ODE-08

Stability Theory

Stability and Lyapunov Methods

Analyze the long-term behavior of dynamical systems: equilibrium points, stability concepts, Lyapunov's direct method, linearization, and phase plane analysis.

Learning Objectives

Identify and classify equilibrium points
Distinguish stability, asymptotic stability, and instability
Construct and use Lyapunov functions
Apply Lyapunov's stability theorems
Linearize nonlinear systems about equilibria
Classify 2D equilibria using eigenvalues
Sketch phase portraits
Apply LaSalle's invariance principle

1. Equilibrium Points

Definition 8.1: Equilibrium Point

For the autonomous system $\dot{x} = f(x)$ , a point $x^*$ is an equilibrium point(also called fixed point, critical point, or stationary point) if:

f(x^*) = 0

At an equilibrium, the system remains stationary: if $x(0) = x^*$ , then $x(t) = x^*$ for all $t$ .

Example 8.1: Finding Equilibria

Problem: Find equilibria of $\dot{x} = x(1-x)$ .

Solution:

Set $f(x) = x(1-x) = 0$ . This gives $x = 0$ or $x = 1$ .

Equilibrium points: $x^* = 0$ and $x^* = 1$ .

2. Stability Concepts

Definition 8.2: Lyapunov Stability

An equilibrium $x^*$ is Lyapunov stable (or simply stable) if for every $\epsilon > 0$ , there exists $\delta > 0$ such that:

|x(0) - x^*| < \delta \implies |x(t) - x^*| < \epsilon \quad \forall t \geq 0

Intuitively: solutions starting close stay close forever.

Definition 8.3: Asymptotic Stability

An equilibrium $x^*$ is asymptotically stable if:

It is Lyapunov stable, AND
There exists $\delta > 0$ such that $|x(0) - x^*| < \delta$ implies $\lim_{t \to \infty} x(t) = x^*$

Intuitively: solutions starting close not only stay close but converge to $x^*$ .

Definition 8.4: Instability

An equilibrium $x^*$ is unstable if it is not Lyapunov stable. That is, there exists $\epsilon > 0$ such that for any $\delta > 0$ , some solution starting within $\delta$ of $x^*$ eventually leaves the $\epsilon$ -ball.

Remark:

Stability ⊃ Asymptotic stability: stable does not imply asymptotically stable. Example: the center $\dot{x} = y, \dot{y} = -x$ has stable (but not asymptotically stable) origin—solutions orbit forever.

3. Lyapunov's Direct Method

Lyapunov's method determines stability without solving the ODE, using an energy-like function.

Definition 8.5: Lyapunov Function

A continuously differentiable function $V: D \to \mathbb,{R}$ is a Lyapunov functionfor $\dot{x} = f(x)$ at $x^* = 0$ if:

$V(0) = 0$
$V(x) > 0$ for $x \neq 0$ (positive definite)
$\dot{V}(x) = \nabla V \cdot f(x) \leq 0$ (non-increasing along trajectories)

Theorem 8.1: Lyapunov Stability Theorem

If there exists a Lyapunov function $V$ with $\dot{V} \leq 0$ in a neighborhood of the origin, then $x^* = 0$ is Lyapunov stable.

If additionally $\dot{V} < 0$ for $x \neq 0$ (negative definite), then $x^* = 0$ is asymptotically stable.

Proof of Theorem 8.1:

Given $\epsilon > 0$ , let $B_\epsilon$ be the $\epsilon$ -ball around origin. Let $m = \min_,{|x|=\epsilon}, V(x) ,> 0$ (since $V$ is positive definite).

Choose $\delta$ so that $|x| ,< \delta$ implies $V(x) ,< m$ . If $x(0)$ starts with $|x(0)| ,< \delta$ , then $V(x(0)) ,< m$ .

Since $\dot{V} \leq 0$ , we have $V(x(t)) \leq V(x(0)) ,< m$ for all $t \geq 0$ . Thus $|x(t)| ,< \epsilon$ (otherwise $V \geq m$ ), proving stability.

For asymptotic stability with $\dot{V} < 0$ , $V(x(t))$ strictly decreases, forcing $x(t) \to 0$ . ∎

Algorithm 8.1: Constructing Lyapunov Functions

Common choices:

1. Quadratic: $V(x) = x^T P x$ where $P$ is positive definite

2. Energy-based: For mechanical systems, $V = T + U$ (kinetic + potential)

3. Sum of squares: $V(x) = x_1^2 + x_2^2 + \cdots$

4. Weighted quadratic: $V = ax_1^2 + bx_2^2$ with $a, b ,> 0$

Verification:

Compute $\dot{V} = \nabla V \cdot f(x)$ and check if $\dot{V} \leq 0$ (or < 0)

Example 8.2: Lyapunov Function Analysis

Problem: Analyze stability of $x^* = 0$ for $\dot{x} = -x^3$ .

Solution:

Try $V(x) = x^2$ . Then:

\dot{V} = 2x \cdot \dot{x} = 2x \cdot (-x^3) = -2x^4 < 0 \text{ for } x \neq 0

Since $V ,> 0$ for $x \neq 0$ and $\dot{V} < 0$ , the origin is asymptotically stable.

Example 8.3: 2D System with Lyapunov Function

Problem: Show that $(0, 0)$ is asymptotically stable for:

\dot{x} = -x + y^2, \quad \dot{y} = -y - x^2

Solution:

Try $V(x, y) = x^2 + y^2$ . Then:

\dot{V} = 2x\dot{x} + 2y\dot{y} = 2x(-x + y^2) + 2y(-y - x^2)

= -2x^2 + 2xy^2 - 2y^2 - 2x^2 y = -2(x^2 + y^2) + 2xy^2 - 2x^2 y

For small $|x|, |y|$ , the leading term $-2(x^2 + y^2)$ dominates, so $\dot{V} < 0$ near origin.

The origin is asymptotically stable.

4. Linearization and Local Stability

Theorem 8.2: Linearization Theorem

Consider $\dot{x} = f(x)$ with equilibrium $f(x^*) = 0$ . The linearization about $x^*$ is:

\dot{y} = Ay, \quad A = Df(x^*) = \left[\frac{\partial f_i}{\partial x_j}\right]_{x^*}

where $y = x - x^*$ .

If all eigenvalues of $A$ have $\text{Re}(\lambda) < 0$ : $x^*$ is asymptotically stable
If any eigenvalue has $\text{Re}(\lambda) > 0$ : $x^*$ is unstable
If some eigenvalue has $\text{Re}(\lambda) = 0$ and none positive: inconclusive

2D Equilibrium Classification

Stable Node: Two negative real eigenvalues

Unstable Node: Two positive real eigenvalues

Saddle Point: One positive, one negative eigenvalue (unstable)

Stable Focus: Complex eigenvalues with negative real part (spiral in)

Unstable Focus: Complex eigenvalues with positive real part (spiral out)

Center: Pure imaginary eigenvalues (ellipses, marginally stable)

Example 8.4: Linearization Analysis

Problem: Classify equilibria of $\dot{x} = x - x^2, \dot{y} = -y$ .

Solution:

Equilibria: $x - x^2 = x(1-x) = 0$ and $-y = 0$ give $(0, 0)$ and $(1, 0)$ .

Jacobian: $A = \begin,{pmatrix}, 1 - 2x & 0 \\ 0 & -1 \end,{pmatrix}$

At $(0, 0)$ : $A = \begin,{pmatrix}, 1 & 0 \\ 0 & -1 \end,{pmatrix}$ , eigenvalues $\lambda = 1, -1$ → Saddle (unstable)

At $(1, 0)$ : $A = \begin,{pmatrix}, -1 & 0 \\ 0 & -1 \end,{pmatrix}$ , eigenvalues $\lambda = -1, -1$ → Stable Node

5. LaSalle's Invariance Principle

Theorem 8.3: LaSalle's Invariance Principle

Let $\Omega$ be a compact set that is positively invariant for $\dot{x} = f(x)$ . Let $V$ be a Lyapunov function with $\dot{V} \leq 0$ on $\Omega$ . Define:

E = \{x \in \Omega : \dot{V}(x) = 0\}

Let $M$ be the largest invariant set contained in $E$ . Then every solution starting in $\Omega$ approaches $M$ as $t \to \infty$ .

Remark:

LaSalle's principle is powerful when $\dot{V}$ is only negative semidefinite. If the only invariant set in $E$ is the equilibrium, we still get asymptotic stability.

Example 8.5: Damped Pendulum

Problem: Analyze $\ddot{\theta} + c\dot{\theta} + \sin\theta = 0$ (damped pendulum, $c ,> 0$ ).

Solution:

Let $x_1 = \theta, x_2 = \dot,{\theta}$ . Equilibrium: $(\theta, \dot,{\theta},) = (0, 0)$ .

Energy: $V = \frac,{1}{2},x_2^2 + (1 - \cos x_1)$ (kinetic + potential)

\dot{V} = x_2 \dot{x}_2 + \sin x_1 \cdot x_2 = x_2(-cx_2 - \sin x_1) + x_2 \sin x_1 = -cx_2^2 \leq 0

Note $\dot{V} = 0$ when $x_2 = 0$ . But if $x_2 = 0$ and $x_1 \neq 0$ , then $\dot,{x},_2 = -\sin x_1 \neq 0$ , so the trajectory leaves the set $x_2 = 0$ .

The largest invariant set in $\{\dot{V} = 0\}$ is just $(0, 0)$ .

By LaSalle: $(0, 0)$ is asymptotically stable.

6. Instability Theorems

Theorem 8.4: Chetaev's Instability Theorem

Let $V$ be continuously differentiable. If there exists an open set $U$ with $0 \in \partial U$ such that:

$V(x) > 0$ and $\dot{V}(x) > 0$ for $x \in U$
$V(x) = 0$ on $\partial U \cap B_r(0)$ for some $r ,> 0$

Then $x^* = 0$ is unstable.

Practice Quiz

Stability and Lyapunov Methods Quiz

Questions

Correct

Accuracy

An equilibrium point

x^*

\dot{x} = f(x)

satisfies:

Easy

Not attempted

Lyapunov stability means that solutions starting near

x^*

Easy

Not attempted

Asymptotic stability means:

Easy

Not attempted

A Lyapunov function

V(x)

for stability must satisfy:

Medium

Not attempted

The derivative of

V

along solutions,

\dot{V}

, equals:

Medium

Not attempted

For the linear system

\dot{x} = Ax

, the origin is asymptotically stable if and only if:

Medium

Not attempted

Linearization about an equilibrium

x^*

involves computing:

Medium

Not attempted

If the linearization has eigenvalues with zero real parts:

Medium

Not attempted

In the phase plane, a saddle point has:

Hard

Not attempted

LaSalle's invariance principle is useful when:

Hard

Not attempted

Frequently Asked Questions

How do I find a Lyapunov function?

There's no universal method, but common strategies include: (1) Start with

V = x^T x

V = x^T P x

, (2) Use physical energy for mechanical systems, (3) For polynomial systems, try polynomial

V

with undetermined coefficients. Finding Lyapunov functions is generally an art.

What if linearization is inconclusive?

When eigenvalues have zero real parts (centers, non-hyperbolic cases), use: (1) Lyapunov functions, (2) Center manifold theory, (3) Normal form theory, or (4) Higher-order analysis. The nonlinear terms determine stability in these marginal cases.

What is the difference between local and global stability?

Local asymptotic stability means solutions converge to equilibrium if starting sufficiently close. Global asymptotic stability means all solutions converge, regardless of initial condition. Global stability requires a Lyapunov function that works on all of state space (often with

V \to \infty

|x| \to \infty

How do saddle points affect the phase portrait?

Saddles have stable and unstable manifolds—1D curves where trajectories approach or leave the saddle. These manifolds organize the phase portrait, acting as separatrices that divide regions with different long-term behaviors.

Can a system be stable but not asymptotically stable?

Yes! The classic example is a center (pure imaginary eigenvalues): trajectories orbit around the equilibrium forever without converging. They stay close (stable) but don't approach the equilibrium (not asymptotically stable). This is marginal or neutral stability.

When should I use Lyapunov vs. linearization?

Linearization is simpler when eigenvalues have non-zero real parts (hyperbolic equilibria). Use Lyapunov when: (1) linearization is inconclusive, (2) you need global stability, (3) you want explicit convergence estimates, or (4) for nonlinear control design.