---
title: Population Models, Limit Cycles, and Chaos
module: Nonlinear Systems and Stability
moduleNumber: 8
lessonNumber: 3
order: 803
summary: >
  The phase-plane methods apply directly to interacting-population models.
  Competing species either coexist or drive one another to extinction, decided by
  a single inequality among the interaction constants; the Lotka-Volterra
  predator-prey system produces closed population cycles. Limit cycles and the
  Poincaré-Bendixson theorem, the van der Pol oscillator, and the Lorenz
  equations with their strange attractor carry the theory into chaos.
topics: [Nonlinear Systems and Stability]
sources:
  - book: Boyce
    ref: "Ch. 9 §9.4 Competing Species; §9.5 Predator-Prey Equations; §9.7 Periodic Solutions and Limit Cycles; §9.8 Chaos and Strange Attractors: The Lorenz Equations"
  - book: Simmons
    ref: "Ch. 10 §57 Volterra's Prey-Predator Equations; Ch. 11 §64 Periodic Solutions; The Poincaré-Bendixson Theorem; §65 More about the van der Pol Equation"
draft: false
---

Two species sharing an environment obey a pair of coupled nonlinear rate
equations with several critical points. The [phase
plane](/differential-equations/nonlinear/phase-plane-autonomous-stability) and the
[linearization and Liapunov
tools](/differential-equations/nonlinear/locally-linear-and-liapunov) answer
whether the species coexist, which one survives, and whether the populations
settle or cycle, with no closed-form solution. The same methods reach phenomena
with no linear analogue: the self-sustaining oscillation of a limit cycle and the
chaotic wandering of the Lorenz attractor.

## Competing species

Two species that consume a common food supply, without preying on each other,
each grow logistically in isolation and each depress the other's growth. Writing
$x$ and $y$ for the two populations gives the competition system[^boyce-comp]

$$
\frac{\d x}{\d t} = x(\epsilon_1 - \sigma_1 x - \alpha_1 y), \qquad
\frac{\d y}{\d t} = y(\epsilon_2 - \sigma_2 y - \alpha_2 x),
$$

with all six constants positive: $\epsilon_i$ are the growth rates, $\sigma_i$
measure each species' self-limitation, and $\alpha_i$ measure the interference
from the competitor. Only nonnegative $x, y$ are meaningful, and because the
coordinate axes are themselves trajectories, a state that starts in the first
quadrant stays there.

Setting both right-hand sides to zero gives up to four critical points: the
origin $(0, 0)$, the two single-species states $(\epsilon_1/\sigma_1, 0)$ and
$(0, \epsilon_2/\sigma_2)$, and — when the nullclines cross in the interior — a
**coexistence** point with both populations positive. Because each right-hand
side is a quadratic polynomial, the system is locally linear at every critical
point, and the Jacobian classifies each one.

$$
J(x, y) = \begin{pmatrix}
\epsilon_1 - 2\sigma_1 x - \alpha_1 y & -\alpha_1 x \\
-\alpha_2 y & \epsilon_2 - 2\sigma_2 y - \alpha_2 x
\end{pmatrix}.
$$

### Coexistence or exclusion

The interior critical point $(X, Y)$, when it exists, determines the outcome, and
its character reduces to a single comparison. Evaluating the Jacobian there and using
$\epsilon_1 - \sigma_1 X - \alpha_1 Y = 0$ and its counterpart, the eigenvalues
come from

$$
r^2 + (\sigma_1 X + \sigma_2 Y)\,r + (\sigma_1 \sigma_2 - \alpha_1 \alpha_2)XY = 0.
$$

The product of the eigenvalues is $(\sigma_1 \sigma_2 - \alpha_1 \alpha_2)XY$, so
its sign determines the type.[^boyce-comp2]

- **Weak competition, $\sigma_1 \sigma_2 > \alpha_1 \alpha_2$.** The eigenvalues
  are real and negative: the coexistence point is an asymptotically stable node,
  the three boundary points are unstable, and every trajectory in the first
  quadrant converges to coexistence. The two species share the environment.
- **Strong competition, $\sigma_1 \sigma_2 < \alpha_1 \alpha_2$.** The eigenvalues
  are real with opposite signs: the coexistence point is a saddle. The two
  single-species nodes are asymptotically stable, and the two trajectories
  entering the saddle form a **separatrix** dividing the quadrant into two basins.
  One species drives the other to extinction; which one wins depends on the
  starting populations.

