--- title: Locally Linear Systems and Liapunov's Method module: Nonlinear Systems and Stability moduleNumber: 8 lessonNumber: 2 order: 802 summary: > Near a critical point a nonlinear system looks linear, and the linear part is the Jacobian. The linearization fixes the type and stability of the nonlinear critical point in every case except a center or a repeated eigenvalue. Liapunov's direct method settles those cases and bounds the basin of attraction by constructing an energy-like function, without solving the system. topics: [Nonlinear Systems and Stability] sources: - book: Boyce ref: "Ch. 9 §9.3 Locally Linear Systems; §9.6 Liapunov's Second Method" - book: Simmons ref: "Ch. 11 §61 Stability by Liapunov's Direct Method; §62 Simple Critical Points of Nonlinear Systems; §63 Nonlinear Mechanics; Conservative Systems" draft: false --- 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, $$ \mathbf{x}' = A\mathbf{x} + \mathbf{g}(\mathbf{x}), $$ where $A$ is a constant matrix with $\det A \neq 0$, so that the origin is an isolated critical point. The question is how small $\mathbf{g}$ must be for the trajectories of $\mathbf{x}' = A\mathbf{x}$ to approximate those of the full system near the origin. > **Definition (Locally linear system).** The system $\mathbf{x}' = A\mathbf{x} + > \mathbf{g}(\mathbf{x})$ is **locally linear** near the origin if > $\mathbf{g}$ has continuous first partial derivatives and > $$ > \frac{\|\mathbf{g}(\mathbf{x})\|}{\|\mathbf{x}\|} \to 0 \quad \text{as } \mathbf{x} \to \mathbf{0}. > $$ > The nonlinear term vanishes faster than $\mathbf{x}$ itself; it is negligible > compared to the linear term 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 $F(x_0, y_0) = G(x_0, y_0) = 0$, $$ F(x, y) = F_x(x_0, y_0)(x - x_0) + F_y(x_0, y_0)(y - y_0) + \eta_1, $$ and likewise for $G$, with the remainders $\eta_i$ satisfying $\eta_i / \|\mathbf{x} - \mathbf{x}_0\| \to 0$. The linear coefficients assemble into the **Jacobian matrix**.[^boyce-jac] > **Definition (Jacobian matrix).** For $\mathbf{f} = (F, G)$, the matrix of > first partial derivatives > $$ > J(x, y) = \begin{pmatrix} F_x & F_y \\ G_x & G_y \end{pmatrix}. > $$ > The linear system approximating $\mathbf{x}' = \mathbf{f}(\mathbf{x})$ near a > critical point $(x_0, y_0)$ is $\mathbf{u}' = J(x_0, y_0)\,\mathbf{u}$, with > $\mathbf{u} = \mathbf{x} - \mathbf{x}_0$. $$ % caption: Linearization evaluates the Jacobian at the critical point; the % resulting constant matrix is the coefficient matrix of the local linear system. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize, box/.style={draw, minimum width=34mm, minimum height=15mm, align=center}] \definecolor{acc}{HTML}{4A6FA5} \node[box] (nl) at (0,0) {nonlinear system\\$x'=F(x,y)$\\$y'=G(x,y)$}; \node[box] (jac) at (5.4,0) {Jacobian at $(x_0,y_0)$\\entries $F_x,\ F_y,\ G_x,\ G_y$}; \node[box, draw=acc, text=acc] (lin) at (10.8,0) {local linear system\\$\mathbf{u}'=J(x_0,y_0)\,\mathbf{u}$}; \draw[->, acc, thick] (nl) -- (jac) node[midway, above, font=\scriptsize, black!70] {differentiate}; \draw[->, acc, thick] (jac) -- (lin) node[midway, above, font=\scriptsize, black!70] {evaluate}; \end{tikzpicture} $$ > **Worked example.** Linearize the damped pendulum at its two equilibrium > types. With $x = \theta$ and $y = \theta'$, > $$ > x' = y, \qquad y' = -\omega^2 \sin x - \gamma y, > $$ > the critical points are $(\pm n\pi, 0)$. Writing $F = y$ and > $G = -\omega^2 \sin x - \gamma y$, the Jacobian is > $$ > J(x, y) = \begin{pmatrix} 0 & 1 \\ -\omega^2 \cos x & -\gamma \end{pmatrix}. > $$ > At the origin $(0, 0)$, $\cos x = 1$, so > $$ > \mathbf{u}' = \begin{pmatrix} 0 & 1 \\ -\omega^2 & -\gamma \end{pmatrix}\mathbf{u}, > \qquad r_{1,2} = \frac{-\gamma \pm \sqrt{\gamma^2 - 4\omega^2}}{2}. > $$ > For small damping $\gamma^2 < 4\omega^2$ the eigenvalues are complex with > negative real part: the origin is an asymptotically stable spiral, the bob > settling into decaying oscillations. At the inverted position $(\pi, 0)$, > $\cos x = -1$, and the eigenvalues become > $r_{1,2} = (-\gamma \pm \sqrt{\gamma^2 + 4\omega^2})/2$, one positive and one > negative: an unstable saddle, the bob balanced on end.