Two-Parameters Bifurcation in Quasilinear Dierential-Algebraic Equations

In this paper, bifurcation of solution of guasilinear dierential-algebraic equations (DAEs) is studied. Whereas basic principle that quasilinear DAE is eventually reducible to an ordinary dierential equation (ODEs) and that this reduction so we can apply the classical bifurcation theory of the (ODEs). The taylor expansion applied to the reduced DAEs to prove that is equivalent to an ODE which is a normal form under some non-degeneracy conditions theorems given in this work deal with the saddle node,transcritical and pitchfork bifurcation with two-parameters. Some illustrated examples are given to explain the idea of the paper.


INTRODUCTION
Nearly all DAEs arising in scientific or engineering problems are quasilinear.Thisarticle presents bifurcationin guasilinear di fferentialalgebraic equations (DAEs) differfrom ordinary differential equations (ODEs).Over the years several approaches havebeen introduced for the study of local existence and uniquenessquestions for DAEs.While they exhibit major technical differences and are based on different assumptions,all these approaches agreewith the basic principle that a DAE is even tually reducibleto an ODE and that this reduction should be donevia a recursiveprocess .The bifurcation in guasilinear para meterized DAEs form it will be investigated. Accordingly, we shall assume that for some open interval I ⊂ Rand open subset U n ⊂ Rn the mappings A : U n×I ×I £(Rn) and G : U n×I ×I Rn are of class C∞. And proves a bifurcation theorem based on assumptions on theTaylor coefficients. Since we will impose conditions on these coefficients we will be ableto show that the system undergoes saddle node ,transcritical and pitchfork bifurcationthat is a little more akin to bifurcation ODE .
A simple comparison of the areas of the sciences in which DAEs are involved withthose in which examples of bifurcation in ODEs arise [1], [3] that bifurcation of periodicsolutions occurs from (0,0)and [4] Our exposition is based on Jepson, A. and Spence [2] and the references therein reveals a considerable overlap and suggests that an appropriate variant of the bifurcationtheorem should be available in the DAE setting.it is important note that all theorems and condition s for Bifurcation to be occurredin the reduced DAEs will be given in terms of A and G in (1,1) and thiswillbe extension of the bifurcation theory to DAEs of index one.
In the index one case, our coal is it use the reduction of (1.1) to an ODE In withthe reduction (1.1) method given in [5] then apply classical bifurcation theory to the reduced ODEs.
This paper is organized as follows: Section 2 deals with the problem of reducingparameterized families of DAEs simultaneously.Since reduction of DAEs to ODE formleads to implicit rather than explicit ODEs, it is important to rephrase some of thehypotheses of the classical bifurcation theorems in that setting. This is done in section

Reduction of Parametrized DAEs [5]
The bifurcation in guasilinear parameterized DAEs form (1.1) will be investigatedand DAEs will be reduced to an equivalent parameterized ODEs. Then classical bifurcation theory can be applied. In the reduction peroses we will follow the methodof reduction given in [5]. So the following theorem is an essential in our work, which summarized the reduction of DAEs (1.1). only if x(t) = ϕ(µ, ξ(t)) , ∀ t ∈ J , and ξ : J is a solution of the system A 1(µ, ξ) and ∈ £( R n )is an arbitrary linear projection onto rge A (µ, x)) ( R r 1 ).or Fixed µ ∈ I × I and ∈ , the DAE (1.1) reduces to the form 3) The fixed but arbitrary µ = (µ1, µ2) ∈ I × I are also given by the solutions of thenon parametrized D A E For the sake of argument, assume that the D A E (1.1) withµ= has index one at ∈ (so that ( , ) ∈ W1).
With the previous notation, this means that the operator ( , ) where ϕ( , ) = , has full rank r 1 and hence is invertible.By continuityA 1(µ, ξ) remains invertible for (µ, ξ) near ( , ) ∈ R × R r 1 and it thus follows fromTheorem 2.1, that in the vicinity of ( , ) ∈ W1 , the parameterized DA E (1.1) is equivalent to the explicit parameterized ODE: To motivate the discussion in the next section, suppose also =0 , =0

Local two-Parameter Bifurcations of Equilibrium Points
We will now consider the general guasilinear parametrized DAEs equation and µ = (µ1, µ2), and prove a bifurcation theorems based on assumptions on the Taylorexpansion of G. We assume that G (x, µ)for all value of µ ,A (x, µ) is independent of xand µ.

Saddle-Node bifurcation
The saddle-node bifurcation can take place in any system and is, in fact, a very typicalbifurcation to happen when a parameter is varied. Maybe because this bifurcation isso typical, it has a lot of other names. The saddle-node bifurcation is also called foldbifurcation, tangent bifurcation, limit point bifurcation, or turning point bifurcation.
the following theorem related to this kind of bifurcation.
where G∈ 3 has at =0the eguilibrium x=0 ,and (0,0,0)≠ 0.Assume that the following two non-degeneracy conditions are satisfied: (i) Proof. According to the reduction processes mentioned in (Section 2) D A E will be reduced to ODEs: where G 1(ξ, µ) and A 1(ξ, µ) −1 from theorem 2.1 reduced to G (x, µ) and A (x, µ). Then by Taylor expansion about (0,0,0)we have: Next we remove the linear term w.r.t ξ by introducing a new variable z:  Next assume y = z then we have: Substituting z = we get = + .
Suppose that = and = Then we get the normal form = O( )

Trans-critical bifurcation
If two curves of fixed points intersect at the origin in the µ − x plain, both existed on either side of µ = 0 then the origin is called a transcritical bifurcation ( T C B ) point see [8].

Pitchfork bifurcation
If two curves of fixed points intersect at the origin in the µ − x plain and only oneexists in both sides of µ = 0, moreover, the other curve of fixed points lays entirely toone side of µ = 0, then the origin is called a