The biological reading is direct. The $\sigma$'s measure how much each population
inhibits itself; the $\alpha$'s measure how much it inhibits its rival. When
self-limitation dominates cross-limitation, competition is too weak to exclude,
and coexistence is stable. When cross-limitation dominates, competition is strong
and the two cannot share — the principle of competitive exclusion.

| Regime | Interior point | Boundary nodes | Outcome |
| --- | --- | --- | --- |
| $\sigma_1 \sigma_2 > \alpha_1 \alpha_2$ (weak) | Stable node | Unstable | Coexistence |
| $\sigma_1 \sigma_2 < \alpha_1 \alpha_2$ (strong) | Saddle | Stable | Exclusion; winner set by initial state |

> **Worked example.** Two Boyce competition systems show both outcomes. For
> $$
> x' = x(1 - x - y), \qquad y' = y(0.75 - 0.5x - y),
> $$
> the constants give $\sigma_1\sigma_2 - \alpha_1\alpha_2 = (1)(1) - (1)(0.5) =
> 0.5 > 0$ (weak competition). The interior critical point is $(1/2, 1/2)$, where
> the Jacobian is
> $$
> J = \begin{pmatrix} -0.5 & -0.5 \\ -0.25 & -0.5 \end{pmatrix}, \qquad
> r = \tfrac{1}{4}\!\left(-2 \pm \sqrt{2}\right) \approx -0.15,\ -0.85.
> $$
> Both eigenvalues are negative: an asymptotically stable node, and every
> first-quadrant trajectory converges to coexistence. Changing only the second
> species' coefficients,
> $$
> x' = x(1 - x - y), \qquad y' = y(0.5 - 0.75x - 0.25y),
> $$
> gives $\sigma_1\sigma_2 - \alpha_1\alpha_2 = 0.25 - 0.75 = -0.5 < 0$ (strong
> competition). Now the interior point $(1/2, 1/2)$ is a saddle, the two
> single-species states $(1, 0)$ and $(0, 2)$ are asymptotically stable nodes,
> and the survivor is set by the initial populations.[^boyce-comp3]

$$
% caption: Weak competition: the two nullclines cross at an interior node that
% draws in every first-quadrant trajectory, so both species survive.
\begin{tikzpicture}[scale=1.05, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (-0.2,0)--(4.2,0) node[right, black] {$x$};
\draw[->, black] (0,-0.2)--(0,4.2) node[above, black] {$y$};
% x-nullcline (solid) and y-nullcline (dashed), crossing at (1.27,1.27)
\draw[acc, thick] (0.0,2.0)--(3.5,0.0);
\node[acc, anchor=south west, font=\scriptsize] at (2.35,0.62) {$x$ nullcline};
\draw[acc, thick, dashed] (0.0,3.5)--(2.0,0.0);
\node[acc, anchor=west, font=\scriptsize] at (1.2,2.35) {$y$ nullcline};
% critical points: boundary (unstable) dots and interior stable node
\fill[black] (0,0) circle (1.5pt);
\fill[black] (3.5,0) circle (1.7pt);
\fill[black] (0,3.5) circle (1.7pt);
\fill[black] (1.27,1.27) circle (2.1pt);
\node[anchor=south west, font=\scriptsize] at (1.4,1.34) {stable node};
% trajectories converging to the interior node
\draw[black, ->] (3.2,2.9) .. controls (2.4,2.15) .. (1.55,1.5);
\draw[black, ->] (0.45,0.45) .. controls (0.85,0.9) .. (1.1,1.15);
\draw[black, ->] (3.0,0.5) .. controls (2.15,0.9) .. (1.5,1.18);
\end{tikzpicture}
$$

