Stability of autonomous systems

For an Autonomous system ${\mathbf{x}}^{\prime }=\mathbf{F}(\mathbf{x})$ with critical point ${\mathbf{x}}_0$ (where $\mathbf{F}({\mathbf{x}}_0)=0$), three flavors of stability characterize how nearby trajectories behave:

• Stable: nearby trajectories stay nearby for all time.
• Asymptotically stable: nearby trajectories actually converge to the critical point.
• Unstable: at least some nearby trajectories drift away.

Formal definitions

Stable

A critical point ${\mathbf{x}}_0$ is stable if for every $\epsilon >0$ there exists $\delta >0$ such that every solution $\mathbf{x}(t)$ satisfying $\Vert \mathbf{x}(0)-{\mathbf{x}}_0\Vert <\delta$ also satisfies $\Vert \mathbf{x}(t)-{\mathbf{x}}_0\Vert <\epsilon$ for all $t>0$.

In words: start close enough to ${\mathbf{x}}_0$, and you stay close to ${\mathbf{x}}_0$ forever. The further-away tolerance $\epsilon$ can be made as small as you want, with a corresponding starting tolerance $\delta$.

Asymptotically stable

A critical point ${\mathbf{x}}_0$ is asymptotically stable if:

1. ${\mathbf{x}}_0$ is stable (above).
2. There exists ${\delta }_1>0$ such that any solution starting with $\Vert \mathbf{x}(0)-{\mathbf{x}}_0\Vert <{\delta }_1$ satisfies

${\lim }_{t\to \infty }\mathbf{x}(t)={\mathbf{x}}_0$

Stability + actual convergence. Trajectories starting close enough not only stay close but eventually reach the critical point.

Unstable

${\mathbf{x}}_0$ is unstable if it’s not stable: there exists $\epsilon >0$ such that for every $\delta >0$, you can find an initial condition with $\Vert \mathbf{x}(0)-{\mathbf{x}}_0\Vert <\delta$ and a time $\tau >0$ where $\Vert \mathbf{x}(\tau )-{\mathbf{x}}_0\Vert >\epsilon$.

In words: no matter how close you start, you can drift arbitrarily far away.

For linear systems with constant coefficients

For ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with $\det A\ne 0$, the critical point $(0,0)$ is:

• Asymptotically stable if all eigenvalues of $A$ are real and negative, OR complex conjugate with negative real part.
• Stable but not asymptotically stable if eigenvalues are purely imaginary (centers).
• Unstable if any eigenvalue is real and positive, has opposite-sign real eigenvalues (saddle), or is complex with positive real part.

This is one of the cleanest results in dynamical systems: eigenvalues alone classify stability.

Eigenvalues	Type	Stability
Both real, both $<0$	Stable node	Asymp. stable
Both real, both $>0$	Unstable node	Unstable
Real, opposite signs	Saddle	Unstable
Complex, $\text{Re}<0$	Stable spiral	Asymp. stable
Complex, $\text{Re}>0$	Unstable spiral	Unstable
Pure imaginary	Center	Stable, not asymp.

For visualizations, see Phase plane behaviour.

Isolated equilibria

A critical point ${\mathbf{x}}_0$ is isolated if there’s a circle around it containing no other critical points. For a linear system ${\mathbf{x}}^{\prime }=A\mathbf{x}$ with $\det A\ne 0$, the origin is the unique critical point — automatically isolated.

If $\det A=0$, you might have a line of critical points (e.g., the line $y=-x$ if $A$ has rank 1). Each point on that line is a critical point, and none is isolated — circles around any one contain others.

Lemma: If an autonomous system has finitely many critical points, each is isolated.

Isolated critical points are easier to analyze — the eigenvalue methods work cleanly. Non-isolated equilibria require more care.

For nonlinear systems

For ${\mathbf{x}}^{\prime }=\mathbf{F}(\mathbf{x})$ at an equilibrium ${\mathbf{x}}_0$, linearize by computing the Jacobian $D\mathbf{F}({\mathbf{x}}_0)$. If the linearized system ${\mathbf{x}}^{\prime }=D\mathbf{F}({\mathbf{x}}_0)\mathbf{x}$ has a hyperbolic equilibrium (no eigenvalues on the imaginary axis), then the nonlinear system has the same stability type at ${\mathbf{x}}_0$.

The exception: when the linearization has eigenvalues on the imaginary axis (centers), the nonlinear system can behave differently — the linear theory doesn’t determine stability. Special techniques like Lyapunov’s method are needed.

For the linearization technique, see Locally linear system. For the energy-like-function approach to stability proofs, see Lyapunov’s method.

Why this matters

Stability tells you whether a system “settles down” or “blows up” near a particular state. Engineering applications:

• Control systems: design controllers to make the desired operating point asymptotically stable.
• Mechanical systems: a stable equilibrium of a robot’s pose is a “rest position”; an unstable one (like a pendulum balanced upside down) requires constant correction.
• Population dynamics: stable equilibria are sustainable populations; unstable ones are populations doomed to crash or explode.
• Electrical circuits: stable equilibria are steady-state operating points; unstable ones produce oscillations or runaway behavior.

The stability classification is the qualitative answer to “what does this system do in the long run?”