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:
