Back to the on-screen lesson ·
The driven damped pendulum, the logistic map, fixed points and period doubling, the Feigenbaum constant, Lyapunov exponents, strange attractors and prediction horizons.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
By the end of this lesson you will be able to find and test fixed points of a map, follow a period-doubling cascade, use the Feigenbaum constant, and compute how far ahead a chaotic system can be predicted.
You know damped and driven linear oscillators, phase space and fixed points, and that small deviations from a saddle grow exponentially. Every system solved so far in this course has been regular, with motion that can be written in closed form or at least repeats. This lesson shows that a simple nonlinear system, driven hard enough, can move in a way that never repeats and cannot be predicted far ahead, though its equations are exact.
| Term | What it means |
|---|---|
| Nonlinear system | One whose equations contain powers or functions of the unknowns, such as $\sin\phi$. |
| Driven damped pendulum | $\ddot{\phi} + 2\beta\dot{\phi} + \omega_0^2\sin\phi = \gamma\omega_0^2\cos\omega t$, a standard example of chaos. |
| Logistic map | $x_{n+1} = rx_n(1 - x_n)$, a one-line model with the same route to chaos. |
| Period doubling | A stable cycle giving way to one of twice the period as a parameter increases. |
| Bifurcation | A qualitative change in long-run behavior at a threshold value of a parameter. |
| Lyapunov exponent | $\lambda$ in $\delta(t) = \delta_0e^{\lambda t}$; positive means sensitive dependence on starting conditions. |
| Feigenbaum constant | $\delta = 4.669$, the universal ratio of successive period-doubling intervals. |
| Strange attractor | A fractal set in phase space on which a dissipative chaotic system moves. |
The driven damped pendulum, $\ddot{\phi} + 2\beta\dot{\phi} + \omega_0^2\sin\phi = \gamma\omega_0^2\cos\omega t$, differs from the driven oscillator of unit 2 only by $\sin\phi$ in place of $\phi$. For a gentle drive, $\gamma \ll 1$, it behaves the same way, settling to a motion at the drive's period. As $\gamma$ grows past about $1.066$ (for Taylor's standard parameters), the settled motion takes two drive periods to repeat, then four, then eight, the thresholds crowding closer each time, until beyond about $1.083$ the motion never repeats at all: chaos.
Chaotic motion has sensitive dependence on initial conditions. Two pendulums started with angles differing by a millionth of a radian separate as $\delta(t) \approx \delta_0e^{\lambda t}$, where the Lyapunov exponent $\lambda$ is positive, until within a few dozen cycles they are swinging unrelatedly. The equations are deterministic, but any error in knowing the start, however small, grows exponentially, so the motion can be predicted only a limited time ahead.
Another way: picture
Knead dough with a drop of food coloring in it. Each fold stretches the drop and lays it over itself. Two specks of color that started side by side soon end up in distant parts of the loaf, though every fold was exactly determined. The phase space of a chaotic system is kneaded the same way: stretched, which separates neighbors, and folded, which keeps the motion bounded.
Another way: steps
The simplest system with the same behavior is a single line of arithmetic. The logistic map $x_{n+1} = rx_n(1 - x_n)$, with $x$ between $0$ and $1$, was proposed as a model of a population in successive generations. For $1 < r < 3$, every start settles to the fixed point $x^* = 1 - 1/r$. At $r = 3$ the slope of the map at that point reaches $-1$, the fixed point turns unstable, and a stable two-cycle is born.
As $r$ rises further the two-cycle gives way to a four-cycle at $3.449$, an eight-cycle at $3.544$, and so on, the thresholds crowding toward $r_\infty = 3.5699$. Beyond it, for most values of $r$, the iterates never repeat and neighboring starts separate exponentially. The mathematical biologist Robert May at Princeton drew attention to this map in 1976, as a warning that simple deterministic models can behave in ways that look like noise.
Near a fixed point, write $x_n = x^ + \epsilon_n$. Expanding the map to first order, $\epsilon_{n+1} = f'(x^)\epsilon_n$, so the deviation is multiplied by the slope at every step. If $|f'(x^)| < 1$, deviations shrink and the fixed point is stable; if $|f'(x^)| > 1$, they grow.
For the logistic map, $f'(x^) = 2 - r$. It lies between $-1$ and $1$ exactly when $1 < r < 3$. At $r = 3$ the slope passes through $-1$, and a negative slope near $-1$ means the deviation flips sign each step while barely shrinking: the iterates alternate on either side of $x^$. Past the threshold that alternation stabilizes into a genuine two-cycle, the first period doubling. The same test, applied to the two-cycle as a fixed point of the twice-applied map, finds the next doubling.
