In mathematics, stability theory studies the stability of solutions of differential equations and dynamical systems, that is, it examines how the solutions differ under small modifications of the initial conditions.

Stability is very important in physics and applied sciences, since in general in practical problems the initial conditions are never known with complete precision, and predictability requires that small initial deviations do not generate qualitatively very different behaviors in the short term. When the difference between two solutions with close initial values ​​can be limited by the difference in initial values, the temporal evolution of the system is said to be stable.

Because every differential equation can be reduced to a system of equivalent first-order differential equations, the study of the stability of solutions of differential equations can be reduced to the study of the stability of systems of differential equations. Consider for example a nonlinear autonomous system of equations given by:.

Where is the state vector of the system, an open set containing the origin and a continuous function. Without loss of generality, it can be assumed that the origin is an equilibrium point (if the equilibrium point were another point, a variable change can be made and the function f can be redefined to coincide with the origin):

1. • The origin point is stable in the Lyapunov sense, if for each , there exists such that, if , then , for any .

2. • The origin point is asymptotically stable if it is stable in the Lyapunov sense and exists such that if , then .

3. • The origin point is exponentially stable if it is asymptotically stable and if there are such that if , then , for .

Conceptually, the above definitions can be interpreted as:

1. • Stability in the Lyapunov sense of an equilibrium point means that solutions that start "close enough" to an equilibrium point (at a distance from them) remain "close enough" forever (at most at a distance from each other). Note that this should be true for anyone one chooses.