with the application of the Tikhonov theorem to chemical systems, similarly bo books by Romanovskii, Stepanova and Chernavskii, which additionally provides information on the more general Shoshitishvili theorem (covering essentially a part of Guckenheimer s results). Information on an application of the centre manifold theorem can be found in a book by Carr or by Guckenheimer and Holmes. In the Carr book, the Tikhonov theorem is compared with the centre manifold theorem. 

From the above analysis a conclusion can be drawn that, because the stationary state (x2, y2, z2) can have a sensitive state corresponding at most to the Hopf bifurcation, the centre manifold theorem may be applied to the system of equations (6.98) thus reducing it to a system of equations in two variables in which the Hopf bifurcation would appear. 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

The centre manifold method permits us to split the system (5.29) into the subsystem having a sensitive state and the subsystem fulfilling assumptions of the Grobman-Hartman theorem. Note that in the linearized system (5.30) the separation took place automatically as a result of linearity. The phase portrait of the linearized system is of the form shown in Fig. 73. 

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