[^boyce-pend] $$ % caption: Phase portrait of the pendulum: hanging equilibria are stable % spirals, inverted equilibria are saddles, and the saddle separatrices divide % back-and-forth swinging from over-the-top rotation. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \definecolor{sep}{HTML}{B4532A} \draw[->, black] (-4.4,0)--(4.4,0) node[right, black] {$x$}; \draw[->, black] (0,-1.75)--(0,1.75) node[above, black] {$y$}; % critical points: spiral sinks at 0 and +/-3.2; saddles at +/-1.6 \fill[black] (0,0) circle (1.6pt); \fill[black] (3.2,0) circle (1.6pt); \fill[black] (-3.2,0) circle (1.6pt); \fill[black] (1.6,0) circle (1.9pt); \fill[black] (-1.6,0) circle (1.9pt); \node[anchor=north, font=\scriptsize] at (0,-0.14) {spiral sink}; \node[anchor=south, font=\scriptsize] at (1.6,0.12) {saddle}; % inward spiral at the origin \draw[acc, ->, domain=0:400, samples=80, variable=\t] plot ({(0.66 - 0.0014*\t)*cos(\t)}, {0.55*(0.66 - 0.0014*\t)*sin(\t)}); % saddle separatrix arms at the right saddle \draw[sep, thick] (0.85,1.05) .. controls (1.35,0.45) .. (1.6,0.0); \draw[sep, thick] (1.6,0.0) .. controls (1.85,-0.45) .. (2.35,-1.05); \draw[sep, thick] (2.35,1.05) .. controls (1.85,0.45) .. (1.6,0.0); \draw[sep, thick] (1.6,0.0) .. controls (1.35,-0.45) .. (0.85,-1.05); % saddle separatrix arms at the left saddle \draw[sep, thick] (-0.85,1.05) .. controls (-1.35,0.45) .. (-1.6,0.0); \draw[sep, thick] (-1.6,0.0) .. controls (-1.85,-0.45) .. (-2.35,-1.05); \draw[sep, thick] (-2.35,1.05) .. controls (-1.85,0.45) .. (-1.6,0.0); \draw[sep, thick] (-1.6,0.0) .. controls (-1.35,-0.45) .. (-0.85,-1.05); % over-the-top rotation trajectories \draw[acc, ->] (-4.3,1.25) .. controls (-1.0,0.75) and (1.0,0.75) .. (4.3,1.25); \draw[acc, <-] (-4.3,-1.25) .. controls (-1.0,-0.75) and (1.0,-0.75) .. (4.3,-1.25); \end{tikzpicture} $$ ## 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.[^boyce-thm] > **Theorem (Linearization).** Let $r_1, r_2$ be the eigenvalues of $J(x_0, > y_0)$ for a locally linear system. The type and stability of the critical point > of the nonlinear system agree with those of the linear system in every case > **except** the two sensitive ones below, where the nonlinear terms can change > the outcome. Both exceptions sit at the borderline configurations of the [trace–determinant plane](/differential-equations/nonlinear/phase-plane-autonomous-stability): - **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. | Linear eigenvalues | Linear type | Nonlinear type | Nonlinear stability | | --- | --- | --- | --- | | $r_1 > r_2 > 0$ | Node | Node | Unstable | | $r_1 < r_2 < 0$ | Node | Node | Asymptotically stable | | $r_2 < 0 < r_1$ | Saddle | Saddle | Unstable | | $r_1 = r_2 > 0$ | Node | Node or spiral | Unstable | | $r_1 = r_2 < 0$ | Node | Node or spiral | Asymptotically stable | | $\lambda \pm i\mu,\ \lambda \neq 0$ | Spiral | Spiral | matches sign of $\lambda$ | | $\pm i\mu$ (center) | Center | Center **or** spiral | Indeterminate | 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. $$ % caption: Near an asymptotically stable spiral, the nonlinear portrait (left) % and its linearization (right) share the same inward-spiralling character. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % ---- nonlinear (left): a skewed spiral ---- \begin{scope}[xshift=0cm] \draw[black] (-1.5,0)--(1.5,0); \draw[black] (0,-1.5)--(0,1.5); \draw[acc, ->, domain=0:610, samples=110, variable=\t] plot ({(1.3 - 0.00196*\t)*cos(\t)+0.18*sin(2*\t)}, {(1.3 - 0.00196*\t)*sin(\t)}); \fill[black] (0,0) circle (1.5pt); \node[anchor=north] at (0,-1.85) {nonlinear system}; \end{scope} % ---- linear (right): a clean spiral ---- \begin{scope}[xshift=5.4cm] \draw[black] (-1.5,0)--(1.5,0); \draw[black] (0,-1.5)--(0,1.5); \draw[acc, ->, domain=0:610, samples=110, variable=\t] plot ({(1.3 - 0.00196*\t)*cos(\t)}, {(1.3 - 0.00196*\t)*sin(\t)}); \fill[black] (0,0) circle (1.5pt); \node[anchor=north] at (0,-1.85) {linearization}; \end{scope} \draw[black, ->] (1.9,0)--(3.4,0) node[midway, above, font=\scriptsize] {zoom in}; \end{tikzpicture} $$ The indeterminate center row remains. The undamped pendulum $x' = y$, $y' = -\omega^2 \sin x$ 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.[^boyce-liap] Consider the general autonomous system $x' = F(x, y)$, $y' = G(x, y)$ 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 $x'$ and $y'$: $$ \dot V(x, y) = V_x\, F(x, y) + V_y\, G(x, y). $$ 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. > **Definition (Definite functions).** On a domain $D$ containing the origin, a > function $V$ with $V(0,0) = 0$ is **positive definite** if $V > 0$ everywhere > else in $D$, and **negative definite** if $V < 0$ everywhere else. Replacing > the strict inequalities by $\geq$ and $\leq$ gives **positive semidefinite** > and **negative semidefinite**. > **Theorem (Liapunov stability).** Suppose $V$ is positive definite with > continuous first partials on $D$. If $\dot V$ is negative definite on $D$, the > origin is **asymptotically stable**. If $\dot V$ is only negative semidefinite, > the origin is **stable**. A function $V$ meeting these conditions is a > **Liapunov function**. > **Theorem (Liapunov instability).** If $V(0,0) = 0$, every neighborhood of the > origin contains a point where $V > 0$, and $\dot V$ is positive definite on > $D$, then the origin is **unstable**. ### 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 $\mathbf{T} = (x', y')$ is tangent to the path, and $$ \dot V = \nabla V \cdot \mathbf{T}. $$ If $\dot V \leq 0$, 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.[^boyce-geom] $$ % caption: The gradient of $V$ points outward, normal to the level curve; if the % trajectory's velocity points inward, $V$ decreases and the origin is stable. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % nested level curves \draw[black] (0,0) ellipse (2.4 and 1.7); \draw[black] (0,0) ellipse (1.55 and 1.1); \draw[black] (0,0) ellipse (0.8 and 0.57); \fill[black] (0,0) circle (1.5pt); \node[black, anchor=west, font=\scriptsize] at (2.45,-0.05) {$V=c_2$}; \node[black, anchor=west, font=\scriptsize] at (1.6,0.72) {$V=c_1$}; % a point on the middle curve, at (0.954, 0.867) \fill[acc] (0.954,0.867) circle (1.6pt); % outward gradient (normal) \draw[acc, ->, thick] (0.954,0.867) -- (1.574,1.587); \node[acc, anchor=south west, font=\scriptsize] at (1.554,1.527) {$\nabla V$}; % inward-pointing velocity, into the curve \draw[black!70, ->, thick] (0.954,0.867) -- (0.004,0.717); \node[black!70, anchor=north east, font=\scriptsize] at (0.054,0.747) {$\mathbf{T}$}; \end{tikzpicture} $$ ### 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 $V = ax^2 + bxy + cy^2$ is standard, and an algebraic test decides definiteness.[^boyce-quad] > **Theorem (Definiteness of a quadratic form).** $V(x, y) = ax^2 + bxy + cy^2$ > is positive definite if and only if > $$ > a > 0 \quad\text{and}\quad 4ac - b^2 > 0, > $$ > and negative definite if and only if $a < 0$ and $4ac - b^2 > 0$. > **Worked example.** For $x' = -x - xy^2$, $y' = -y - x^2 y$, try > $V = ax^2 + cy^2$ with $b = 0$. Then > $$ > \dot V = (2ax)(-x - xy^2) + (2cy)(-y - x^2 y) > = -2a x^2 - 2c y^2 - (2a + 2c)x^2 y^2. > $$ > Any $a, c > 0$ makes $V$ positive definite and $\dot V$ negative definite, so > the origin is asymptotically stable. The linearization $\mathbf{u}' = > -\mathbf{u}$ reaches the same conclusion, but Liapunov's method delivers it > directly and also bounds the basin of attraction. ## Conservative systems and the energy function > **Worked example.** For the undamped pendulum $x' = y$, $y' = -(g/L)\sin x$, > take the total energy as the Liapunov function, > $$ > V(x, y) = \tfrac{1}{2} m L^2 y^2 + mgL(1 - \cos x), > $$ > the sum of kinetic and potential parts, positive definite on > $-\pi/2 < x < \pi/2$. Its rate of change along trajectories is > $$ > \dot V = (mgL \sin x)(y) + (mL^2 y)\!\left(-\tfrac{g}{L}\sin x\right) = 0. > $$ > The energy is constant, so $\dot V$ is negative semidefinite, and the > stability theorem makes the origin **stable** — the result the linearization > could not supply, since the origin is a center there.[^boyce-cons] The level > curves $V = c$ are the closed trajectories, small ellipses near the origin > corresponding to the pendulum's periodic swings. $$ % caption: A conservative system's trajectories are level curves of the energy; % adding damping tilts the energy downhill so trajectories cross curves inward. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % ---- left: conservative, energy constant ---- \begin{scope}[xshift=0cm] \draw[->, black] (-1.9,0)--(1.9,0) node[right, black, font=\scriptsize] {$x$}; \draw[->, black] (0,-1.6)--(0,1.6) node[above, black, font=\scriptsize] {$y$}; \foreach \r in {0.5,0.9,1.3} \draw[acc] (0,0) ellipse ({\r*1.3} and \r); \draw[acc, ->] (0.65,0.02)--(0.65,-0.02); \fill[black] (0,0) circle (1.4pt); \node[anchor=north] at (0,-1.95) {conservative: energy constant}; \end{scope} % ---- right: damped, energy falls ---- \begin{scope}[xshift=5.6cm] \draw[->, black] (-1.9,0)--(1.9,0) node[right, black, font=\scriptsize] {$x$}; \draw[->, black] (0,-1.6)--(0,1.6) node[above, black, font=\scriptsize] {$y$}; \draw[black] (0,0) ellipse (1.69 and 1.3); \draw[black] (0,0) ellipse (1.04 and 0.8); \draw[acc, ->, domain=0:560, samples=100, variable=\t] plot ({(1.5 - 0.00232*\t)*cos(\t)*1.3}, {(1.5 - 0.00232*\t)*sin(\t)}); \fill[black] (0,0) circle (1.4pt); \node[anchor=north] at (0,-1.95) {damped: energy decreases}; \end{scope} \end{tikzpicture} $$ With damping the calculation changes. For the damped pendulum the same energy gives $\dot V = -cL\,y^2 \leq 0$: 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.[^boyce-basin] > **Theorem (Region of asymptotic stability).** If $V$ is positive definite, > $\dot V$ is negative definite, and $D_K = \{ V(x, y) < K \}$ is a bounded > domain containing the origin, then every trajectory starting in $D_K$ > approaches the origin as $t \to \infty$. 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. [^boyce-jac]: **Boyce**, §9.3 — the Taylor-expansion derivation of the local linear system and the Jacobian matrix; Simmons, §62, treats simple critical points of nonlinear systems. [^boyce-pend]: **Boyce**, §9.3, Example 3 — the Jacobian of the pendulum at the origin and at the inverted position $(\pi, 0)$. [^boyce-thm]: **Boyce**, §9.3, Theorem 9.3.3 and Table 9.3.1 — agreement of nonlinear and linear classification except for centers and repeated roots. [^boyce-liap]: **Boyce**, §9.6 — Liapunov's second method as a generalization of the energy principles for conservative systems; Simmons, §61, on the direct method. [^boyce-geom]: **Boyce**, §9.6, Figure 9.6.1 — the gradient-versus-tangent argument, $\dot V = \nabla V \cdot \mathbf{T}$, and the trapping of trajectories inside level curves. [^boyce-quad]: **Boyce**, §9.6, Theorem 9.6.4 and Example 4 — the definiteness criterion for a quadratic form and its use in constructing Liapunov functions. [^boyce-cons]: **Boyce**, §9.6, Examples 2–3 — the energy function as a Liapunov function for the undamped pendulum; Simmons, §63, on conservative systems. [^boyce-basin]: **Boyce**, §9.6, Theorem 9.6.3 and Example 5 — the sublevel-set estimate of the region of asymptotic stability.