$$
% caption: Strong competition: the interior saddle sends out a separatrix that
% splits the first quadrant into the basins of the two single-species nodes.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\definecolor{sep}{HTML}{B4532A}
\draw[->, black] (-0.2,0)--(4.4,0) node[right, black] {$x$};
\draw[->, black] (0,-0.2)--(0,4.2) node[above, black] {$y$};
% nullclines (schematic straight lines)
\draw[acc, thick] (0.0,3.4) -- (3.6,0.0);
\node[acc, anchor=east, font=\scriptsize] at (2.35,0.32) {$x'=0$};
\draw[acc, thick, dashed] (0.0,1.5) -- (3.9,3.6);
\node[acc, anchor=west, font=\scriptsize] at (2.75,1.35) {$y'=0$};
% critical points
\fill[black] (0,0) circle (1.7pt);
\fill[black] (3.6,0) circle (2.0pt); \node[anchor=north, font=\scriptsize] at (3.6,-0.12) {node};
\fill[black] (0,3.35) circle (2.0pt); \node[anchor=east, font=\scriptsize] at (-0.1,3.35) {node};
\fill[black] (1.65,1.72) circle (2.2pt); \node[anchor=south west, font=\scriptsize] at (1.75,1.78) {saddle};
% separatrix through the saddle
\draw[sep, very thick] (0.35,0.15) .. controls (1.0,0.85) and (1.3,1.25) .. (1.65,1.72)
  .. controls (2.1,2.35) and (2.7,3.4) .. (3.1,4.05);
\node[sep, anchor=west, font=\scriptsize] at (2.75,3.55) {separatrix};
% a couple of trajectories converging to the nodes
\draw[black, ->] (2.3,0.9) .. controls (2.9,0.6) .. (3.45,0.12);
\draw[black, ->] (0.9,2.4) .. controls (0.6,2.9) .. (0.12,3.25);
\end{tikzpicture}
$$

## Predator and prey

Now let one species eat the other. The prey $x$ grows in the predator's absence;
the predator $y$ starves in the prey's absence; encounters, proportional to the
product $xy$, feed the predator and cost the prey. These assumptions give the
**Lotka–Volterra equations**:[^boyce-pp]

$$
\frac{\d x}{\d t} = x(a - \alpha y), \qquad \frac{\d y}{\d t} = y(-c + \gamma x),
$$

with $a, c, \alpha, \gamma > 0$. There are two critical points: the origin, and
the coexistence point $(c/\gamma,\ a/\alpha)$. Linearizing at the origin gives
eigenvalues $a > 0$ and $-c < 0$, a **saddle** — a small population of either
species does not simply die out. At the coexistence point the Jacobian is

$$
\begin{pmatrix} 0 & -\alpha c/\gamma \\ \gamma a/\alpha & 0 \end{pmatrix},
\qquad r = \pm i\sqrt{ac},
$$

pure imaginary: a **center** for the linear system. This is the delicate case the
[linearization theorem](/differential-equations/nonlinear/locally-linear-and-liapunov)
leaves undecided, so we go back to the full system. Dividing the two equations
eliminates $t$,

$$
\frac{\d y}{\d x} = \frac{y(-c + \gamma x)}{x(a - \alpha y)},
$$

which separates and integrates to the conserved relation

$$
a \ln y - \alpha y + c \ln x - \gamma x = C.
$$

For each $C$ this is a closed curve around $(c/\gamma, a/\alpha)$, so the
coexistence point is a center for the nonlinear system too: predator and prey
populations cycle indefinitely.[^boyce-pp2]

> **Worked example.** Boyce's predator-prey system is
> $$
> x' = x(1 - 0.5y), \qquad y' = y(-0.75 + 0.25x),
> $$
> so $a = 1$, $\alpha = 0.5$, $c = 0.75$, $\gamma = 0.25$. The critical points are
> the origin and $(c/\gamma,\ a/\alpha) = (3, 2)$. At $(3, 2)$ the Jacobian is
> $$
> \begin{pmatrix} 0 & -1.5 \\ 0.5 & 0 \end{pmatrix}, \qquad
> r = \pm\tfrac{\sqrt{3}}{2}\,i = \pm i\sqrt{ac},
> $$
> a linear center. Dividing the two equations and separating produces the
> conserved quantity
> $$
> 0.75 \ln x + \ln y - 0.5y - 0.25x = C,
> $$
> whose level curves are closed loops around $(3, 2)$, so the nonlinear system
> keeps the center. The small-oscillation period is
> $2\pi/\sqrt{ac} = 2\pi/\sqrt{0.75} \approx 7.26$, the orbits run
> counterclockwise, and the predator peak trails the prey peak by a quarter
> cycle.[^boyce-pp4]

