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Math Help - Characterizing/classifying with system of equations

  1. #1
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    Characterizing/classifying with system of equations

    "  x_1' = x_1 -\alpha x_2

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

     <br />
1< \alpha < \infty, 1 \leq \beta < \infty<br />

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

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

    c) Show there are two equilibrium states if 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.
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  2. #2
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    Quote Originally Posted by seadog View Post
    "  x_1' = x_1 -\alpha x_2

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

     <br />
1< \alpha < \infty, 1 \leq \beta < \infty<br />

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

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

    c) Show there are two equilibrium states if 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 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?
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