Dynamical Systems Lab 9 - Routes to Chaos
24 Apr 2026
Chaos is not a destination that a system leaps to abruptly. In most systems there is a precisely structured route — a sequence of bifurcations that systematically destroy order before chaos takes hold. The best understood route is period doubling: at each step the orbit’s period doubles, the doubling intervals shrink at a universal ratio, and at the accumulation point chaos begins. The ratio is the Feigenbaum constant $\delta \approx 4.669$, independent of the specific equations.
The logistic map: derivation from population dynamics
Consider a population $N_t$ with discrete generations. The simplest model with density-dependent regulation is
\[N_{t+1} = r\, N_t\!\left(1 - \frac{N_t}{K}\right)\]where $r > 0$ is the intrinsic growth rate and $K$ the carrying capacity. Non-dimensionalise by $x_t = N_t / K$:
\[x_{t+1} = r\, x_t(1-x_t)\]This is the logistic map, popularised by Robert May (1976) as a model of insect populations. It has a remarkably simple form but contains the full complexity of chaotic dynamics. The state space is $x \in [0,1]$ and the map is well-defined (it maps $[0,1]$ to $[0,1]$) for $r \in [0,4]$.
The logistic map is also the universal representative of unimodal maps — maps on an interval with a single quadratic maximum. Any such map undergoes qualitatively identical period-doubling dynamics.
Fixed points and their stability
Period-1 fixed points: Solve $x^* = rx^(1-x^)$, giving
\[x^* = 0 \quad \text{and} \quad x^* = 1 - 1/r \equiv x_1^* \quad (r > 1)\]| Stability: $ | F’(x^*) | = | r(1-2x^*) | $. |
| - At $x^* = 0$: $ | F’(0) | = r$. Stable for $r < 1$. | ||
| - At $x_1^*$: $ | F’(x_1^*) | = | 2-r | $. Stable for $1 < r < 3$. |
Period-2 cycle: The period-2 orbit satisfies $F^2(x) = x$, $F(x) \neq x$. The points of the 2-cycle are the solutions of
\[F^2(x) - x = 0 \quad \text{after dividing out the fixed points}\]After algebra, the two 2-cycle points are
\[x_{\pm} = \frac{r+1 \pm \sqrt{(r+1)(r-3)}}{2r}\]Real for $r \geq 3$. They are born at $r = 3$ in a period-doubling bifurcation of $x_1^*$.
| Stability of the 2-cycle: the cycle is stable when $ | (F^2)’(x_-) | < 1$. The chain rule gives |
After expanding: $(F^2)’(x_\pm) = -r^2 + 2r + 4$. This equals $+1$ at $r = 3$ (birth of the cycle) and $-1$ at $r = 3.449…$, where the 2-cycle undergoes its own period-doubling bifurcation.
The cobweb construction
The cobweb diagram is the standard graphical method for iterating a 1D map:
- Start at $(x_0, 0)$ on the $x$-axis.
- Draw a vertical line up to the curve $y = F(x_0)$, landing at $(x_0, x_1)$ where $x_1 = F(x_0)$.
- Draw a horizontal line to the diagonal $y = x$, landing at $(x_1, x_1)$ — this puts $x_1$ on the $x$-axis.
- Repeat: vertical to curve, horizontal to diagonal.
The cobweb’s long-run pattern reveals the attractor: - Converging to a point: fixed point (the cobweb spirals or monotonically approaches one $x$-value). - Bouncing between two levels: period-2 cycle (a rectangle is traced). - Cycle of $p$ values: period-$p$ cycle ($p$-gon in the cobweb). - Band filling: chaos (the cobweb wanders densely through an interval).
The cobweb is a complete solution method — by following the construction, one can predict the entire future of the orbit for any initial condition and any $r$.
The period-doubling cascade and Feigenbaum’s discovery
Let $r_n$ denote the value of $r$ at which the $n$-th period-doubling bifurcation occurs:
| $n$ | $r_n$ | Period | Interval $r_n - r_{n-1}$ | Ratio |
|---|---|---|---|---|
| 1 | 3.0000 | $1 \to 2$ | — | — |
| 2 | 3.4495 | $2 \to 4$ | 0.4495 | — |
| 3 | 3.5441 | $4 \to 8$ | 0.0946 | 4.75 |
| 4 | 3.5644 | $8 \to 16$ | 0.0203 | 4.66 |
| 5 | 3.5688 | $16 \to 32$ | 0.0044 | 4.62 |
| $\infty$ | 3.5699… | Chaos onset | 0 | 4.6692… |
Mitchell Feigenbaum (1975) discovered numerically that the ratio of successive interval lengths converges to a universal constant:
\[\delta = \lim_{n\to\infty}\frac{r_n - r_{n-1}}{r_{n+1} - r_n} \approx 4.6692016...\qquad \text{(Feigenbaum constant)}\]This constant is the same for every smooth unimodal map with a quadratic maximum — the logistic map, $r\sin(\pi x)$, the Hénon map (Lab 7), and many others.
Renormalisation group derivation of $\delta$
Feigenbaum’s key insight was to view the period-doubling cascade as a fixed point of a renormalisation operator.