$$
% caption: Lotka-Volterra orbits are closed loops around the coexistence point
% (left); in time the prey peak leads the predator peak by a quarter cycle (right).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\definecolor{prd}{HTML}{B4532A}
% ---- phase portrait ----
\begin{scope}
  \draw[->, black] (-0.2,0)--(4.3,0) node[right, black] {$x$ (prey)};
  \draw[->, black] (0,-0.2)--(0,3.9) node[above, black] {$y$ (pred.)};
  \foreach \s in {0.55,0.95,1.35} \draw[acc] (2.2,1.7) ellipse ({\s*1.35} and \s);
  \draw[acc, ->] (2.2,{1.7+1.35}) -- (2.17,{1.7+1.35});
  \fill[black] (2.2,1.7) circle (1.8pt);
  \node[anchor=west, font=\scriptsize] at (2.32,1.5) {equilibrium};
  \node[anchor=north] at (2.1,-0.55) {closed orbits};
\end{scope}
% ---- time series ----
\begin{scope}[xshift=6.2cm]
  \draw[->, black] (-0.2,0)--(4.5,0) node[right, black] {$t$};
  \draw[->, black] (0,-0.2)--(0,2.9) node[above, black] {pop.};
  \draw[acc, thick, domain=0:4.2, samples=90] plot (\x, {1.4 + 1.0*sin(180*\x/1.4)});
  \draw[prd, thick, dashed, domain=0:4.2, samples=90] plot (\x, {1.4 + 1.0*sin(180*\x/1.4 - 90)});
  \node[acc, anchor=south, font=\scriptsize] at (0.7,2.5) {prey};
  \node[prd, anchor=south, font=\scriptsize] at (1.75,2.5) {predator};
  \node[anchor=north] at (2.1,-0.55) {prey leads predator};
\end{scope}
\end{tikzpicture}
$$

Near the coexistence point the linear approximation gives sinusoidal oscillations
of period $2\pi/\sqrt{ac}$, independent of amplitude, with the predator lagging
the prey by a quarter cycle. Averaged over a full cycle the populations equal the
equilibrium values $c/\gamma$ and $a/\alpha$.[^boyce-pp3] The mechanism is a feed
back loop: abundant prey feed a predator boom, heavy predation crashes the prey,
the starved predators decline, and the released prey recover. Cyclic
predator–prey data — the lynx and snowshoe hare records of the Hudson's Bay
Company, with their nine-to-ten-year period — show exactly this out-of-phase
oscillation.

Because a center is structurally fragile, the Lotka–Volterra cycle must be read
with caution. The smallest change to the model, such as a self-limiting prey
term, can turn the neutral center into a spiral, so real populations rarely
trace the perfect closed loops the idealized equations predict.

## Limit cycles

The predator–prey orbits are closed curves, but each sits in a nested family, and
a perturbation slides the state onto a neighboring loop. A **limit cycle** is
isolated instead: a closed trajectory that nearby trajectories spiral
_toward_.

> **Definition (Limit cycle).** A closed trajectory such that other, non-closed
> trajectories spiral toward it (from inside, outside, or both) as $t \to
> \infty$ or $t \to -\infty$. It is **asymptotically stable** (orbitally stable)
> if all nearby trajectories on both sides approach it, **unstable** if all
> depart, and **semistable** if one side approaches and the other departs.

