ON THE APPLICATION OF NEWTON ’ S AND CHORD METHODS TO

Thispaper isconcernedwith the applicationsofNewton's andchordmethods in thecomputations of the bifurcation solutions in a neighborhood of a simple bifurcation point for prescribed values of the


INTRODUCTION
Many problems in applications are formulated as (x,X)-o, where G:H x fft H is a smooth (nonlinear) map satisfying a(0,)-0; H is a real Hilbert space and , . is a real parameter which often represents some physical quantity.
The solution curve F0: xo(.)-0 is called the trivial solution curve.A point (0,)E F is called a bifurcation point if there exists a smooth curve Fl:xl xl(.) of nontrivial solutions of (1.1)which is defined in some neighborhood of (0, ko) and passing through it.It follows from the Implicit Function Theorem that a necessary condition for (0, :k0) to be a bifurcation point is that the Frechet derivative G,,(0, ko) is singular.
On the other hand the singularity of Gx(0, k0) is not a sufficient condition for (), ) to be a bifurcation point.Some sufficient conditions for (0,ko) E Fo to be a "simple" bifurcation point will be stated in Section 2. This paper is concerned with the numerical computations of the nontrivial solution curves of (1.1) in a neighborhood of a simple bifurcation point (0, k0).Many excellent analytical and numerical treatments of this problem exist in the literature.The reader is referred to [1]- [7], and the references therein for an extensive account of the subject.In almost all the previous numerical works the parameter .is treated as a variable and is determined along with the "state" variable x.However, in applications .represents a physical parameter and it is often required to determine the state variable x for some given values of the parameter ..I n this paper we examine the applications of Newton's and chord methods in solving (1.1) for x while .is kept fixed near a simple bifurcation point The rest of the paper is organized in three sections.In Section 2 we present some well known pre- liminary results, regarding the solution set of (1.1) which are based upon the Implicit Function Theorem and state a convergence theorem for Newton's and chord methods due to G. Moore [7].In Section 3 we present and prove the convergence of some numerical schemes for computing the nontrivial solution curves of (1.1) in a neighborhood of a simple bifurcation point (0,0) for given values of the parameter X.In (1.1) (1.2) section 4 we illustrate the use of the schemes developed in Section 3 by applying them to a finite dimensional numerical example.

PRELIMINARIES
All the results and proofs stated in this section are well known and we present them here for com- pleteness.We present some sufficient conditions for a point (0,) of re to be a bifurcation point and state a basic convergence theorem for Newton's and chord methods.
We assume that the Frechet derivative G .G,(0, ) satisfies, for some R, the conditions (a) N(G) is one-dimensional spanned by , (, ) 1, Co) N(G) is one-dimensional spanned by V , 0, where G G,(0, o) and the notations L*, N(L) and R (L) denote the adjoint operator, the null and the range spaces of a linear operatorL, respectively.Under the assumptions (a)-(e) of(2.I), (0, 0) is a bifurcation point.To see this we decompose H as H -<)RCG).
(2.6) Define h (e, I) g( I).Since h (0, 0) 0 and h,(O, O) a ,, O, we can apply the Implicit Function Theorem to conclude the existence and uniqueness of a smooth function ILt(e) defined in some neighborhood of e 0 such that (.()) 0.
for each I in thal neighborhood.
Fuhcrmorc, if condition ) is replaced by () [17(){Au(),)(u--[F(u,)-F(v,)]}ll -()llU-Vll for where (6) 6) as 6 0, cn crc cxism 0 < 6 and a continuous mapping U*() om (0,) into Ht such at U*( 6)is c iquc lution of F(U,6)= 0 in N()(U( 6)) and c chord itcratcs u" u" :(u(), )(u', ), converge to U*(6) for each 6 NEWTON'S AND CHORD METHODS In [6], Decker and Keller introduced a method for constructing the bifurcation branches near a simple bifurcation point (0, Ao).In that paper the equation G(e# + w, ku + it) 0, (3.1) was replaced by an "inflated" system [C + ,zo + )] F(y,e)-[(#,w> -0, (3.2) where y "(/6H x[R.Using the initial guess y-0, they proved the convergence of Newton's and chord iterates of 0.2) to a nontrivial solution y() of 0.1) for each given e in 0 I1 h, provided e0 is small enough.Thus, in this method both the state variable x and the bifurcation parameter k are treated as unknowns and are approximated as functions of the parameter e.This method cannot be used in practical problems where it is required to approximate the state variable x for some given values of k.On the other hand it is not possible in general to parameterize x using the bifurcation parameter k; for example; in the case when nontrivial solutions only occur at the bifurcation point.This shows that some additional assumptions are to be imposed.
In addition to conditions (a)-(e) of (2.1) we will assume that either (Non-degenerate case) E ,, 0 (3.3) a (Degenerate case) E 0, E ,, 0, ss < 0, (3.4) hold, and show that the methods of [6] can be modified to approximate the nontrivial solutions of (3.1) with k being fixed near the bifurcation point Let H, denote the Hilbert space H x 9] with inner product and define F: H, x R Ht by for U () 6Hx, Z 6 R. For fixed k, using (2.7) and (2.8) we define the initial guesses uO(k).().-a(k-kO)Eok6, for the non-degenerate case (3.3) and degenerate case 0.4), respectively.
To ex.mine e convergence of Newton's .hachord iter.teswe need to estim.te is done in e following mma.

3.3
For small enough 0 we have where y is as in mma 3.1.Proof: We note at for e non-degenerate case E0 0 and for the degenerate case F(U,.)-O We will denote F(U,.),Fu(U,.)and Fu(U,Xo) by F(.), Fu() and F, respectively.
Furthermore, H can be decomposed into H N(:) (H(:), It follows from property (iv) above that in order to examine the rate at which F?()ll tends to infinity as : tends to zero it is enough to examine the behavior of the two eigenvalues of B(:).Those eigenvalues are studied in the following lemma.

NUMERICAL EXAMPLE
In this section we illustrate the use of the numerical schemes developed in Section 3 by applying them to approximate the nontrivial solutions of the (finite dimensional in a neighborhood of the simple bifurcation point (0,0), for several prescribed values of the bifurcation parameter . .
Using the same notations as in Section 3 we note that the inflated system corresponding to (4.1) is

W1
W "I" (W 4" E) 4" (W "I" E)W2 (W "I" E) w and, for a given .near 0, the initial guesses for the nontrivial solutions are given by (4.1)

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.

First
Round of Reviews May 1, 2009