APPLICATIONS OF THE EULER CHARACTERISTIC IN BIFURCATION THEORY

SLAWOMIR RYBICKI Let f : R" x Rk -. R" be a continuous map such that f(O,,\) = 0 for all A E Rk . In this article we formulate, in terms of the Euler characteristic of algebraic sets, sufficient conditions for the existente of bifurcation points of the equation f(x, A) =0. Moreover we apply these results in bifurcation theory to ordinary differential equations . It is worth to point out that in the last paragraph we show how to verify, by computer, the assumptions of the theorems of this paper.


Introduction
In [A] Alexander has defined an invariant which nontriviality implies the existente of a bifurcation point of a continuous map f : R' x R k --> R' such that f (0, A) = 0 for all A E Rk .This invariant is an element of the group 7rk_1(GL(n)) .Generally it is dificult to verify if this invariant is a nontrivial element in 7rk-I(GL(n)) .
Krasnosielski in [K] has proved a 1-parameter bifurcation theorem which is a very useful tool in bifurcation theory.This theorem gives sufficient conditions for the existente of a bifurcation point of f in the case k = 1 .Many authors have proved generalizations of the classical Krasnosielski theorem (see [MA], [R1], [R2]) .
We are interested in formulating sufficient conditions for the existence .ofbifurcation points of f in case the dimension of the parameter space is greather than one and when the Alexander invariant can not be applied .
In [S1], [S2] and [W] the authors Nave proved very interesting formulas to a computation ofthe Euler characteristic of algebraic sets in terms of the Brouwer topological degree of suitable maps.
In the first part of this paper (using these formulas) we formulate and prove sufficient conditions for the existente of bifurcation points of f .
Namely, we define (Def.1.4.) a set of essential maps (Ess (n, k)) and prove that 0 -E R is not an isolated bifurcation point of any essential map f (Prop. 1 .1 .).
We also define (Def.1.3.) a set of regular maps (Reg (n, k)) and show how to verify if a regular map is an essential map (Prop.1.3.).
In Proposition 1.4.we formulate sufficient condition for the existence of a bifurcation point of a homogeneous map f (for definition of a homogeneous map, see Def. 1.2.) .
As the last case we consider a set of even maps (Even (n, k), Def.1.5.) .In Proposition 1 .5 .we give sufficient conditions for the existence of a bifurcation point of even map .
Notice that the assumptions of Propositions 1.4.,1.5.,1 .6 .are expressed in terms of the Brouwer topological degree of polynomial maps.From this point of view it is important to compute the Brouwer topological degree for polynomial maps.
Nierenberg has formulated in [N] án integral definition of the Brouwer topological degree. .We have written a computer program which computes the topological degree for polynomial maps, in a version given by Nierenberg.
The important question is how to verify that f E Reg (n, k).In other words we must verify if 0 E Rk is an isolated point in IP -1(0) .There are computer algorithms which allow us to check if 0 E Rk is an isolated zero of the polynomial map, T : Rk --> Rk .There algorithms are based on the Eisenbud and Levine results (see [E.L .]) .There is a computer program, writen by Andrzej Lecki from University of Gdansk, which is based on such kinds of algorithms .Using this program we can verify if f E Reg (n, k) .
Acknowledgement.The author wishes to thanks to Andrzej Lecki for several helpful comments.
In the second paragraph we apply topological results of this article to the bifurcation theory for ordinary differential equations (Th.2.1., 2.2., 2.3.) .
In the last part of this paper we show how to verify by computer the assumptions of the theorems of this páper .