A limit cycle is a self-sustaining oscillation with a fixed amplitude set by the
system itself, not by the initial conditions — unlike the center, whose amplitude
is whatever the starting point dictates. The cleanest example is

$$
x' = x + y - x(x^2 + y^2), \qquad y' = -x + y - y(x^2 + y^2).
$$

In polar coordinates this decouples into

$$
\frac{\d r}{\d t} = r(1 - r^2), \qquad \frac{\d\theta}{\d t} = -1.
$$

The radial equation has $r' > 0$ for $r < 1$ and $r' < 0$ for $r > 1$: the unit
circle $r = 1$ is an asymptotically stable limit cycle, and every trajectory
except the origin spirals onto it.[^boyce-lc]

$$
% caption: An asymptotically stable limit cycle: trajectories from inside spiral
% outward and from outside spiral inward, both approaching the closed orbit.
\begin{tikzpicture}[scale=1.1, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\definecolor{cyc}{HTML}{B4532A}
\draw[black] (-2.1,0)--(2.1,0); \draw[black] (0,-2.1)--(0,2.1);
% the limit cycle
\draw[cyc, very thick] (0,0) circle (1.3);
\node[cyc, anchor=south west] at (0.95,0.95) {limit cycle};
% inner trajectory spiralling out toward r=1.3
\draw[acc, ->, domain=0:430, samples=90, variable=\t]
  plot ({(0.28 + 0.00238*\t)*cos(\t)}, {(0.28 + 0.00238*\t)*sin(\t)});
% outer trajectory spiralling in toward r=1.3
\draw[acc, ->, domain=0:360, samples=80, variable=\t]
  plot ({(2.0 - 0.00194*\t)*cos(\t+30)}, {(2.0 - 0.00194*\t)*sin(\t+30)});
\fill[black] (0,0) circle (1.5pt);
\end{tikzpicture}
$$

### Existence and nonexistence of closed orbits

Three theorems bracket the question of when closed trajectories can occur, all
for $x' = F(x, y)$, $y' = G(x, y)$ with continuous first partials.[^boyce-pb]

> **Theorem (Enclosed critical point).** A closed trajectory must enclose at
> least one critical point. If it encloses exactly one, that point cannot be a
> saddle.

> **Theorem (Bendixson's negative criterion).** On a simply connected domain
> where $F_x + G_y$ (the divergence of the field) keeps one sign, there is no
> closed trajectory.

> **Theorem (Poincaré–Bendixson).** Let $R$ be a closed, bounded region
> containing no critical point. If a trajectory enters $R$ and never leaves, then
> $R$ contains a closed trajectory, and the trapped trajectory is periodic or
> spirals toward a periodic one.

The first two are tests for _non_-existence: a region with no critical point, or
with only a saddle, or across which the divergence keeps its sign, holds no
closed orbit. The Poincaré–Bendixson theorem is the positive result — build an
annular trapping region with inflow on both boundaries and no equilibrium inside,
and a limit cycle must live there. This is a phenomenon peculiar to the plane; it
has no analogue in three dimensions, where chaos first becomes
possible.

### The van der Pol oscillator

The archetype of a system with a limit cycle is the van der Pol equation,
governing a triode oscillator's current $u$:[^boyce-vdp]

$$
u'' - \mu(1 - u^2)u' + u = 0, \qquad \mu \geq 0.
$$

Its damping coefficient $-\mu(1 - u^2)$ is the mechanism. For large $u$ it is
positive and drains energy, shrinking large oscillations; for small $u$ it is
negative and pumps energy, growing small ones. Between the two lies a single
oscillation of intermediate size that all others approach — an asymptotically
stable limit cycle. Written as a system with $x = u$, $y = u'$,

$$
x' = y, \qquad y' = -x + \mu(1 - x^2)y.
$$

The origin is the only critical point: an unstable spiral for $0 < \mu < 2$, an
unstable node for $\mu \geq 2$, so every trajectory is pushed away from it, while
far out the trajectories turn inward — the Poincaré–Bendixson picture. As $\mu$
grows the limit cycle deforms from a near-circle into a sharply cornered
relaxation oscillation, holding amplitude near $2$ while its period lengthens.
When a parameter crossing turns a stable spiral into an unstable one and spawns a
surrounding limit cycle, the transition is a **Hopf bifurcation**.

