Abstract
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.
Highlights
Most 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
[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
The saddle-node bifurcation is called foldbifurcation, tangent bifurcation, limit point bifurcation, or turning point bifurcation. from theorem (1.1) it follow that near (0,0) that DAEs (1.1) reduced to the system
Summary
Most 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.
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