4.1 Informal Introduction to Stability

In general, it can be difficult to obtain an explicit solution of nonlinear differential equations modeling a dynamical system. However, often we do not need to know the explicit solution of the modeling equations. Instead, we are interested in the behavior of the solution as time tends to infinity. Furthermore, in many models of physical systems a "small" change in the initial conditions results in a "small" change in the solution. Specifically, if a system is perturbed from its rest, or equilibrium position, then it starts to move. We can roughly say that the equilibrium position is stable if the system does not go far from this position for small initial perturbations. This concept may be illustrated, as in Willems [300], Brogan [35], or Barnett [20], by means of a ball resting in equilibrium on a sheet of metal bent into various shapes with cross sections depicted in Figure 4.1. This figure illustrates four possible scenarios. If we neglect frictional forces, a small perturbation from the equilibrium leads to the following:

1. Oscillatory motion about equilibrium irrespective of the magnitude of the initial perturbation. In this case the equilibrium position is globally stable.

2. The new position is also an equilibrium,...