$$
% caption: The van der Pol relaxation oscillation for large $\mu$: long slow
% drifts punctuated by fast jumps, a waveform far from sinusoidal.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (-0.2,0)--(6.6,0) node[right, black] {$t$};
\draw[->, black] (0,-1.7)--(0,1.7) node[above, black] {$u$};
\draw[acc, thick]
  (0,1.15) .. controls (0.9,1.32) and (1.5,1.27) .. (1.8,1.1)
  -- (1.95,-1.1)
  .. controls (2.7,-1.28) and (3.3,-1.24) .. (3.65,-1.07)
  -- (3.8,1.12)
  .. controls (4.6,1.3) and (5.2,1.25) .. (5.55,1.1)
  -- (5.7,-1.1) .. controls (6.1,-1.22) .. (6.4,-1.16);
\node[black!70, anchor=south, font=\scriptsize] at (0.95,1.3) {slow drift};
\node[black!70, anchor=west, font=\scriptsize] at (2.0,0.12) {fast jump};
\end{tikzpicture}
$$

## Chaos: the Lorenz equations

Everything so far is two-dimensional, and in the plane the trajectories'
inability to cross forces order: a trapped trajectory must approach an equilibrium
or a closed orbit, and nothing wilder is possible. In three dimensions that
constraint lifts, and a deterministic system can behave unpredictably. The
canonical example is the **Lorenz equations**, a drastic reduction of atmospheric
convection:[^boyce-lorenz]

$$
\frac{\d x}{\d t} = \sigma(-x + y), \qquad
\frac{\d y}{\d t} = rx - y - xz, \qquad
\frac{\d z}{\d t} = -bz + xy,
$$

with $\sigma = 10$ and $b = 8/3$ standard, and $r$ (proportional to the imposed
temperature difference) the control parameter. Here $x$ measures the intensity of
convective motion and $y, z$ the temperature variations. The system has one
critical point at the origin for $r \leq 1$; as $r$ passes $1$ the origin loses
stability and two new critical points $P_1, P_2$ appear at $(\pm\sqrt{b(r-1)},\
\pm\sqrt{b(r-1)},\ r - 1)$, representing steady convection rolls turning one way
or the other.

The eigenvalues at $P_1$ and $P_2$ stay in the stable half-plane until a second
threshold. For $\sigma = 10$, $b = 8/3$, that threshold is $r_2 = 470/19 \approx
24.74$: beyond it, $P_1$ and $P_2$ each acquire a pair of complex eigenvalues
with positive real part, and all three critical points are unstable.[^boyce-lorenz2]
Although all three critical points are unstable, every solution stays bounded,
drawn toward a set of zero volume with an intricate layered structure: a
**strange attractor**.

| $r$ range | Critical points | Behavior |
| --- | --- | --- |
| $0 < r < 1$ | $P_0$ only, stable | all trajectories reach the origin |
| $1 < r < r_2$ | $P_0$ unstable; $P_1, P_2$ stable | settle to one steady convection roll |
| $r > r_2 \approx 24.74$ | all three unstable | chaotic wandering on the strange attractor |

$$
% caption: Schematic $xz$-projection of a Lorenz trajectory for $r=28$: it loops
% erratically around each unstable point $P_1$, $P_2$, never repeating or settling.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% ---- left wing: outward spiral around P1 ----
\begin{scope}[xshift=-1.35cm, yshift=0.2cm]
  \draw[acc, domain=20:520, samples=120, variable=\t]
    plot ({(0.16 + 0.0023*\t)*cos(\t)}, {(0.16 + 0.0023*\t)*sin(\t)});
  \fill[black] (0,0) circle (1.6pt);
  \node[anchor=north, font=\scriptsize] at (0,-0.3) {$P_1$};
\end{scope}
% ---- right wing: outward spiral around P2 ----
\begin{scope}[xshift=1.35cm, yshift=0.2cm]
  \draw[acc, domain=20:520, samples=120, variable=\t]
    plot ({-(0.16 + 0.0023*\t)*cos(\t)}, {(0.16 + 0.0023*\t)*sin(\t)});
  \fill[black] (0,0) circle (1.6pt);
  \node[anchor=north, font=\scriptsize] at (0,-0.3) {$P_2$};
\end{scope}
% crossover strands linking the two wings
\draw[acc] (-0.15,1.55) .. controls (0.0,2.05) and (0.0,2.05) .. (0.15,1.55);
\draw[acc] (-1.1,-0.75) .. controls (0.0,-1.35) .. (1.1,-0.75);
\draw[->, black] (2.9,1.0)--(3.6,1.0) node[right, black, font=\scriptsize] {$x$};
\draw[->, black] (2.9,1.0)--(2.9,1.9) node[above, black, font=\scriptsize] {$z$};
\end{tikzpicture}
$$

