Chapter 5 · Maps, bifurcations, and the routes to chaos

Part II — From regular motion to chaos Knob: $r$, $x_0$, the parameter window live notebook

Overview

Every forecast model contains parameters nobody can measure exactly: a mixing length, an autoconversion threshold, an entrainment rate. Suppose one is uncertain by half a percent. How much does that matter?

Usually, half a percent. But there are parameter values where the answer is qualitatively different — where an arbitrarily small change turns a steady state into an oscillation, or an oscillation into chaos. Those values are bifurcations, and this chapter is about how they are organised.

The vehicle is deliberately absurd:

x_{n+1} = r\,x_n(1 - x_n), \qquad 0 \le x \le 1 .

One variable, one parameter, no time step, no derivatives — just multiplication. It has no atmosphere in it anywhere. And yet the numbers describing its route to chaos are the same numbers found in convecting fluids, dripping taps and nonlinear circuits. That portability is the claim chapter 3 makes about why a three-variable model teaches something true about a $10^9$ -variable one. This chapter is where it stops being a claim.

The model

A fixed point satisfies $x^* = 1 - 1/r$ , and attracts when $|f'(x^*)| = |2 - r| < 1$ . At $r = 3$ that multiplier reaches $-1$ and the fixed point gives way to a 2-cycle; the 2-cycle gives way to a 4-cycle at 3.449, and so on. The notebook makes the stability calculation graphical (the cobweb), then sweeps the whole family at once.

chaoslib.maps provides the machinery: bifurcation_points and map_lyapunov_exponent are vectorised over the parameter axis, so a 1400-point sweep costs the same as a single value — which is why almost everything in this chapter runs live in the browser rather than being precomputed.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Four results

The Lyapunov exponent is the diagnosis, and the diagram alone is not. For a 1-D map chapter 7’s definition collapses to $\lambda(r) = \langle\ln|f'_r(x)|\rangle$ — no tangent linear model, no re-orthonormalisation, one average. Measured $\lambda(4) = 0.69315$ against the exact $\ln 2$ , since at $r = 4$ the map is conjugate to $x \mapsto 2x \bmod 1$ . Its sign separates periodic from chaotic parameters where the picture cannot: the period-3 window sits well above the accumulation point $r_\infty = 3.5699457$ and has $\lambda < 0$ across three quarters of its width.

“Chaotic” is a property of a parameter value, not of a system. Inside that period-3 window is a complete cascade $3 \to 6 \to 12 \to 24$ , and inside its chaotic band are further windows of period 9, 12, 30 and 36. The chaotic and periodic parameter sets are interleaved at every scale, so no finite sampling of parameter space establishes which side a given model sits on.

$\delta = 4.669$ belongs to no particular map. The notebook locates superstable parameters — where the $2^n$ -cycle contains the critical point, so its multiplier is exactly zero and the parameter solves the smooth equation $f_r^{2^n}(x_c) = x_c$ — and does it for three families sharing no algebra: logistic $r x(1-x)$ , sine $r\sin(\pi x)$ , cubic $r x(1-x^2)$ , with critical points $1/2$ , $1/2$ , $1/\sqrt3$ and first superstable parameters 2, $1/2$ , $3/2$ . Three unrelated cascades, one ratio: 4.669191, 4.664075, 4.669038 against $\delta = 4.669201609$ .

The self-similarity is quantitative too. The period-3 window’s own $3\cdot 2^n$ cascade has superstable parameters 3.831874, 3.844569, 3.848345, 3.849198, 3.849383, 3.849423, whose spacing ratios converge to the same $\delta$ from a cascade spanning 0.0175 in $r$ rather than 1.57. That identity is the renormalisation structure which makes $\delta$ universal.

A regime can end with nothing changing. Below the tangent bifurcation at $r_c = 1 + 2\sqrt2$ the 3-cycle does not exist, but $f^3$ passes close to the diagonal, and an orbit entering that channel is nearly 3-periodic for tens of iterations before bursting out. Mean laminar length diverges as $(r_c - r)^{-0.4965}$ against the predicted $-1/2$ , measured over a 33-fold range of $r_c - r$ . Nothing external changes when a burst begins — and the diverging timescale is the mechanism behind critical-slowing-down early-warning indicators, taken up in chapter 27.

Exercises

Analytic. Show that the 2-cycle born at $r = 3$ is $x_\pm = [(r+1) \pm \sqrt{(r-3)(r+1)}]/2r$ , and that it loses stability at $r = 1 + \sqrt6$ . Then show that at $r_c = 1 + 2\sqrt2$ the third iterate satisfies $f^3(x) = x$ and $(f^3)'(x) = +1$ simultaneously, and explain why that makes the 3-cycle’s birth invisible to a root-finder.

Computational. Measure $\delta$ from a fourth unimodal family of your own choosing. Then explain, from the size of the successive gaps, why double precision limits the logistic cascade to about nine levels — and why that is a limitation of arithmetic rather than of the mathematics.

Exploratory. Set the parameter window to the period-3 range and read $\lambda$ ; then set $r = 3.83$ in Section 1 and count the cycle. Explain why the bifurcation diagram alone could mislead you here and $\lambda$ could not.

Chapter 5 · Maps, bifurcations, and the routes to chaos

Overview

The model

Four results

Exercises

Further reading