next up previous contents
Next: 1.2.4 Pairs of non-linear Up: 1.2.3 Pairs of linear Previous: 1.2.3.1 Stability analysis

1.2.3.2 Nullclines

We defined Nullclines as the curves on which we have either $ \dot{v}_1 = 0$ or $ \dot{v}_2 = 0$. These curves are a great graphical tool for finding equilibrium points, analyzing their stability and drawing qualitatively the direction field in the phase plane. We draw the nullclines and the corresponding direction field for three examples in which the equilibrium point is at (1,1):
  1. Example 1:

    $\displaystyle \left \{ \begin{array}{lc} \dot{v}_1(t) = v_1(t) - v_2(t)  \dot{v}_2(t) = v_1(t) + v_2(t) - 2  \end{array} \right.$    

    The equilibrium point is unstable in this case.

  2. Example 2:

    $\displaystyle \left \{ \begin{array}{lc} \dot{v}_1(t) = -v_1(t) + v_2(t)  \dot{v}_2(t) = - v_1(t) - v_2(t) + 2  \end{array} \right.$    

    The equilibrium point is stable in this case.

  3. Example 3:

    $\displaystyle \left \{ \begin{array}{lc} \dot{v}_1(t) = -v_1(t) + v_2(t)  \dot{v}_2(t) = v_1(t) + v_2(t) - 2  \end{array} \right.$    

    The equilibrium point is semi-stable in this case. (There is a direction in which the equilibrium point is stable.)



Eran Borenstein
2007-05-04