Let \(x^*\) be an equilibrium of the autonomous system \(\dot x=f(x)\). Stability concerns all nearby trajectories; attraction concerns their long-time limits. These are logically distinct.
Lyapunov, asymptotic, and exponential stability
Definition. \(x^*\) is Lyapunov stable if for every \(\varepsilon>0\) there is \(\delta>0\) such that \[ \|x_0-x^*\|<\delta \Longrightarrow \|\phi_t(x_0)-x^*\|<\varepsilon \quad(t\ge0). \] It is asymptotically stable if it is stable and nearby trajectories converge to \(x^*\). It is exponentially stable if \[ \|\phi_t(x_0)-x^*\| \le M e^{-\alpha t}\|x_0-x^*\| \] locally.
Linearization principle
Theorem. Suppose \(f\in C^1\), \(f(x^*)=0\), and \(A=Df(x^*)\).
- If every eigenvalue of \(A\) has negative real part, \(x^*\) is locally exponentially stable.
- If \(A\) has an eigenvalue with positive real part, \(x^*\) is unstable.
Proof of the stable case. Translate \(x^*=0\) and write \[ f(x)=Ax+r(x),\qquad \|r(x)\|=o(\|x\|). \] Choose a positive definite \(P\) solving \[ A^\top P+PA=-I. \] For \(V(x)=x^\top Px\), \[ \dot V=-\|x\|^2+2x^\top Pr(x). \] Near zero, the remainder is at most \(\tfrac12\|x\|^2\), so \(\dot V\le-cV\). Integration gives exponential decay. For an unstable eigenvalue, an invariant-cone or unstable-manifold argument produces arbitrarily small initial points that leave a fixed neighborhood.
The inconclusive center case
If all eigenvalues have nonpositive real parts and some have zero real part, linearization alone is inconclusive. The scalar systems \[ \dot x=-x^3,\qquad \dot x=x^3 \] both have zero derivative at the origin; the first is asymptotically stable and the second unstable. Nonlinear terms on the center manifold decide.
Orbital stability
A periodic orbit cannot be asymptotically stable as an individual phase point because time translation gives a neutral tangent direction. Orbital stability measures distance to the whole orbit. Floquet multipliers of the Poincaré map determine transverse stability; one trivial multiplier equals one for the flow direction.
Input-to-state stability
For a forced system \[ \dot x=f(x,u), \] input-to-state stability asks for \[ \|x(t)\| \le\beta(\|x_0\|,t)+\gamma(\|u\|_\infty), \] with \(\beta\) decaying in time. This separates decay of initial conditions from persistent forcing and is the nonlinear analogue of stable variation-of-constants bounds.