ON THE NUMERICAL SOLUTION OF PERTURBED BIFURCATION PROBLEMS

Some numerical schemes, based upon Newton’s and chord methods, for the computations of the perturbed bifurcation points as well as the solution curves through them, are presented. The "initial" guesses for Newton’s and chord methods are obtained using the local analysis techniques and proved to fall into the neighborhoods ofcontraction for these methods. In applications the "perturbation" parameter represents a physical quantity and it is desirable to use it to parameterize the solution curves near the perturbed bifurcation point. In this regard, it is shown that, for certain classes of the perturbed bifurcation problems, Newton’s and chord methods can be used to follow the solution curves in a neighborhood of the perturbed bifurcation point while the perturbation parameter is kept fixed.

Properties (1.2) and 1.3 respectively imply that the "trivial" solution x 0 solves the "unperturbed" problem G(x,.,O) =0, (1.5) for each k R and that it is not a solution of (1.1) for any % R when x : 0. Properties (a)-(c) of (1.4) imply that G is a Fredholm operator with zero index and together with the property (d) of (1.4) imply that (0, , 0) is a bifurcation point of (1.5).Observe also that property (e) of (1.4) implies that the zero eigenvalue of GI' has algebraic multiplicity one.
The parameters and x in (1.1) are usually called the bifurcation parameter and the perturbation parameter respectively.Whcn : :g: 0, the solution set of (1.1) is completely different from that of (1.5) and in this case (1.1) is usually termed a perturbed bifurcation problem.Many excellent analytical and numerical treatments of such problems exist in the literature.The reader is referred to [4], [6], [7], [9], 10], 11 and the references therein for an extensive account of the subject.
In this paper we concentrate on the numerical aspect of (1.1) and in particular on the applications of Newton's and chord methods to this problem.In this respect our techniques go along the same lines as those of Decker and Keller [3] for the bifurcation problem (1.5).The difficulties arising in the applications of Newton's and chord methods to (1.1) in the neighborhood of (0, o, 0) are due to the nonuniqueness of the solution, the singularity (or the near singularity) of G, and the non-availability of the "close enough" initial guesses for these methods.
The rest of this paper is organized in four sections.Some preliminary results regarding the solution set of (1.1) and based upon the Implicit Function Theorem are presented in Section 2. Also in Section 2 we state the basic convergence theorem for Newton's and chord methods which is used in the later sections.In Section 3 we present and prove the convergence of some numerical schemes for computing the perturbed bifurcation points and the solution curves through them.In Section 4 we show that Newton' s and chord methods can be used to compute all solution curves of (1.1) near (0, , 0) for certain types of problems, while the perturbation parameter a: is kept fixed.In Section 5 we apply the schemes developed in Sections 3 and 4 to a numerical example.

PRELIMINARIES
In this section we study the behavior of the bifurcation point (0,),,o, 0) of (1.5) under the perturbation x : 0. It is shown that there is locally a family of "limit points" through each of which there is a family of solutions of (1.1).The proof of these statements is based upon the Implicit Function Theorem.We also state the basic convergence result for Newton's and chord methods which will be used in the later sections.
PROOF.We observe that if y (y,, y2,r,,r,_) e Y satisfies F, (0, O)y =0, where F, (0, 0) is defined by (2.3), then the components of y satisfy the systetn of equations From (a) it follows that rb 0, and therefore condition (f) of (1.4) implies that r 0, and hence yl A , for some constantA.Now by (c),A must be zero and hence yl 0. This reduces (b) to Gy2 + rlG,x O, which in turn implies that ra 0. Since a e 0 by condition (d) of (1.4), it follows that r 0, and that y A , for some constant A. This together with (d) implies that Y2 0. Thus, y 0. It follows that F, (0, 0) is one to one.Since F,.(0, 0) is clearly onto, it follows from the Open Mapping Theorem that it has a bounded inverse.This enables us to apply the Implicit Function Theorem to equation (2.2) and obtain the desired conclusions.
We summarize the conclusions of the above paragraph in the following theorem.
THEOREM 2.2.There exists a unique smooth curve (x(e), k(e), x(e)) defined on el < Co, for some e0>0, of solutions of (1.1) passing through the bifurcation point (0,k0,0).Furthermore, for : 0, (x(e), :k(e), x(e)) is a limit point through which there passes a unique smooth solution branch F(:) of(1.1).REMARK 2.3.The use of the Implicit Function Theorem in the treatment of the perturbed bifurcation problems of the type considered in this paper was suggested in [7] where a different technique was used.

