Solving System of ODEs: Finding Fixed Points & Stability

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In summary, the conversation discusses the task of finding the fixed points and determining their stability for a system of differential equations. The fixed point is (1,1) and the Jacobian is used to calculate the eigenvalues and eigenvectors, which determine the type of fixed point. The system is then graphed using a StreamPlot to visualize the behavior of the trajectories.
  • #1
The question asks me to consider the system of differential equations:

[itex]\frac{dx}{dt} = 1 - 2x + x^2y [/itex]
[itex]\frac{dy}{dt} = x-x^2y[/itex]

It asks me to find the fixed point(s), and determine their stability, also to draw the phase plane.

So to find the fixed points, I set both equations equal to zero:
[itex]1 - 2x + x^2y = 0 \Rightarrow x^2y = 2x-1[/itex]
[itex]x-x^2y = 0 \Rightarrow x^2y = x[/itex]

Setting them equal to each other:

[itex]2x-1 = x \Rightarrow x=1[/itex] and from this we can deduce [itex]y=1[/itex]. So the fixed point is (1,1).

Now this is where I'm not really sure on what to do and get confused by my lecture notes. I believe I have to use some Jacobian to work out the stability yet I have no clue why I'm doing this (I know eigenvalues/eigenvectors are involved somehow). Setting [tex]f(x,y)= 1 - 2x + x^2y[/tex] and [tex]g(x,y)= x-x^2y[/tex] I worked out the Jacobian at (1,1) to be:

\begin{equation}

\left|
\begin{array}{cc}
0 & 1 \\
-1 & -1 \end{array}
\right|

\end{equation}

(Sorry about the lack of detail about this part but I'm new to latex and struggled with doing matrices, the jacobian is fx and fy on the first row and gx and gy on the second row.

Following lecture notes, I work out the 'trace' (T) of this matrix which is simply (a11+a22) which is -1 in this case, and also work out the determinant which is 1.

Since [itex]T^2 - 4D = -3 < 0[/itex], we have complex eigenvalues and I believe this means the phase plane will be a spiral, and since T < 0 this means the general trajactory will be moving towards the fixed point (stable). But do I need to work out the eigenvectors to see exactly what this spiral will look like? I'm not sure if I've worked out enough to complete the question. Also any help on actually explaining why I'm working out the Jacobian stuff would be great.

Thanks
 
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  • #2
The Jacobian is just the 2D Taylor series for the right-hand sides truncated to the linear terms. That's how you "linearize" the non-linear system and for values close to the fixed-point, the linear version is usually a good approximation to the non-linear version. I think you're correct in the type of fixed-point you have: if the real part of the eigenvalues are negative, then the fixed-points are sinks. Then do a StreamPlot of the system in Mathematica to get the slope field or rather do one or two manually first to get the idea down, then StreamPlot.
 
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1. What is a fixed point in a system of ODEs?

A fixed point is a point in a system of ODEs where all of the derivatives are equal to zero. This means that the values of the variables at this point do not change over time.

2. How do you find fixed points in a system of ODEs?

To find fixed points, you set all of the derivatives in the system of ODEs equal to zero and solve for the values of the variables. These values will be the fixed points of the system.

3. What is stability in relation to fixed points in a system of ODEs?

Stability refers to the behavior of the system around a fixed point. A fixed point is considered stable if the system returns to that point after small perturbations, while it is considered unstable if the system moves away from the fixed point after small perturbations.

4. How do you determine the stability of a fixed point in a system of ODEs?

The stability of a fixed point can be determined by analyzing the eigenvalues of the Jacobian matrix at that point. If all eigenvalues have negative real parts, the fixed point is stable. If any eigenvalues have positive real parts, the fixed point is unstable.

5. What are some applications of solving systems of ODEs and finding fixed points?

Solving systems of ODEs and finding fixed points is important in many fields, including biology, chemistry, physics, and economics. It can be used to model and predict the behavior of complex systems, such as population dynamics, chemical reactions, and economic systems.

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