5.4 Phase plane and classification

We’ve spent three lessons looking at time-series plots — graphs of $x$ against $t$ . There is a different way to picture the same motion, one that hides time but exposes geometry, and once you’ve seen it you can read off the qualitative behaviour of a second-order linear system at a glance.

That alternative is the phase plane. Plot the position $x$ and the velocity $v = \dot x$ on perpendicular axes, and watch trajectories in this $(x, v)$ plane evolve. The move rests on a small bookkeeping trick that re-frames second-order ODEs as systems of first-order ones, which is also where this lesson begins.

ODEs as matrix systems

A second-order ODE can be rewritten as a first-order system of two coupled equations by introducing the velocity $v = \dot x$ as a second variable. Take the damped oscillator,

\ddot x \;+\; 2\gamma\, \dot x \;+\; \omega_0^2\, x \;=\; 0,

and set $v \equiv \dot x$ . Then $\dot v = \ddot x = -\omega_0^2 x - 2\gamma v$ . Stack the two equations together:

\frac{d}{dt}\begin{pmatrix} x \\ v \end{pmatrix} \;=\; \underbrace{\begin{pmatrix} 0 & 1 \\ -\omega_0^2 & -2\gamma \end{pmatrix}}_{A} \begin{pmatrix} x \\ v \end{pmatrix}.

The first row says $\dot x = v$ — the definition of $v$ . The second row is the original ODE solved for $\ddot x$ . The whole second-order equation has become a single matrix–vector equation $\dot{\mathbf{x}} = A\, \mathbf{x}$ .

What are the eigenvalues of $A$ (refresher →)? Solve $\det(A - \lambda I) = 0$ :

\det\!\begin{pmatrix} -\lambda & 1 \\ -\omega_0^2 & -2\gamma - \lambda \end{pmatrix} \;=\; \lambda(\lambda + 2\gamma) + \omega_0^2 \;=\; \lambda^2 + 2\gamma\lambda + \omega_0^2 \;=\; 0.

That is the same characteristic polynomial we have been solving all along — the matrix’s eigenvalues are the $\lambda_\pm$ from the substitute- $e^{\lambda t}$ trick. The two pictures are one object:

The algebraic view (substitute $e^{\lambda t}$ , solve a polynomial) and the matrix view (eigenvalues of $A$ ) are two ways of writing the same step.
The phase plane is just the $\mathbf{x} = (x, v)$ plane of this matrix system, with the flow generated by $A$ .

The payoff of doing the rewrite is geometric. Once a second-order ODE is recast as $\dot{\mathbf{x}} = A\,\mathbf{x}$ , the state $\mathbf{x} = (x, v)$ lives in a 2-D plane and traces out a curve there as time runs. The matrix $A$ defines a flow — at each point of the plane it tells you the instantaneous direction $\dot{\mathbf{x}}$ that the state moves in.

The phase plane

regime: underdamped

damping γ = 0.30 frequency ω₀ = 1.00

Slide damping from 0 (closed orbits) through γ < ω₀ (inward spirals — underdamped) to γ > ω₀ (overdamped node — no oscillation, monotone return).

Each curve in the interactive above is one solution, starting from a different initial condition. The grey arrows show the local flow direction at every point — the direction that $\dot{\mathbf{x}} = A\mathbf{x}$ points there. The damping slider walks you through the regimes from 5.3:

Undamped ( $\gamma = 0$ ): closed ellipses. Energy is conserved; the state cycles around the origin forever.
Underdamped: inward spirals. Energy slowly drains; the spirals tighten toward the equilibrium at the origin.
Critical / overdamped: direct paths to the origin with no rotation. Energy drains faster than the system can complete an orbit.

The phase plane is the same dynamical system shown differently. The time-domain plot tells you when $x$ does what; the phase plane tells you how $x$ and $v$ relate as the state evolves. Both views are useful — the phase plane is especially powerful for nonlinear systems, where time-series plots can be hard to interpret but the geometry of the flow is often immediately legible.

Classification of equilibria

The phase plane above is for the damped oscillator specifically. But every 2-D linear system $\dot{\mathbf{x}} = A\mathbf{x}$ has an equilibrium (a “fixed point”) at the origin, and its character — how nearby trajectories behave — is set entirely by the eigenvalues of $A$ . These are the same eigenvalues we have been calling $\lambda_\pm$ , viewed now for a generic matrix rather than the specific damped-oscillator one.

There are six canonical types. They correspond exactly to the six root patterns we already met as outputs of the characteristic equation in 5.2:

Trajectories approach the origin from all directions — overdamped behaviour. Both eigenvalues live on the negative real axis.

Flip through the gallery. For each classification you get the phase portrait on the left and the eigenvalues in the complex plane on the right.

