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The Lumenary · Advanced undergraduate

Dynamical systems and chaos

Name: ______________________

Date: ______________________

You will learn to

Find fixed points of maps and flows, prove the derivative test for attracting and repelling fixed points, and classify the equilibria of planar linear systems by trace and determinant
Use linearisation (Hartman–Grobman) and Lyapunov functions to decide stability, and recognise when linearisation is inconclusive
Use trapping regions and the Poincaré–Bendixson theorem to prove that limit cycles exist, and explain why planar flows cannot be chaotic
Analyse the logistic map — fixed points, the period-2 orbit, period doubling and Feigenbaum's constant — distinguishing numerical evidence from proof
Prove that the doubling map has sensitive dependence and dense periodic points, and transfer this to the logistic map at $r = 4$

1.
The map $f(x) = x^3$ has three fixed points. How many of them are attracting?
2.
Classify the equilibrium at the origin of $\dot x = x + 2y$ , $\dot y = 3x + 2y$ .
- A.Saddle
- B.Unstable node
- C.Unstable spiral
- D.Centre
3.
For the logistic map with $r = 2.5$ , how do orbits starting near the fixed point $x^* = 1 - 1/r = 0.6$ behave?
- A.They converge to $0.6$ , alternating sides, with the distance roughly halving each step.
- B.They converge to $0.6$ from one side, monotonically.
- C.They converge to a 2-cycle.
- D.They move away from $0.6$ .
4.
How many points of the circle have minimal period exactly $3$ under the doubling map $D(x) = 2x \bmod 1$ ?
5.
On the torus $\R^2/\Z^2$ , the flow $\dot x = 1$ , $\dot y = \sqrt2$ has no equilibria, every orbit is bounded, and every orbit is dense — so no orbit is periodic or approaches a periodic orbit. Which hypothesis of the Poincaré–Bendixson theorem fails?
- A.The phase space is not (an open subset of) the plane.
- B.The vector field is not $C^1$ .
- C.The orbit does not stay in a compact set.
- D.The compact set contains an equilibrium.
6.
In Mathlib, which statement says that p lies on an orbit of f of period exactly 2 (periodic with period 2, but not a fixed point)?
- A.Function.minimalPeriod f p = 2
- B.Function.IsPeriodicPt f 2 p
- C.f (f p) = p
- D.∀ n, f^[n] p ≠ p
7.
Consider $\dot x = -y - x(x^2 + y^2)$ , $\dot y = x - y(x^2 + y^2)$ . Which statement about the origin is correct?
- A.Its linearisation is a centre, but the origin is asymptotically stable: in polar coordinates $\dot r = -r^3$ .
- B.It is a centre, because its linearisation is a centre.
- C.It is unstable, because the eigenvalues $\pm i$ are not in the left half-plane.
- D.Hartman–Grobman shows the flow is conjugate to rotation near the origin.
8.
Write lyapunov(r, x0, n, transient) that estimates the Lyapunov exponent of the logistic map $f_r(x) = rx(1 - x)$ : iterate transient times from x0 without recording, then return the average of $\ln|f_r'(x_k)| = \ln|r(1 - 2x_k)|$ over the next n iterates $x_k$ .
Computational — Practice
9.
Which statements are true? Choose all that apply.
- A.The logistic map at $r = 4$ is deterministic, and its orbits have the closed form $x_n = \sin^2(2^n\pi\theta_0)$ .
- B.A positive Lyapunov exponent computed in floating point proves that a map is chaotic.
- C.Since the logistic map at $r = 3.83$ has a period-3 orbit, it has periodic orbits of every period.
- D.A planar autonomous flow can have a strange attractor.
- E.An irrational rotation of the circle is chaotic in Devaney's sense, because every orbit is dense.
10.
For which values of $r$ is the period-2 orbit $\{p_+, p_-\}$ of the logistic map attracting?
- A. $3 < r < 1 + \sqrt6$
- B. $3 < r < 4$
- C. $1 < r < 3$
- D. $3 < r < 3.5699\ldots$ (up to the accumulation point)

