Locally linear system

A locally linear system is a nonlinear ODE system that can be approximated near a critical point by a linear system. Linearization is the standard technique for analyzing nonlinear systems’ local behavior — replace the system with its linear part near the equilibrium, then apply linear stability theory.

Definition

The system

${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{g}(\mathbf{x})$

with $A\in {\mathbb{R}}^{2\times 2}$ and $\mathbf{g}(\mathbf{x})=\left[\begin{array}{c}F(x,y)\\ G(x,y)\end{array}\right]$ is locally linear in a neighborhood of the critical point $(0,0)$ if:

1. $F(x,y)$ and $G(x,y)$ are continuous in a disk around $(0,0)$.
2. $(0,0)$ is an isolated critical point of ${\mathbf{x}}^{\prime }=A\mathbf{x}+\mathbf{g}(\mathbf{x})$.
3. $\det A\ne 0$ (so the origin is also an isolated critical point of the linear system ${\mathbf{x}}^{\prime }=A\mathbf{x}$).
4. The nonlinear part is small near the origin:

${\lim }_{\Vert \mathbf{x}\Vert \to 0}\frac{\Vert \mathbf{g}(\mathbf{x})\Vert }{\Vert \mathbf{x}\Vert }=0$

Condition 4 is the key: it says the nonlinear correction $\mathbf{g}$ vanishes faster than $\mathbf{x}$ as you approach the critical point. So very close to the origin, $A\mathbf{x}$ dominates and the system behaves like its linearization.

Why we linearize

Near an equilibrium, the nonlinear behavior is approximately linear. If $\mathbf{F}({\mathbf{x}}_0)=0$, then by Taylor expansion:

$\mathbf{F}(\mathbf{x})=D\mathbf{F}({\mathbf{x}}_0)(\mathbf{x}-{\mathbf{x}}_0)+\mathcal{O}(\Vert \mathbf{x}-{\mathbf{x}}_0{\Vert }^2)$

The first-order (linear) term is the Jacobian $D\mathbf{F}({\mathbf{x}}_0)$ acting on the displacement. The higher-order terms are the nonlinear corrections, which vanish faster than the linear part.

For points very close to ${\mathbf{x}}_0$, the higher-order terms are negligible — and the local dynamics are determined by the eigenvalues of $D\mathbf{F}({\mathbf{x}}_0)$.

This is the Hartman-Grobman theorem (in its formal version): hyperbolic equilibria of nonlinear systems are locally topologically equivalent to their linearizations. As long as no eigenvalue lies on the imaginary axis, linear theory tells you the truth about the local nonlinear behavior.

Worked example

Consider:

$\{\begin{array}{l}{x}^{\prime }=x-x^2-xy\\ y^{\prime }=\frac{1}{2}y-\frac{3}{4}xy-\frac{1}{4}{y}^2\end{array}$

Identify linear and nonlinear parts:

${\mathbf{x}}^{\prime }=\left[\begin{array}{cc}1& 0\\ 0& 1/2\end{array}\right]\mathbf{x}+\left[\begin{array}{c}-x^2-xy\\ -\frac{3}{4}xy-\frac{1}{4}{y}^2\end{array}\right]$

Verify locally linear at $(0,0)$:

1. $F=-x^2-xy$ and $G=-\frac{3}{4}xy-\frac{1}{4}{y}^2$ are continuous everywhere — ✓.
2. $(0,0)$ is isolated for the full system (other critical points are far from origin) — ✓.
3. $\det A=\frac{1}{2}\ne 0$, so origin is isolated for linear part too — ✓.
4. Switch to polar: $x=r\cos \theta$, $y=r\sin \theta$. Then $F=-r^2({\cos }^2\theta +\cos \theta \sin \theta )$, so $\Vert \mathbf{g}\Vert /\Vert \mathbf{x}\Vert =O(r)\to 0$ as $r\to 0$ — ✓.

So the system is locally linear at $(0,0)$. Its local behavior is determined by the eigenvalues of $A=\left[\begin{array}{cc}1& 0\\ 0& 1/2\end{array}\right]$ — both positive ($1$ and $1/2$). So the origin is an unstable node of the nonlinear system too. (Some introductory texts call this an “improper node,” but in standard dynamical-systems terminology “improper” specifically denotes the defective repeated-eigenvalue case where the matrix isn’t diagonalizable. Distinct eigenvalues $1$ and $1/2$ give a plain node; use “improper” only if your course’s textbook does.)

When linearization fails

If $D\mathbf{F}({\mathbf{x}}_0)$ has eigenvalues on the imaginary axis (a center, in 2D), the linear theory is inconclusive. The nonlinear system might be a center, a stable spiral, or an unstable spiral — depending on the higher-order terms.

This is the failure mode of linearization. For these cases, you need other tools:

• Lyapunov’s method — energy-like functions to determine stability directly.
• Center manifold theory (advanced) — restrict to the unstable directions and analyze them carefully.
• Numerical simulation as a sanity check.

Pure pendulum (without damping) has a center at the rest position. The nonlinear correction (the difference between $\sin \theta$ and $\theta$) doesn’t change the center qualitatively — but for slightly different systems with similar linearizations, the nonlinear terms can flip stability.

Multiple equilibria

For a nonlinear system with several critical points, linearize around each one separately. Each linearization gives a local picture. The full phase portrait is built by stitching together local pictures from all equilibria.

For the linear classification of equilibria, see Phase plane behaviour. For the energy method that handles cases linearization can’t, see Lyapunov’s method.