Name	Eigenvalues	Connection to our chapter
Stable node (sink)	two real, both negative	overdamped regime (5.3)
Stable spiral	complex conjugates, negative real part	underdamped regime (5.3)
Centre	pure imaginary	undamped SHM (5.3)
Saddle	real, opposite signs	the saddle case in the trick carousel (5.2)
Unstable spiral	complex conjugates, positive real part	oscillator with negative damping; out of scope for acoustics but standard in fluid instabilities
Unstable node (source)	two real, both positive	exponential-growth case from the carousel (5.2)

The classification names are useful jargon — you’ll see stable spiral and saddle throughout dynamical-systems and control-theory literature — but the underlying math is the same algebra we’ve been doing. Where the eigenvalues live in the complex plane determines how the flow behaves near the equilibrium:

Left half-plane → stable (trajectories return to the origin).
Right half-plane → unstable (trajectories fly off).
Imaginary axis → the marginal boundary, where centres live.

Initial conditions

A second-order linear ODE has a two-parameter family of solutions; two pieces of data fix a single member of that family. Usually we are given one of:

Initial conditions: $x(0)$ and $\dot x(0)$ . Two numbers at the same time, two free constants in the general solution.
Boundary conditions: values at two distinct points in space (for spatial ODEs that come out of separating variables in a PDE — see Foundations Ch 6 — PDEs).

Which is appropriate is dictated by the problem. Initial-value problems (an ODE in time) take initial conditions; boundary-value problems (an ODE in space, e.g. a vibrating string fixed at both ends) take boundary conditions.

Existence and uniqueness. For any linear ODE with continuous coefficients, every initial-value problem has a unique solution defined on the interval where the coefficients are continuous — so the two pieces of data above always specify the answer without ambiguity. This is not automatic for nonlinear ODEs, where solutions can fail to exist, can become non-unique, or can blow up in finite time. For the constant-coefficient linear case the chapter covers, you can safely assume the solution exists and is unique.

Linearity and superposition

The defining property of a linear ODE is

L[\,a x_1 + b x_2\,] \;=\; a\, L[x_1] + b\, L[x_2],

where $L$ is the differential operator (whatever combination of derivatives appears on the left-hand side). Two consequences we use constantly:

Solutions of the homogeneous equation can be added together to give new solutions. (This is what lets us write the general solution as a linear combination of the $e^{\lambda_k t}$ .)
The response to a sum of inputs is the sum of the individual responses. This is what makes Fourier analysis an engineering tool: any periodic input can be decomposed into sinusoidal components, each driven independently, and the total response is the sum.

Nonlinear equations don’t give you this gift. The pendulum’s exact equation,

\ddot \theta \;+\; (g/L)\, \sin\theta \;=\; 0,

fails superposition because $\sin(\theta_1 + \theta_2) \neq \sin\theta_1 + \sin\theta_2$ . Two solutions to the exact pendulum equation cannot in general be added to make a third. Linearisation around small angles — replacing $\sin\theta$ with $\theta$ — restores linearity at the cost of small-amplitude accuracy. Most of acoustics is exactly this kind of linearisation around equilibrium (Sound 4.5 — Linearisation and the wave equation).

What we use linear ODEs for

The cash value of this chapter, across the bookshelf:

Damped, driven oscillators in Sound Ch 2 — every formula in 5.3 maps onto a mass-spring with friction, or equivalently an RLC circuit.
Separation of variables in the wave equation (Sound 4.5) reduces the PDE to second-order ODEs in space and in time. Each is solved by the same characteristic-equation move.
Reflection at a boundary uses the homogeneous solution to match incident and reflected fields.
Resonance bandwidth in Sound Ch 8 is a re-reading of the forced-oscillator amplitude curve.
The cochlear filter at each place on the basilar membrane is, to a good approximation, a damped driven oscillator with its own $\omega_0$ and $\gamma$ — see Hearing Ch 4.
Stability of control loops, of fluid flows, of population dynamics — all reduce, near equilibrium, to the linear-ODE classification above. Eigenvalues in the left half-plane mean stable; in the right, unstable. The phase-plane portraits we just toured are how engineers and biologists see their systems.

ODEs we don’t develop here

Two large families of ODEs sit outside this chapter’s scope but are worth knowing exist:

Variable-coefficient ODEs with special-function solutions. Separating variables in the wave equation in cylindrical coordinates produces an ODE in the radial direction whose solutions are Bessel functions $J_n, Y_n$ ; in spherical coordinates the angular part produces Legendre polynomials $P_\ell$ and the radial part spherical Bessel functions $j_\ell, y_\ell$ . These appear in Sound 6.2 — Spherical waves and Sound 6.3 — Cylindrical waves. Their derivations use the Frobenius series-solution method, which we don’t develop — when you need them, look up the function and the recurrence.
The Laplace transform. An alternative tool for linear ODEs with initial conditions, which converts differentiation into multiplication by $s$ and turns IVPs into algebraic equations on $s$ . We don’t use it here because Fourier methods (refresher) cover the same ground for the steady-state acoustic problems on the bookshelf. The Laplace transform is the standard tool in control theory and transient-circuit analysis; if you encounter either, learn it then.

That is the chapter. Linear constant-coefficient ODEs come down to a polynomial and its roots; the position of those roots in the complex plane is the answer.