Answer key — Dynamical systems and chaos

1.
The fixed points are $0, \pm1$ . Since $f'(0) = 0$ , the point $0$ is attracting (every orbit with $|x_0| < 1$ converges to $0$ ); since $f'(\pm 1) = 3 > 1$ , both $\pm 1$ repel. So exactly one fixed point attracts.
2.
$\tau = 3$ , $\Delta = 1\cdot2 - 2\cdot3 = -4 < 0$ . The eigenvalues $\frac{3 \pm \sqrt{9 + 16}}{2} = 4, -1$ have opposite signs: a saddle. Orbits approach along the eigenvector for $-1$ , namely $(1, -1)$ , and leave along the eigenvector for $4$ , namely $(2, 3)$ .
3.
$f_{2.5}'(0.6) = 2.5(1 - 1.2) = -0.5$ . Since $|-0.5| < 1$ the fixed point attracts, and since the multiplier is negative the deviation $x_n - 0.6$ changes sign at each step, roughly halving: the cobweb is an inward square spiral.
4.
$D^3(x) = x$ iff $7x \in \Z$ , giving $x = 0, \frac17, \dots, \frac67$ . The only fixed point of $D$ is $0$ ( $2x \equiv x$ means $x \in \Z$ ). So $6$ points have minimal period $3$ , forming the two orbits $\frac17 \to \frac27 \to \frac47$ and $\frac37 \to \frac67 \to \frac57$ .
5.
Everything holds except planarity. The proof of Poincaré–Bendixson rests on the Jordan curve theorem, which fails on the torus (a meridian does not separate it). Correspondingly, the irrational flow on the torus has bounded, recurrent, non-periodic orbits.
6.
Function.minimalPeriod f p = 2. The statements IsPeriodicPt f 2 p and f (f p) = p are equivalent to each other and hold at fixed points as well; to get period exactly 2 one adds ¬ IsFixedPt f p, and since 2 is prime this combination is equivalent to minimalPeriod f p = 2 (Function.minimalPeriod_eq_prime).
7.
The Jacobian at $0$ is $\begin{pmatrix}0 & -1\\1 & 0\end{pmatrix}$ , eigenvalues $\pm i$ : a linear centre, non-hyperbolic. But $r\dot r = x\dot x + y\dot y = -(x^2 + y^2)^2$ , so $\dot r = -r^3$ and $\dot\theta = 1$ : orbits spiral into the origin, ever more slowly ( $r(t) = r_0/\sqrt{1 + 2r_0^2t}$ ). Asymptotically stable.
8.
```
import math

def lyapunov(r, x0, n, transient):
    x = x0
    for _ in range(transient):
        x = r * x * (1 - x)
    total = 0.0
    for _ in range(n):
        total += math.log(abs(r * (1 - 2 * x)))
        x = r * x * (1 - x)
    return total / n
```
At $r = 2.5$ the orbit sits at the fixed point $0.6$ and every term is $\ln|2 - r| = \ln 0.5$ . At $r = 4$ the answer approaches $\ln 2$ — the floating-point orbit is not the true orbit after about 50 steps, but its statistics are still right, as shadowing theory explains. (If an iterate hits exactly $x = \frac12$ the logarithm is $-\infty$ ; for these starting values it does not happen.)
9.
True: the closed form at $r = 4$ , and Sharkovskii's consequence of a period-3 orbit. False: numerical Lyapunov exponents are evidence rather than proof; planar flows cannot have strange attractors; and an irrational rotation, though transitive, lacks periodic points and sensitive dependence.
10.
The multiplier is $4 + 2r - r^2$ (Theorem 2(b), checked in Lean). It is $< 1$ iff $r^2 - 2r - 3 > 0$ , i.e. $r > 3$ , and $> -1$ iff $r^2 - 2r - 5 < 0$ , i.e. $r < 1 + \sqrt6 \approx 3.449$ . At $r = 1 + \sqrt 6$ the multiplier reaches $-1$ and the next period doubling occurs.