THE NUMERICAL COMPUTATION OF THE PERTURBED BIFURCA- TION POINTS AND THE SOLUTION BRANCHES THROUGH THEM
In this section we demonstrate the application of Newton' and chord methods in the computations of the perturbed bifurcation points of (1. l) and the solution branches through them.
The computations of the perturbed bifurcation points of (1. l) using the Newton' s and chord iterates involve the application of Theorem 2.4 to the equations (2.2).As an "initial" guess we take ytO 0. It follows from the proof of Lemma 2.1 that F, (0, c) has a bounded inverse, for e in el < e-o, where e0 > 0 is small enough.Using similar arguments as those in the proof of Lemma 2.1, we can show that rl(e), L(e) and g(e) of Theorem 2.4 are 0(e), 0(1) and 0(c) as c 0, respectively.This proves the following theorem.
THEOREM 3.1.There exists Co > 0 such that for 0 < 1 < e0 both Newton's and chord iterates with initial guess y(O) 0 converge to the unique solution y(c) of equation (2.2).
We now consider the application of Theorem 2.4 in computing the solution branches through the perturbed bifurcation points.To this end let us assume that the unique solution y(e) (y, y;,r, r) of (2.2) corresponding to some c in 0 < [[ < c0 has been determined.With x r being fixed we want to NUMERICAL SOLUTION OF PERTURBED BIFURCATION PROBLEMS compute (x, %,) near (x',)'), where x" =+ y and " Xo + r, such that G (x %,, "r.) =0.
Under some additional assumptions it is shown in the next section that Newton's and chord methods can be used to compute all the solution curves in some neighborhood of (0, 2%, 0).The initial guesses in these cases are obtained using the singular perturbation methods [8].

THE NUMERICAL COMPUTATIONS OF THE SOLUTION CURVES OF SOME PERTURBED BIFURCATION PROBLEMS
This section is concerned with the applications of Newton's and chord methods in computing all the solution curves of (1.1) in some neighborhood of (0, , 0) for a given small value of the perturbation parameter x : 0. We will consider the following two cases.
Let H denote the Hilbert space H x R and define F (')'H IxRxR-+Hl, i=1,2, by where A is a solution of (4.4) for the given value of , and F2(U,,:) G(A.d+A 8-w, +w,+z+l.t,:3 where A is determined by (4.8)for the given value of, and in (4.9)and (4.10)U HI.
For R, the linear operator satisfies the properties (Lemma 5.6 [3]) (i) L, is a Fredholm operator with zero index, (ii, N,L,, is one-dimensional spanned by o (?/, (4.13) N(L,) is one-dimensional spanned by (iii) (iv) the algebraic multiplicity of the zero eigenvalue of L, is two.
PROOF.It is enough to show that each eigenvalue a of B,(, ) has the form a C ': + 0() for some constant C, 0. But the solvability condition of the equation resulting by differentiating the first equation of (4.15) with respect to and setting 0 gives ,:0, ) A a , O, where b:(, ) is the (l,2)-ent W of B,(, ).This shows that has the desired form.
THEOREM 4.3.For each C > 0 there exists 8(C) > 0 such that for each in gl C d each in 0 < I1 8(C) Newton's and chord iterates with initial guess U 0 converge to a unique solution of (4.11) for each real solution A of (4.4) or (4.8) (depending upon whether or 2) for the given value of .

NUMERICAL EXAMPLE
In this section we apply the schemes developed in Sections 3 and 4 to the following (finite Equation (2.1), whIch determines the biflrcation curve of (5.1), reduces to 3 E', + r8 + r, y:++ The numerical results obtained by applying the schemes of Section 3 where y Y2 Y22 to (5.2) are presented in Table 5.1.
The numerical results obtained for the solution branch through the perturbed bifurcation point corresponding to 8 0.1 are presented in Table 5.2.
Finally, we apply the schemes of Section 4 to approximate the solution branch of (5.1) which does not pass through the perturbed bifurcation point.
The numerical results of Tables 5.2 and 5.3 determine approximations to all the solution branches near the perturbed bifurcation point when "t .002,which corresponds to 0.1 in Table 5. 1.In Tables 5.1-5.3,N and C denote the number of iterations needed for the convergence of Newton' s and chord methods respectively.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning (a) N(Gx) is one-dimensional spanned by , (, ) 1, (b) N(G*) is one-dimensional spanned by g v: 0, (c) R(Gx) N(G*) and R(Gx')= N(Gx)1, G,y, + G, y2 + r,G, + r2G, O, (c) (,, y.> 0, (d) <,, y.,) 0.

•
Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation