ODE Chapter 7: Stability Theory

A small push hits a system. Does it return to rest, drift away, or break entirely? That single question decides whether bridges survive storms, ecosystems recover from droughts, and economies bounce back from crises. Stability theory answers it – and it does so without ever solving the differential equation. We will learn to read the destiny of a system off the geometry of its phase plane.

What You Will Learn

• Three precise notions: Lyapunov stable, asymptotically stable, unstable
• Linearization via the Jacobian and the Hartman-Grobman theorem
• Lyapunov’s direct method – proving stability with energy-like functions
• LaSalle’s invariance principle for borderline cases
• Trace-determinant classification of all 2D linear systems
• Four canonical bifurcations: saddle-node, transcritical, pitchfork, Hopf
• Worked applications: pendulum, predator-prey, inverted pendulum control

Prerequisites

• Chapter 6: linear systems, eigenvalues, phase portraits
• Multivariable calculus: partial derivatives, Jacobian matrix

A Visual Tour Before the Theory

Stability is, at heart, a geometric statement about how trajectories move in phase space. Six pictures tell the entire story of 2D linear systems.

For nonlinear systems, the same six pictures still appear – but only locally, near each equilibrium. The damped pendulum and the Lotka-Volterra predator-prey model both show this beautifully:

Stability Defined Precisely

Consider $\mathbf{x}' = \mathbf{f}(\mathbf{x})$ with equilibrium $\mathbf{x}^*$ (so $\mathbf{f}(\mathbf{x}^*) = \mathbf{0}$).

Intuition: nearby trajectories stay nearby forever.

Asymptotically stable. Lyapunov stable and $\mathbf{x}(t) \to \mathbf{x}^*$ as $t \to \infty$. Intuition: nearby trajectories not only stay nearby but eventually return.

Unstable. Not Lyapunov stable.

The basin of attraction is the set of all initial conditions that converge to $\mathbf{x}^*$. Asymptotic stability is a local property; the basin tells you how local.

Why two definitions? A center (closed orbits) is Lyapunov stable but not asymptotically stable – trajectories stay close but never settle. The Lotka-Volterra model is the classic example.

Linearization: The Jacobian Method

Hartman-Grobman theorem

If every eigenvalue of $J$ has nonzero real part (a hyperbolic equilibrium), then the nonlinear system is locally topologically equivalent to its linearization $\mathbf{u}' = J\mathbf{u}$.

• All $\operatorname{Re}(\lambda) < 0$: asymptotically stable
• Any $\operatorname{Re}(\lambda) > 0$: unstable
• Purely imaginary eigenvalues: linearization fails – use Lyapunov methods

Example: damped pendulum

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import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

gamma, omega0 = 0.3, 1.0
def pendulum(X, t):
    x, y = X
    return [y, -gamma*y - omega0**2*np.sin(x)]

fig, ax = plt.subplots(figsize=(10, 6))
t = np.linspace(0, 50, 2000)
for x0 in np.linspace(-3*np.pi, 3*np.pi, 15):
    for y0 in np.linspace(-3, 3, 5):
        sol = odeint(pendulum, [x0, y0], t)
        ax.plot(sol[:,0], sol[:,1], 'b-', linewidth=0.3, alpha=0.5)
for n in range(-2, 3):
    ax.plot(n*np.pi, 0, 'go' if n%2==0 else 'rx', markersize=10)
ax.set(xlabel='theta', ylabel='omega', title='Damped pendulum phase portrait')
ax.grid(True, alpha=0.3); plt.tight_layout(); plt.show()

Lyapunov’s Direct Method

The big idea

Stability without solving the ODE. Construct an energy-like scalar function $V(\mathbf{x})$ and show it decreases along trajectories.

Requirements.

1. $V(\mathbf{x}^*) = 0$ and $V(\mathbf{x}) > 0$ otherwise (positive definite)
2. $\dot V = \nabla V \cdot \mathbf{f}(\mathbf{x}) \leq 0$ along orbits

Stability theorems.

Why it works – the picture

Trajectories cross level sets of $V$ inward. Since $V$ has a minimum at $\mathbf{x}^*$, they cannot escape arbitrarily-small level sets, and (with strict descent) they slide all the way to the bottom.

How to find $V$

• Physical energy: kinetic + potential for mechanical systems
• Quadratic forms: $V = \mathbf{x}^T P \mathbf{x}$ where $P$ solves the Lyapunov equation $A^T P + PA = -Q$
• Trial: start with $V = x^2 + y^2$, compute $\dot V$, adjust coefficients

Example: pendulum energy

The hanging position is stable. LaSalle’s principle (next) upgrades this to asymptotic.

