# Thread: Characterizing/classifying with system of equations

1. ## Characterizing/classifying with system of equations

"$\displaystyle x_1' = x_1 -\alpha x_2$

$\displaystyle x_2' = \beta (x_1 - x_2 - G)$

$\displaystyle 1< \alpha < \infty, 1 \leq \beta < \infty$

a) Show that if $\displaystyle G = G_0$ is constant, there is an equilibrium state. Classify the equilibrium state and show that the system oscillates when $\displaystyle \beta$ = 1.

b) Suppose $\displaystyle G=G_0+kx_1$ where k>0. Show that there is no equilibrium state if $\displaystyle k \geq (\alpha -1)/\alpha$. How does the system behave?

c) Show there are two equilibrium states if $\displaystyle G=G_0+kx_1^2$. Characterize the equilibrium points."

I'm pretty confused by how my professor worded this problem. Can someone just clarify or paraphrase what it's asking for? I'm particulary wondering what he meant by "Classify the equilibrium state..." in part a and "Characterize the equilibrium points..." in part c.

"$\displaystyle x_1' = x_1 -\alpha x_2$

$\displaystyle x_2' = \beta (x_1 - x_2 - G)$

$\displaystyle 1< \alpha < \infty, 1 \leq \beta < \infty$

a) Show that if $\displaystyle G = G_0$ is constant, there is an equilibrium state. Classify the equilibrium state and show that the system oscillates when $\displaystyle \beta$ = 1.

b) Suppose $\displaystyle G=G_0+kx_1$ where k>0. Show that there is no equilibrium state if $\displaystyle k \geq (\alpha -1)/\alpha$. How does the system behave?

c) Show there are two equilibrium states if $\displaystyle G=G_0+kx_1^2$. Characterize the equilibrium points."

I'm pretty confused by how my professor worded this problem. Can someone just clarify or paraphrase what it's asking for? I'm particulary wondering what he meant by "Classify the equilibrium state..." in part a and "Characterize the equilibrium points..." in part c.
Your equilibrium or critical points (as sometimes they're called), is when $\displaystyle x_1' = 0,\; x_2' = 0$. Then we ask, if we move away from the critical point will we return (asymptotically stable), will we stay a certain distance away (stable) or go away further away from the critical point (unstable). To determine this we look at the eigenvalues of the coefficient matrix (if linear) or look at the linearized system (if nonlinear). Care must be taken in the latter because if one of the eignevalues has zero real part, then the linear system can't predict the stablilty of the nonlinear system.

Make sense?