Nonlinear Systems Stability Analysis

If r_y > 0 in N’, the equilibrium state is unstable. If r_y ≤ 0 in N’ and xr(x, 0) < 0 for all x in N’, the equilibrium state is stable. This stability is asymptotic provided that the strict inequality r_y < 0 holds. If r_y is indefinite in every neighborhood of the equilibrium state, this state may be either stable or unstable.

The last statement of the theorem will be clarified as follows. The equilibrium state of the following system is unstable:

$\begin{array}{l}\dot{x}=y,\\ \dot{y}=x+xy.\end{array}$

(2.193)

This is a consequence of the fact that one of the characteristic roots of the linearized system corresponding to (2.193) is positive. On the other hand, the equilibrium state of the following system:

$\begin{array}{l}\dot{x}=y,\\ \dot{y}=-x+xy,\end{array}$

(2.194)

is stable. The stability of the equilibrium state ...