Splitting lemma

When the matrix Vi has a zero determinant at a critical point, then this is a degenerate point. The degeneracy may turn out, however, to occur only due to a part of variables of the function T(x). This is revealed by vanishing of only some of the eigenvalues of the matrix of second derivatives (Hessian) at the critical point x = 0. For a function dependent on two variables, given by (2.30), upon transformation (2.33) we obtain 

As the coefficient at y2 vanishes, transformation (2.35) cannot be performed, for the four unknown coefficients a, b, c, d in equation (2.37) cannot be determined by means of the three coefficients A20, Allt A02. 

On the other hand, the function V may be split into two parts, and NM see equation (2.26), only the function NM having a degenerate critical point. Transformation (2.35) applied to the function (2.37) reduces it to a simpler form (but not so simple as (2.36)) 

Such a reduction is feasible due to a sufficient number of equations resulting from non-vanishing of 

Splitting a function into the components FNM and KM will be discussed using as an example the function V dependent on two variables. 

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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. 

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. 

Proof. Inserting the expression for - given in Lemma 3 into the left-hand side of (34) and multiplying the equation thus obtained by the inverse of the nonsingular matrix H we arrive at the system of matrix equations, whose left-hand sides are the linear combinations of the linearly independent matrices E, Syield system of Eqs. (39). The assertion is proved. 

From (3.3) follows that U(t) tends to a limit U(oo) 0 and, similarly, V(t) tends to F(oo). Our second lemma asserts that unless W is decomposable or of splitting type at least one of these limiting values must vanish, so that ultimately all w(t) have the same sign or are zero. The sign is determined by the initial values, because... 

Thus W is reducible. By writing the analog of eq. (3.2) for V(t) one also finds Wwl/ = 0, so that W is of splitting type and also excluded from the lemma. 

For the parameters (r, q) = (3,5), (5,3), in Lemma 8.2.3, we will prove that the vertex-split Icosahedron is such a unique finite polycycle. ... 

Lemma 8.23 ([DSS06]) All finite elliptic non-extensible (r, q)-polycycles are two vertex-splittings (of Octahedron and Icosahedron see first two in Figure 8.3) and five Platonic r, q) (with a face deleted). 

Proof. By analogy with the previous lemma, we can split the set of 7-gonal faces into the following types ... 

Lemma 1.5. — Let be a Nisnevich covering of a noetherian scheme S. Then there exists a splitting sequence for... 

When passing through the critical value a, the torus splits into two tori, Ti g and T2, . The separatrix diagram P. goes from the critical surface and, when descending, meets the torus along two circles 71 and 72 (see above). Lemmas... 

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