Discrete and Continuous Dynamical Systems

A system that changes in fixed time steps by $x_{k + 1} = A x_{k}$ , or continuously by $x^{'} (t) = A x (t)$ , has its entire future determined by the matrix $A$ and a starting state. Eigenvalues determine how the state grows, decays, and reorients over time. The method for both cases is the same: write the initial state in an eigenvector basis, and the coupled system decouples into independent scalar equations.

The eigenvector decomposition

Suppose $A$ is diagonalizable with eigenvectors $v_{1}, \dots, v_{n}$ and eigenvalues $λ_{1}, \dots, λ_{n}$ , ordered so that $∣ λ_{1} ∣ \geq ∣ λ_{2} ∣ \geq \dots \geq ∣ λ_{n} ∣$ . Any initial vector expands uniquely as $x_{0} = c_{1} v_{1} + \dots + c_{n} v_{n}$ . Because each $v_{i}$ is an eigenvector,

x_{k} = A^{k} x_{0} = c_{1} (λ_{1})^{k} v_{1} + c_{2} (λ_{2})^{k} v_{2} + \dots + c_{n} (λ_{n})^{k} v_{n} .

Each term evolves on its own, scaled by its eigenvalue raised to the $k$ th power. The long-term behavior is determined by whichever factors $(λ_{i})^{k}$ are largest.

Owl-rat trajectories grow along the dominant eigenvector v1 (ratio 10 to 13); off-line starts bend toward that direction as the v2 component decays.

Decoupling by change of variable

The eigenvector decomposition is a change of variable. With $P = [v_{1} \dots v_{n}]$ and $A = P D P^{- 1}$ , define $y_{k} = P^{- 1} x_{k}$ , equivalently $x_{k} = P y_{k}$ . Substituting into $x_{k + 1} = A x_{k}$ ,

P y_{k + 1} = A P y_{k} = P D P^{- 1} P y_{k} = P D y_{k} ⟹ y_{k + 1} = D y_{k} .

In the $y$ -coordinates the system is diagonal: $y_{i} (k + 1) = λ_{i} y_{i} (k)$ , so each component evolves independently as $y_{i} (k) = (λ_{i})^{k} y_{i} (0)$ . The vector $y_{k}$ is just the coordinate vector of $x_{k}$ in the eigenvector basis. Working on graph paper whose axes point along $v_{1}$ and $v_{2}$ makes the tangled motion of $A$ look like plain axis-parallel scaling.

The change of variable

y = Q x

, with

Q = P^{- 1}

, replaces the coupled map

A

with the diagonal map

D

; each eigencoordinate then evolves on its own.

Classifying the origin (discrete case)

For a $2 \times 2$ real matrix with real eigenvalues, the sequence $x_{0}, x_{1}, x_{2}, \dots$ traces a trajectory in the plane, and the eigenvalue magnitudes classify the origin.

Attractor. Both $∣ λ_{1} ∣, ∣ λ_{2} ∣ < 1$ . Every trajectory tends to $0$ . The direction of slowest approach is the eigenvector of the larger-magnitude eigenvalue; trajectories become tangent to it.
Repeller. Both $∣ λ_{1} ∣, ∣ λ_{2} ∣ > 1$ . Every nonzero trajectory is unbounded and moves away from $0$ , fastest along the eigenvector of the larger-magnitude eigenvalue.
Saddle point. One magnitude exceeds $1$ and the other is below $1$ . Trajectories are drawn in along the eigenvector of the small eigenvalue and pushed out along the eigenvector of the large one.

Phase portraits for a diagonal

A

: attractor (both eigenvalues below 1), repeller (both above 1), and saddle (one of each), with arrows on the eigen-axes.

When the eigenvalues are complex, the $2 \times 2$ system spirals, as developed for complex eigenvalues: modulus below $1$ spirals inward (attractor), above $1$ spirals outward (repeller), equal to $1$ gives closed elliptical orbits.

Complex eigenvalues rotate each step; modulus below 1 spirals the iterates into the origin (spiral sink), modulus above 1 spirals them out (spiral source).

Continuous systems

Many applied problems have several quantities varying continuously and coupled through their rates of change:

x^{'} (t) = A x (t), x (t) = x_{1} (t) ⋮ x_{n} (t), x^{'} (t) = x_{1}^{'} (t) ⋮ x_{n}^{'} (t) .

This is linear: if $u$ and $v$ solve $x^{'} = A x$ , so does $c u + d v$ . The solution set is an $n$ -dimensional space of vector-valued functions, and a fundamental set of $n$ independent solutions is a basis for it.

The standard guess is an exponential along a fixed direction, $x (t) = v e^{λ t}$ . Then $x^{'} (t) = λ v e^{λ t}$ and $A x (t) = A v e^{λ t}$ , so $x^{'} = A x$ holds if and only if $A v = λ v$ : each eigenpair supplies a solution.

The change of variable $y = P^{- 1} x$ decouples the continuous system exactly as in the discrete case: it turns $x^{'} = A x$ into $y^{'} = D y$ , i.e. $y_{i}^{'} = λ_{i} y_{i}$ , whose solution $y_{i} (t) = c_{i} e^{λ_{i} t}$ is elementary calculus.

Classifying the origin (continuous case)

For continuous systems it is the sign of the real part of each eigenvalue that matters, because $e^{λ t} \to 0$ precisely when $Re λ < 0$ . The stability threshold has moved from the unit circle to the imaginary axis.

Eigenvalues of $A$ (real $2 \times 2$ )	Origin	Trajectories
both real, negative	attractor / sink	decay into $0$
both real, positive	repeller / source	flee from $0$
real, opposite signs	saddle point	in along one axis, out along the other
complex, negative real part	spiral sink	spiral inward
complex, positive real part	spiral source	spiral outward
complex, zero real part	center	closed ellipses

A continuous saddle: negative-eigenvalue direction

v_{2}

pulls trajectories in, positive-eigenvalue direction

v_{1}

pushes them out.

Discrete versus continuous

The two settings run in parallel, with one substitution: powers $λ^{k}$ become exponentials $e^{λ t}$ , and the stability threshold moves from the unit circle to the imaginary axis.

	Discrete $x_{k + 1} = A x_{k}$	Continuous $x^{'} = A x$
Solution	$x_{k} = \sum c_{i} λ_{i}^{k} v_{i}$	$x (t) = \sum c_{i} e^{λ_{i} t} v_{i}$
Decay to $0$ requires	$∣ λ_{i} ∣ < 1$	$Re λ_{i} < 0$
Growth requires	$∣ λ_{i} ∣ > 1$	$Re λ_{i} > 0$
Neutral / periodic	$∣ λ_{i} ∣ = 1$	$Re λ_{i} = 0$
Rotation from	complex $λ$	complex $λ$

Both analyses depend on having an eigenvector basis. When the dominant eigenvalue is all that is wanted — the growth rate and limiting direction — it can be estimated directly without solving the characteristic equation, which is the power method of the final lesson.¹²

Lay, Linear Algebra and Its Applications, 5th ed., §5.6 — Discrete Dynamical Systems: the eigenvector decomposition of $x_{0}$ , decoupling by $y = P^{- 1} x$ , and the attractor/repeller/saddle classification of the origin from eigenvalue magnitudes. ↩
Lay, Linear Algebra and Its Applications, 5th ed., §5.7 — Applications to Differential Equations: eigenfunctions $v e^{λ t}$ solving $x^{'} = A x$ , decoupling into $y^{'} = D y$ , complex-eigenvalue spirals, and origin classification by the sign of $Re λ$ . ↩