LaSalle’s Invariance Principle

Sometimes $\dot V \leq 0$ but vanishes on a whole set, not just at $\mathbf{x}^*$. Standard Lyapunov only gives stability, not attraction.

Theorem. Let $E = \{\mathbf{x} : \dot V = 0\}$ and $M$ be the largest invariant subset of $E$. Every bounded trajectory approaches $M$.

If $M = \{\mathbf{x}^*\}$, then $\mathbf{x}^*$ is asymptotically stable.

For the damped pendulum, $\dot V = 0$ requires $\omega = 0$. But on the line $\omega = 0$ the dynamics force $\dot\omega = -\omega_0^2 \sin\theta \neq 0$ unless also $\theta = 0$. So $M = \{(0,0)\}$ and we get asymptotic stability for free.

Trace-Determinant Classification

For $\mathbf{x}' = A\mathbf{x}$ in 2D, set $\tau = \operatorname{tr}(A)$ and $\Delta = \det(A)$. Then $\lambda_{1,2} = \tfrac12(\tau \pm \sqrt{\tau^2 - 4\Delta})$, and:

A single picture compresses this entire table:

Bifurcations: When the Picture Itself Changes

Slowly turn a parameter knob $r$. Equilibria can be born, die, or swap stability. Four “normal forms” capture every codimension-1 bifurcation locally.

Hopf is the mechanism behind every self-sustained oscillation in nature – from heartbeats to the pulsing of variable stars.

Application 1: Lotka-Volterra Predator-Prey

makes every orbit closed. The system has periodic population cycles (right panel of fig 2).

Application 2: Inverted Pendulum Control

The inverted equilibrium is a saddle. Linear feedback $u = -K\mathbf{x}$ shifts the closed-loop eigenvalues into the open left half-plane, converting the saddle into a stable focus.

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import numpy as np
from scipy.linalg import solve_continuous_are
from scipy.integrate import odeint
import matplotlib.pyplot as plt

g, l, m, M_cart = 9.81, 0.5, 0.1, 1.0
A = np.array([[0,1,0,0],[0,0,-m*g/M_cart,0],
              [0,0,0,1],[0,0,(M_cart+m)*g/(M_cart*l),0]])
B = np.array([[0],[1/M_cart],[0],[-1/(M_cart*l)]])

Q = np.diag([10, 1, 100, 1])
R = np.array([[0.1]])
P = solve_continuous_are(A, B, Q, R)
K = np.linalg.inv(R) @ B.T @ P

print("Closed-loop eigenvalues:", np.linalg.eigvals(A - B @ K))

t = np.linspace(0, 5, 500)
sol = odeint(lambda X, t: (A @ X + B @ (-K @ X)).flatten(),
             [0,0,0.2,0], t)

plt.figure(figsize=(8, 4))
plt.plot(t, sol[:,2]*180/np.pi, 'b-', linewidth=2)
plt.xlabel('Time (s)'); plt.ylabel('Angle (degrees)')
plt.title('Inverted pendulum stabilized by LQR')
plt.grid(True, alpha=0.3); plt.tight_layout(); plt.show()

Summary

Exercises

Basic.

1. Determine stability for: (a) $x' = -x,\ y' = -2y$; (b) $x' = y,\ y' = -\sin x - 0.5y$.
2. Use $V = x^2 + y^2$ to analyze $x' = -x + y^2,\ y' = -y$.
3. Find all bifurcation points of $\dot x = rx - x^3$.

Advanced.

4. Prove total energy is a Lyapunov function for the damped pendulum, then apply LaSalle.
5. Analyze the Van der Pol oscillator: show the origin is unstable but a stable limit cycle exists.
6. For Lotka-Volterra competition, derive the conditions for coexistence vs. exclusion.

Programming.

7. Auto-classify 2D linear-system equilibria from trace and determinant; reproduce fig 3.
8. Animate the Hopf bifurcation: sweep $\mu$ and watch the limit cycle grow.

References

• Strogatz, Nonlinear Dynamics and Chaos, CRC Press (2015)
• Khalil, Nonlinear Systems, Prentice Hall (2002)
• Guckenheimer & Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations, Springer (1983)
• Perko, Differential Equations and Dynamical Systems, Springer (2001)

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