Pudo-arclength Continuation
Anonymous and Sarah Brockman
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Abstract
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Pudo-arclength continuation is a continuation scheme forfinding solutions of an equation or systems of equations that depend on a parameterα,such as F(x,α)=0.Standard methods for approximating solutions,such as New-ton’s method,break down at turning point bifurcations or branching points becau the Jacobian matrix becomes singular.Thus,there is a need for more advanced continuation schemes such as pudo-arclength continuation to continuefinding solutions past such a point.
1.Standard Solution Methods
Newton’s method,also called the Newton-Raphson method,is a method offinding solutions to an equation or system of equations of the form F(x)= 0,or in our ca F(x,α)=0for parameterα.Newton’s method for a single
.Newton’s method for systems equation takes the form:x i+1=x i−f(x i)
宁波必去三大景点f (x i)
takes the form:x i+1=x i−J−1(x i)F(x i)where J is the Jacobian matrix. Clearly,if the Jacobian becomes singular,Newton’s method will break down. When we reach a turning point bifurcation or branching point,this will happen,and we cannot u Newton’s method near that point.We need some way of continuing past that point so we can continue using Newton’s method.
Preprint submitted to Math552
Figure1:Example of a Turning point bifurcation
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语无伦次什么意思Figure2:Example of a branching point bifurcation
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2.Background
The pudo-arclength continuation scheme solves the problems mentioned above.The arclength s is ud as a continuation parameter,so x and αare taken to be functions of s .So,now we need to find x and αsuch that F (x (s ),α(s ))=0.The pudo-arclength continuation scheme is called a predictor-corrector method.When a bifurcation point is found,a tangent vector is found at a point slightly before the bifurcation point.This tangent vector can be ud to predict the next values of x and α.Since the tangent predictor is usually not quite accurate,a corrector step is usually needed.For this,we can u the Newton-Raphson scheme mentioned above.We perform iterations of Newton’s method orthogonal to the tangent vector.Figure 3illustrates the behavior of the method well.
Figure 3:Pudo-arclength continuation
3.Implementation
We first implemented this method on the following equation:
f (x )=x −α194e x
1+αe x ,(1)
放下面子which depends on the parameter α.When we attempt to find solutions x of this equation using Newton’s method,we obtain the results en in figure
六年级下册4.Clearly,Newton’s method encounters a problem around α=0.1,when it reaches a bifurcation point.However,we can u the pudo-arclength
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Figure4:Newton’s method for Equation(1)
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Figure6:Newton’s method for Equation(2)
method to continue past this point and produce a smooth solution curve,as en infigure5.
We next applied the pudo-arclength method to the following system of equations,which model a two-component enzyme system:
dx1 dt =(α−x1)+(x2−x1)−ρR(x1),
dx2
dt
=(α+µ−x2)+(x1−x2)−ρR(x2),(2)
where
R(x)=
x
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1+x+κx2
.
Wefixed the parametersµ=0,ρ=100,andκ=1and attempted tofind solutions x=(x1,x2)while varying the parameterα.Like with Equation (1),Newton’s method reaches a bifurcation point when attempting to solve this system(figure6).Applying the pudo-arclength method allows the solution to once again bypass this bifurcation and obtain a smooth solution curve.The results can be en infigure7.
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Figure7:Pudo-arclength method for Equation(2)
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