The defining feature is **sensitive dependence on initial conditions**. Two
solutions starting a hundredth of a unit apart, at $(5, 5, 5)$ and $(5.01, 5, 5)$,
track together until about $t = 10$, then diverge and bear no
resemblance thereafter. The Lorenz equations are deterministic — the
future is fixed by the present — yet arbitrarily small uncertainty in the present
grows until long-range prediction is impossible.[^boyce-lorenz3] This was the
observation that led Lorenz to conclude that detailed long-range weather forecasts
cannot be made, and it gave the field its name.

> **Definition (Strange attractor, chaos).** A **strange attractor** is a bounded
> attracting set of zero volume with fine self-similar structure, toward which
> trajectories converge without ever closing into a cycle. A system whose
> solutions display sensitive dependence on initial conditions — nearby starts
> leading to wildly different futures — is called **chaotic**.

The projection above only hints at the structure; the true trajectory lives in
three dimensions and never crosses itself, the apparent crossings being artifacts
of the flattened view. In the plane the qualitative methods force a trapped
trajectory toward an equilibrium or a closed orbit; in one more dimension the
same equations produce behavior that is deterministic and bounded yet permanently
unpredictable.

[^boyce-comp]: **Boyce**, §9.4 — the competing-species model and its four
    critical points; the coordinate axes as invariant trajectories.
[^boyce-comp2]: **Boyce**, §9.4, equations (38)–(41) — the eigenvalue equation at
    the coexistence point and the $\sigma_1\sigma_2$ versus $\alpha_1\alpha_2$
    criterion for coexistence.
[^boyce-comp3]: **Boyce**, §9.4, Examples 1–2 — the weak-competition system with a
    coexistence node at $(1/2, 1/2)$ and eigenvalues $\tfrac{1}{4}(-2\pm\sqrt 2)$,
    and the strong-competition system whose interior point is a saddle.
[^boyce-pp]: **Boyce**, §9.5 — derivation of the Lotka–Volterra equations and the
    saddle/center classification; Simmons, §57, on Volterra's prey-predator
    equations.
[^boyce-pp2]: **Boyce**, §9.5, equations (21)–(22) — the conserved quantity whose
    level curves are the closed predator–prey orbits.
[^boyce-pp4]: **Boyce**, §9.5, Example 1 — the system $x'=x(1-0.5y)$,
    $y'=y(-0.75+0.25x)$, its center at $(3,2)$ with $r=\pm(\sqrt3/2)i$, and the
    conserved relation $0.75\ln x+\ln y-0.5y-0.25x=C$.
[^boyce-pp3]: **Boyce**, §9.5, equations (23)–(24) — the small-oscillation
    period $2\pi/\sqrt{ac}$, the quarter-cycle phase lag, and the cycle averages.
[^boyce-lc]: **Boyce**, §9.7, Example 1 — the polar reduction $r' = r(1 - r^2)$
    and the unit-circle limit cycle.
[^boyce-pb]: **Boyce**, §9.7, Theorems 9.7.1–9.7.3 — the enclosed-critical-point
    result, Bendixson's negative criterion, and the Poincaré–Bendixson theorem;
    Simmons, §64.
[^boyce-vdp]: **Boyce**, §9.7, Example 2 — the van der Pol equation and its limit
    cycle; Simmons, §65, on the van der Pol equation.
[^boyce-lorenz]: **Boyce**, §9.8 — the Lorenz equations, their derivation from
    convection, and the critical points $P_0, P_1, P_2$.
[^boyce-lorenz2]: **Boyce**, §9.8, equations (9)–(11) — the eigenvalue thresholds
    $r = 1$ and $r_2 = 470/19 \approx 24.74$.
[^boyce-lorenz3]: **Boyce**, §9.8, Figures 9.8.2–9.8.6 — sensitive dependence on
    initial conditions and the strange attractor.