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November 20th, 2018, 02:52 AM   #1
Joined: Nov 2018
From: Baku

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Defining a Jacobian for a given problem

Hello everyone, reaching out my post. I have a trouble of defining a Jacobian matrix for my problem. Basically, I have 4 differential equations to be solved.

$\dot x_1(t)=x_2(t),$

$\dot x_2(t)=p_2(t)−\sqrt 2 x_1(t)e^{-αt},$

$\dot p_1(t)=\sqrt 2p_2(t)e^{-αt}+x_(t)$

$\dot p_2(t)=−p_1(t)$

with initial and boundary values of: $x_1(0)=1, p_2(0)=0,p_1(1)=0,p_2(1)=0$
Following classic shooting method strategy, I suggest some values for $x_2(0), p_1(0)$, solve the system of diff equations using Dormand Prince method, and obtain some values at the boundary. Here I check, if the solution on my boundaries meets the criteria of $p_1(1)=0,p_2(1)=0$. If not, I have to suggest new values for initial $x_2(0), p_1(0)$, to be iterated again. Since this is two-parametric shooting, I can't use simple linear interpolation as in one-parametric shooting, so I have to use Newton's method for that. Here is my question: How to implement Newton's method to iterate towards the right solution? I tried myself, and doesn't work. Can you at least help me with defining a Jacobian, for the given problem of iterative solution to $x_2(0), p_1(0)$
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November 21st, 2018, 10:16 PM   #2
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Its not really true that you can't use linear interpolation as your initial condition. The usual approach is to do one parameter continuation twice.

The basic gist would be to do something like:

1. Let $a = x_2(0), b = p_1(0)$ be your continuation parameters. Begin by fixing $b$ and allowing $a$ to remain free.

2. Now, you do continuation in the free parameter and iteratively shoot until you have satisfied one of the boundary conditions. You can choose to shoot in the direction of the gradient with respect to both boundary conditions (this is pseudo-Newton). The intuition here is like walking to the store by crossing an empty field.

On the other hand, you can simply ignore one boundary condition and continue only in a single direction. This is analgous to walking toward the store by walking east down a road.

3. Ideally, (I'll cover what can go wrong shortly) you will continue in $a$ until you have satisfied one boundary condition. Lets say we have obtained the value $a_0$ for our fixed $b$ for which $p_1(1) = 0$ is satisfied. Now, we fix the first parameter at $a_0$, free the second parameter, and continue in this direction. Intuitively, we have reached the intersection, turned left, and are now heading north toward the store.

4. Generically, this method of iterative continuation one variable at a time will converge to the BVP solution. However, 2 things can go wrong. The first problem is you can choose an initial $a,b$ which lie in the wrong connected component of the the manifold which you are continuing along. However, the only fix for this is a more in depth global analysis of the ODE. Keep in mind, solving this problem via 2-parameter Newton's method will also fail in this case.

The second thing which can happen is you hit a saddle-node bifurcation while doing continuation in one parameter. In this case, the fix is simple. There is no rule which says you have to do continuation in $a$, followed by $b$, and then converge. When you hit a saddle-node during continuation in $a$, you simply fix it and switch to continuation in $b$ until you pass it. Sine the BVP solution lies on the intersection of two 1-dimensional manifolds, you won't hit a saddle-node bifurcation with respect to both parameters simultaneously (this is a co-dimension 2 bifurcation in this setting which is non-generic)
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