how to find stable and unstable spiral point?

Hi I am working on a differential equation given "0 == diff(x,2) + (z.*((1/3)*(diff(x))^3-diff(x)) + x)". To show wether (0,0) is a stable or unstable spiral point when z>0 and <0. also to show (0,0)is the center when z=0.

Answer 1

[Saarthak Gupta]

Hi Yixuan,

I understand you are trying to find the nature of an equilibrium point (0,0) under various conditions of the parameter z, for the second-order differential equation:

The following steps can help you achieve the desired result:

• Convert the second-order differential equation into a system of two first-order differential equations

• Calculate the Jacobian matrix for the system

• Find roots of the characteristic polynomial of the Jacobian at (0,0):
• The roots hence obtained will be in terms of z i.e,

An equilibrium point is called a stable spiral point if the Jacobian matrix has two complex eigenvalues with negative real parts.

We can evaluate both cases, i.e., z > 0 and z < 0 separately. For a case, if the eigenvalues obtained in step 4 are complex with negative real parts, we can conclude that (0,0) is a stable spiral point, otherwise it is unstable.

The Symbolic Math Toolbox can be used to calculate the Jacobian, and to compute the eigenvalues.

• The ‘jacobian’ function can be used to compute the Jacobian matrix.
• The ‘eig’ function can be used to compute the eigenvalues of the Jacobian matrix obtained above.

Please refer to the following MATLAB documentation for more details:

• https://in.mathworks.com/help/symbolic/sym.jacobian.html
• https://in.mathworks.com/help/symbolic/eig.html

Answer 2

[Sam Chak]

@Yixuan Qi, This is actually the Rayleigh Differential Equation. I assume you would like to utilize a graphical approach, as the terms "stable spiral" and "unstable spiral" suggest this. You can check out this link:

https://www.mathworks.com/matlabcentral/answers/1659715-quiver-and-streamline-functions?s_tid=srchtitle

% Rayleigh Differential Equation
title('\mu < 0, Stable spiral')

title('\mu = 0, Center')

title('\mu > 0, Unstable spiral')