How can we investigate the stability of a specific equilibrium in a 3-D flow when the real part of one (or more) of the eigenvalues is zero? I think such systems are called nonhyperbolic.
I know the procedure for 1-D systems, but I couldn’t find the exact method for 3-D. for example, I want to investigate the stability of the origin (0,0,0) for the following system:
x_dot = y
y_dot = -x +z
z_dot = -0.8x^2 +z^2
Hello Sajad,
I don't understand notation x_dot.
Is it derivative of x by time: x'?
If it is then one way of investigation is to transform your system to the third order equation by elimination of y and z (in your case one can get (x''+x)'=(x''+x)^2-0.8x^2 and then using Euler method x(t)=\exp(\int \xi(t) dt) calculate the characteristic equation for the phase function \xi(t). In your case it is complicated nonlinear equation but you can try to find asymptotic solutions, which could be used to find behavior of the original system. Note that first approximations of \xi(t) are eigenvalues of original system.
Thank you Gro. Yes, x_dot is x'.
I will investigate the method you mentioned. However not all the 3-D equations can be written in the form of jerk system.
You may have to Taylor expand the displacements about the origins to higher orders. Look at the Hessian instead of the Jacobian. Of course, an analytical solution may not be possible...
- Badges
- Science topic
- Similar topics
- Biological Science
- Bioecology
- Systems Ecology
- Landscape Ecology
- Connectivity