Results
Denote by X and Y Banach spaces and by f X x Rk -> Y a continuous operator such that f (0, A) = 0 for all a E Rk .Definition 1.1.A point ao E Rk is said to be a bifurcation point of the equation (*) f (x,,\) = 0, if (0, \o) E closure {(x, .\)E X x Rk : f (x, .1)= 0 and x :,A 0} .
The set of bifurcation points óf the equation (*) will be denoted by Bif (f).
Definition 1.2.A map f is said to be a homogeneous map, if -P(A) is a homogeneous polynomial of degree greather than 1 .
The set of homogeneous maps will be denoted by Hom (n, k) .Definition 1.3.A map f : R" x Rk -> R' is said to be a regular map, if 1) f E Hom (n, k), 2) <D has an isolated critical point at the origin.The set of regular maps will be denoted by Reg (n, k) .Define a map IP : Rk --> Rk by the formula and notice that if T(Ao) = 0, then xP(t -Xo) = 0 for all t E R. Remark 1 .1 .The map ~¿ has an isolated critical point at the origin iff IP-1(0) = {0} .
Using this remark we will show that a map considered in Example 3 .2 . is a regular map.
Using the above formula and C.T.C .Wall results we obtain the thesis.
-dimensional manifold or is an empty set .From Proposition 1.2.and from the assumptions it follows that ~¿-1 (0) n SÉ-1 is not empty set .So our proof is completed .
In this part of the paragraph we formulate sufficient conditions for the existence of bifurcation points, in a case when f E Hom (n, k).
If m is an odd number then ob has to change a sign near the origin that is why it is enough to consider only the case of even m.
In the last part of this section we turn to a case when a map f is such that d> (A) is not necessary an homogeneous polynomial.
Definition 1.5.A map f : R'E x Rk -> R", is said to be an even map, if -D(A) is a polynomial such that 4)(-A) = The set of even maps will be denoted by Even (n, k).
Fix f E Even (n, k) and denote by d the degreee of <>.Choose any e > 0. Define a map A, : DÉ+I -, R as follows Remark 1 .2. Nótice that A, (A) is a homogeneous polynomial and that sgn A, (A) = sgn D(P(A)) for A E SÉ, where P : SÉ -> DÉ is the projection given by the formula P(A) = (al, . . ., \k).
A proof of this proposition is a consequence of Proposition 1 .4. and Remark 1.2.
In this section we will use the notations of the first section .Consider a Cl-map g : R x R' x Rk -+ Rn and assume that g(t, x, A) can be expressed in the form  A) is a n x n-matrix such that A(0) = 0, 2) W(t, 0, A) = 0 for all (t, 0, A) E R x Rn x Rk, 3) Dx cp(t, 0, A) = 0 for all (t, 0, A) E R x Rn x Rk.
We are interested in describing the set of bifurcation points of the following boundary value problem where L(A)(x(t)) = x(t) -A(A)x(t) .Notice that zeroes of the operator f are in one-to-one correspondence with solutions of the problem (*) .Let f : R' x Rk ---> R'' be the map defined by f(x, A) = A (A)x.Now we are in a position to formulate the main theorem of our paper.Theorem 2.1 .Iff E Ess (n, k) and E is a sufcciently small positive numóer, then Proof.Notice that L(0) : X --> Y defined by L(0)(x(t)) = x(t) is a Fredholm operator with Fredholm index 0 and that X = Xo ® Xl and Y = Yo ® Y,, where Xo = ker L(0) = Rn = {subspace of constant functions}, (see [M] for more details) .
We begin with the Lyapunov-Schmidt reduction .Let Po(x) = fo x(s) ds and P, (x) = xfó x(s) ds denote the projections of Y onto Yo, Y,, respectively.Then the equation F(x, A) = 0 is equivalent to the system of equations Notice that the map Pl oF : Xo ®Xl ®Rk --+Y, is continuously differentiable near (0, 0, 0) E Xo ® Xl ® Rk, Pl o F(0, 0,,\) = 0, and the FYéchet derivative of Pl o F with respect to xl at (0, 0, 0), Dxl Pl o F(0, 0, 0) is an isomorphism of Xl onto Y, .
Therefore by the implicit function theorem, there is an upen neighbourhood U of (0, 0) E Xo ®R k and xl E Cl (U, XI) such that the zeros of F near (0, 0, 0) are given by (xo, xl (xo, A), A) for (xo, A) E Xo ® Rk .
It is easy to see that xl(xo, A) = 0(Ilxo11) at xo = 0, uniformly for A near 0.
From the above it follows that zeros of F are in one-to-one correspondence with zeros of a finite dimensional map Q : U -+ Yo defined by the formula where T(0, A) = 0 and D,,T(0, A) = 0 .Notice that Q E Ess (n, k) .
The rest is a consequence of Propositions 1.1.and 1 .2.
The next theorems give sufficient conditions for the existente of bifurcation points of the operator F for more general class maps than the class Ess (n, k) .Theorem 2.2.If f E Hom (n, k) and deg (dri, DÉ, 0) ~(-1) k for i = 1, 2, then for any el G e Bif (F) f1 SÉ - , 1 0 .In particular, 0 E Bif (F) .Moreover the topological dimension of the set Bif (F) n SÉ;l is equal to k - 2 .This theorem is a consequence of Proposition 1.4.A proof of this theorem is similar to the proof of Theorem 2.1 .
Theorem 2 .3 .Let f E Even (n, k) .Then for sufficiently small E if deg (d Es, DÉ+1, 0)  (-1)k+1 for i = 1, 2, then Bif (F) n DEk 0 .More-¡ over the topological dimension of the set Bif (F) n DÉ is equal to k -1; in particular, 0 E Bif (f) .This theorem is a consequence of Proposition 1 .5 .A proof of this theorem is similar to the proof of Theorem 2.1 .
For g = (91, . . . .gn ) : (Rn, 0) --, (Rn, 0) we put Q(g) = e.M1, -. . ,gn)- If g is finite, in the sense that Q(g) is finite dimensional real vector space, then 0 E Rn is isolated in g-1 (0) .Consider the following boundary value problem :t (t) = g(t, x(t), x(0) = x (1) and assume that the map g satisfies all the assumptions of the previous paragraph .So we can express g in the form g(t, x, A) = A(A)x+W(t, x, A) .Theorems 2.1 ., 2.2., 2 .3.show that it is enough to examine only the matrix A(A).Example 3 .1.Assume that n = k = 3 and that the matrix A(A) is of the form A1 + A2, A2 + A3, A 1 + A3 A(A) = A 1 + A2 + A3, A l -A2, 2 -A l -3 -A3 A3, A1, 0 Using a computer program we show that deg (dri, D3,0) = 1 for i = 1, 2. From this it follows that deg (dri, DÉ, 0) A ( -1)3, so the assumption of Theorem 2.2. is fulfilled .Applying Theorem 2.2 we claim that there exists e > 0 such that for any el < e the intersection of the set of bifurcation points of the problem (**) with SÉ, is not empty, in particular 0 E Bif (F) .Moreover the topological dimension of this intersection is equal to 1.
Define a map f : R2 x R 2 -> R2 by f (x, A) = A (A)x.
We will show that f E Ess (2,2) .Using a computer program we show that dim Q(YP) = 16, so from the results from singularity theory it follows that T -1 (0) = {0}, that is why f E Reg (2,2) .
So from Theorem 2.1 it follows that Bif (F) (1 D2 consist of exactly six intervals, which emanate from the origin.