Locally Linear Systems and Liapunov's Method

A linear system has an eigenvalue formula for the type and stability of its one critical point; a nonlinear system has no such formula. Near a critical point, though, its direction field is almost indistinguishable from that of a linear system, and two tools exploit this. Linearization replaces the nonlinear system near a critical point by its Jacobian and reads type and stability off the eigenvalues, valid in every case but two. Liapunov's direct method handles those two cases and more, deciding stability by constructing an energy-like function without ever solving the differential equation.

Locally linear systems

Place the critical point of interest at the origin; a substitution $\mathbf{u}=\mathbf{x}-{\mathbf{x}}_0$ always arranges this. Split the system into its linear and nonlinear parts,

where $A$ is a constant matrix with $\det A\ne 0$, so that the origin is an isolated critical point. The question is how small $\mathbf{g}$ must be for the trajectories of ${\mathbf{x}}^{\prime }=A\mathbf{x}$ to approximate those of the full system near the origin.

The condition need not be checked by hand. If $F$ and $G$ have continuous second partial derivatives, Taylor's theorem supplies the linear part automatically. Expanding about a critical point $(x_0,y_0)$ where $F(x_0,y_0)=G(x_0,y_0)=0$,

and likewise for $G$, with the remainders ${\eta }_i$ satisfying ${\eta }_i/\Vert \mathbf{x}-{\mathbf{x}}_0\Vert \to 0$. The linear coefficients assemble into the Jacobian matrix.¹

Validity of the linearization

The central theorem says that the type and stability of the nonlinear critical point match those of the linear system — except in two borderline cases.³

Both exceptions sit at the borderline configurations of the trace–determinant plane:

• Pure imaginary eigenvalues (a center). The linear system predicts closed orbits, but the nonlinear terms may spiral them slowly inward or outward. The center can become a spiral sink, a spiral source, or remain a center; the linearization cannot tell which.
• Equal real eigenvalues (a node). The nonlinear terms may bend the node into a spiral point. The stability is unaffected — a stable node stays stable — but the geometric type is undetermined.

Away from the two sensitive rows, a nonlinear critical point can be classified by a single evaluation of the Jacobian. The trajectories of the full system may look quite different from the linear ones far from the critical point, but the slopes at which trajectories enter or leave the critical point are given correctly by the linear system.

The indeterminate center row remains. The undamped pendulum $x^{\prime }=y$, $y^{\prime }=-{\omega }^2\sin x$ has a center at the origin for the linear system, and the theorem says nothing about the nonlinear system. Resolving it requires a different method.

Liapunov's direct method

Liapunov's method comes from mechanics. For a conservative system, two physical principles govern equilibrium: a rest position is stable when the potential energy has a local minimum there, and the total energy stays constant along any motion. Liapunov abstracted the total energy into an auxiliary function whose behavior along trajectories certifies stability — with no need to solve the system, which is why it is called a direct method.⁴

Consider the general autonomous system $x^{\prime }=F(x,y)$, $y^{\prime }=G(x,y)$ with an isolated critical point at the origin. The rate of change of a function $V(x,y)$along a trajectory is computed by the chain rule, using the differential equations themselves to substitute for $x^{\prime }$ and $y^{\prime }$:

The key is that $\dot{V}$ is obtained without knowing the solution: it is a function of position alone, assembled from $V$ and the right-hand sides. The sign conditions that make $V$ useful have names.

Geometry of the method

The geometry is a family of nested closed curves $V(x,y)=c$, shrinking to the origin as $c\to 0$. The gradient $\nabla V$ is normal to each level curve and points outward, in the direction of increasing $V$. Along a trajectory the velocity vector $\mathbf{T}=(x^{\prime },y^{\prime })$ is tangent to the path, and

If $\dot{V}\le 0$, the angle between $\nabla V$ and $\mathbf{T}$ is obtuse (or right): the motion points inward across each level curve, or at worst along it. A trajectory that starts inside a level curve $V=c$ can never cross to the outside, so it stays trapped near the origin — stability. If $\dot{V}<0$ strictly, the crossing is strictly inward everywhere, and the trapped trajectory is squeezed down to the origin — asymptotic stability.⁵

The quadratic-form criterion

The theorems say nothing about how to find a Liapunov function; no general construction exists. For a physical problem the total energy is the natural first guess. Otherwise a quadratic trial function $V=a{x}^2+bxy+c{y}^2$ is standard, and an algebraic test decides definiteness.⁶

Conservative systems and the energy function

With damping the calculation changes. For the damped pendulum the same energy gives $\dot{V}=-cL{y}^2\le 0$: the energy is nonincreasing, and strictly decreasing except on the axis $y=0$. Along every trajectory the energy decreases to its minimum, the stable rest position — the physical statement that friction converts the center into an asymptotically stable spiral.

Estimating the basin of attraction

Liapunov's method does more than classify a critical point; it bounds the basin of attraction, which linearization cannot address.⁸

The largest sublevel set $\{V<K\}$ on which $\dot{V}$ stays negative definite is a guaranteed region of asymptotic stability, usually a conservative underestimate of the true basin (a better Liapunov function enlarges it) but a rigorous inner bound obtained without any solution formula. Linearization gives no such global information; Liapunov's method applies both when the linear analysis is indeterminate at a center and when the basin, not just the type of a critical point, is what matters.