Help ODE System Stability - Origin Analysis

[Carl140]

Homework Statement

Hello.


I want to study the stability of the origin of the following problem:


dx/dt = -2y


dy/dt = x + 2y


dz/dt = -2z


So the eigenvalues of this 3 x 3 matrix are -2, 1 + i, 1-i.


The eigenvectors are (0,0,1) , (2,-1-i,0), (-2,-1+i,0).


The solution (confirmed with Mathematica) is given by:


x(t) = 2*exp(t)cos(t) * C_1 + 2*exp(t)*sin(t) * C_2 - 2*exp(t)*sin(t) *C_3


y(t) = C_1 * (-exp(t)*¨sin(t) - exp(t)*cos(t)) + C_2 * (exp(t)cos(t) -exp(t)sin(t)) +
C_3 * (-exp(t)cos(t)+exp(t)sin(t) )


z(t) = 2*exp(2*t) *C_3


Where C_1,C_2,C_3 are constants.


How can I find (analytically, not by plotting) if the origin (0,0,0) is stable, asymptotically stable? unstable, a node, a center?


I'm having trouble figuring this out. Can you please help?

 

[Mark44]

Homework Statement

Hello.


I want to study the stability of the origin of the following problem:


dx/dt = -2y


dy/dt = x + 2y


dz/dt = -2z


So the eigenvalues of this 3 x 3 matrix are -2, 1 + i, 1-i.


The eigenvectors are (0,0,1) , (2,-1-i,0), (-2,-1+i,0).


The solution (confirmed with Mathematica) is given by:


x(t) = 2*exp(t)cos(t) * C_1 + 2*exp(t)*sin(t) * C_2 - 2*exp(t)*sin(t) *C_3


y(t) = C_1 * (-exp(t)*¨sin(t) - exp(t)*cos(t)) + C_2 * (exp(t)cos(t) -exp(t)sin(t)) +
C_3 * (-exp(t)cos(t)+exp(t)sin(t) )


z(t) = 2*exp(2*t) *C_3


Where C_1,C_2,C_3 are constants.


How can I find (analytically, not by plotting) if the origin (0,0,0) is stable, asymptotically stable? unstable, a node, a center?


I'm having trouble figuring this out. Can you please help?

Is there a typo in z(t)? One of your eigenvalues is -2, so I would expect to see e^(-2t) in one of your solution functions.

Because x(t) and y(t) both have terms with e^t, I would expect orbits that move away from the origin over time, which would make the origin unstable or a node (I don't recall exactly what these terms mean in the context of phase diagrams. And because z(t) is a decaying exponential function, whatever the orbits are doing, they are going to be heading down to the x-y plane over time. I'm just going off the top of my head here, so take what I'm saying with a grain of salt.