Define the operator $\mathcal{R}$ acting on unimodal maps $f$:
\[(\mathcal{R}f)(x) = \frac{1}{\alpha}f\!\left(f(\alpha x)\right)\]where $\alpha$ is chosen so that $\mathcal{R}f$ is again a unimodal map with the same normalisation (e.g., maximum value 1 at $x=0$). The operator $\mathcal{R}$ squares the map and rescales — it takes a map near its $2^n$-cycle to a map near its $2^{n-1}$-cycle (in the sense of looking at the dynamics on finer scales).
Fixed point of $\mathcal{R}$: There exists a universal function $g^*$ (the Feigenbaum-Cvitanovic fixed-point function) satisfying
\[g^*(x) = \frac{1}{\alpha}g^*\!\bigl(g^*(\alpha x)\bigr), \qquad g^*(0) = 1, \quad \alpha = -g^*(1) \approx -2.5029...\]| Linearising $\mathcal{R}$ around $g^*$: The derivative $D\mathcal{R} | _{g^}$ has a single unstable eigenvalue $\delta \approx 4.669$. This eigenvalue governs the rate at which perturbations grow under successive doublings — the ratio of successive parameter intervals is $1/\delta$ because the bifurcation points are the preimages of $g^$ under $\mathcal{R}$. |
The universality of $\delta$ is thus explained: it is an eigenvalue of a universal linear operator, independent of which particular unimodal map you start with.
Periodic windows and Sharkovskii’s theorem
Within the chaotic regime ($r > r_\infty \approx 3.5699$), there are isolated intervals of $r$ where a stable periodic orbit suddenly appears. These periodic windows arise from saddle-node bifurcations of the iterated map $F^p$.
The largest window is the period-3 window at $r \approx 3.828$. Its existence has a profound implication:
Sharkovskii’s theorem (1964): Order the positive integers as
\[3 \succ 5 \succ 7 \succ \cdots \succ 2\cdot 3 \succ 2\cdot 5 \succ \cdots \succ 4\cdot 3 \succ \cdots \succ 16 \succ 8 \succ 4 \succ 2 \succ 1\]If a continuous map of an interval has a periodic orbit of period $m$, it has periodic orbits of all periods $n$ with $m \succ n$ in this ordering. Since 3 is the largest element, a period-3 orbit implies orbits of all periods. Li and Yorke (1975) popularised this as “Period 3 implies chaos”.
Other routes to chaos
Period doubling is not the only route. Two other classical routes:
Quasi-periodicity route (Ruelle-Takens-Newhouse): Add incommensurate frequencies $\omega_1, \omega_2$ (a 2-torus), then $\omega_3$ (3-torus). Ruelle and Takens (1971) proved that a generic perturbation of a 3-torus produces a strange attractor. This route appears in Taylor-Couette flow and in electronic circuits.
Intermittency route (Pomeau-Manneville, 1980): Near a saddle-node bifurcation of a periodic orbit, the system spends long stretches in a “laminar” near-periodic phase (the ghost of the destroyed orbit) punctuated by brief chaotic bursts. Three types: - Type I: near a saddle-node bifurcation; bursts grow algebraically frequent as $r$ increases. - Type II: near a subcritical Hopf; bursts are accompanied by oscillation. - Type III: near an inverse period-doubling; bursts follow a $1/f$ power spectrum.
Live demo
The left canvas shows the cobweb diagram for the chosen $r$ and initial condition $x_0$, with $N$ iterates drawn. The right canvas shows the full bifurcation diagram (precomputed for 500 values of $r$, 80 orbit points each) with gold dots marking the current $r$’s orbit. Click Sweep r to animate $r$ from 1 to 4 in real time, watching the cascade unfold. The info bar shows the detected period and Lyapunov exponent.
Things to observe from the demo
Fixed point regime ($r = 2.5$): - Cobweb: a spiral converging to the intersection of $y = F(x)$ and $y = x$. - Period = 1. Lyapunov exponent $\lambda < 0$.
First period doubling ($r = 3.2$): - Cobweb: a square — the orbit bounces between exactly two values. - Period = 2. Two gold dots on the bifurcation diagram.
Period-4 ($r = 3.5$): - Cobweb: an octagon-like path through four levels. - Period = 4. Four dots on bifurcation diagram.
Near chaos onset ($r = 3.57$): - Cobweb wanders in a band but the orbit seems to have many closely spaced values. - $\lambda \approx 0$: exactly at the accumulation point.
Period-3 window ($r = 3.83$): - Cobweb converges to a triangle — the three-point cycle. - Period = 3. Three gold dots sharply visible on the bifurcation diagram. - This is Sharkovskii’s window: the most ordered feature inside the chaotic sea.
Full chaos ($r = 4$): - Cobweb fills the entire parabola densely — every part of $[0,1]$ is visited. - $\lambda = \ln 2 \approx 0.693$ — the analytical maximum. - Sweep r: watch the entire cascade in motion. Note the intervals between doublings shrinking at ratio $\approx 4.669$.
Key takeaways
- The logistic map $x_{n+1} = rx_n(1-x_n)$ models both population dynamics and the universal period-doubling route to chaos.
- Fixed-point and period-2 stabilities can be computed exactly; the stability criteria correctly predict the bifurcation values $r = 1, 3, 3.449…$
- The Feigenbaum constant $\delta \approx 4.6692$ is an eigenvalue of the renormalisation operator $\mathcal{R}$, explaining its universality across all unimodal maps.
- Sharkovskii’s theorem: period-3 implies orbits of all periods — the period-3 window is not just one special orbit, it is a gateway to the entire periodic hierarchy.