In 1975, Mitchell Feigenbaum, working with a programmable calculator at Los Alamos National Laboratory, noticed that the intervals between successive period doublings shrink by the same ratio, $\delta = 4.669\ldots$, and then that the ratio was the same for entirely different maps. Any system whose route to chaos is period doubling through a smooth single hump shows this constant.
The driven damped pendulum does: its doublings at drive strengths $1.0663$, $1.0793$ and $1.0821$ give ratios near $4.6$. So do dripping faucets, heated fluids and electronic circuits, all measured in laboratories to agree with $4.669$. Universality means a number computed from a one-line map predicts experiments on real physical systems, one of the great surprises of twentieth-century physics.
Take two trajectories starting $\delta_0$ apart. In regular motion their separation grows at most linearly with time. In chaotic motion it grows exponentially, $\delta(t) = \delta_0e^{\lambda t}$, until it reaches the size of the whole motion. The Lyapunov exponent $\lambda$ measures the rate; its reciprocal is the time for errors to grow by a factor of $e$.
Plot $\ln\delta$ against time for two nearly identical pendulums and the result is a straight line with slope $\lambda$, with small wiggles, until it levels off. For Taylor's pendulum at $\gamma = 1.105$ the slope is about $1$ per drive period. Improving the starting accuracy by a factor of a thousand buys only $\ln 1000/\lambda \approx 7$ more periods of prediction: better measurements help only logarithmically.
A driven damped pendulum loses energy to friction, so its phase-space volumes shrink, and its long-run motion lies on an attractor. For a periodic motion the attractor is a closed curve. For chaotic motion it is a strange attractor: a set of zero volume but infinitely detailed structure, made of layers within layers, a fractal.
The best way to see it is a Poincaré section: record the pendulum's angle and angular velocity once per drive period, at the same phase. A periodic motion gives a few dots; a chaotic one gives an intricate pattern that, magnified, reveals ever finer copies of itself. Hamiltonian chaos, with no friction, is different: volumes are conserved, there is no attractor, and chaotic regions are interleaved with islands of regular motion.
Checking an answer. A fixed point must satisfy $f(x^) = x^$ exactly. Logistic-map iterates must stay between $0$ and $1$ for $r \le 4$. Successive bifurcation intervals must shrink by roughly $4.7$. And a predictability horizon must grow only logarithmically as the starting error shrinks; a linear dependence means the motion was not chaotic.
In 1963 Edward Lorenz, a meteorologist at MIT, restarted a weather simulation from numbers printed to three decimal places instead of six, and found that the forecast diverged completely after a few simulated weeks. His three-equation model became the first famous strange attractor, and his talk asking whether a butterfly's wings in Brazil could set off a tornado in Texas gave the idea its popular name.
Chaos is not limited to the weather. The orbits of the inner planets are chaotic with a Lyapunov time of about five million years, so their positions cannot be predicted precisely beyond a few tens of millions of years, though the solar system as a whole remains stable. Pluto's orbit is chaotic too. Knowing where chaos sets in, and how fast errors grow, tells scientists how far ahead any prediction can be trusted.
The National Weather Service's global models start from millions of observations, but each is uncertain by a little, and the atmosphere is chaotic. Small errors grow, doubling roughly every two days at the scales that matter for daily weather. A starting uncertainty of $0.1$ °C grows to $1.6$ °C, too large for a useful temperature forecast, after four doublings, about eight days.
Halving the starting error adds only one doubling time, two days, to the horizon, which is why forecasts have improved by about a day per decade despite enormous gains in observation and computing. Forecasters now run ensembles, dozens of runs from slightly different starts, and report probabilities: a seventy percent chance of rain is an honest statement of where the ensemble spread has reached.
Chaos is easy to see in the laboratory. A simple circuit of a resistor, an inductor and a diode, driven by a sine wave, shows the full period-doubling cascade on an oscilloscope as the drive voltage is turned up, with thresholds whose spacing gives the Feigenbaum constant within a few percent. Undergraduate labs across the United States run this experiment.
The same mathematics appears in medicine. Heart cells driven by electrical pulses can respond every beat, every other beat, or irregularly as the pulse rate rises, a period-doubling route that cardiologists see in some arrhythmias. Research groups, including ones funded by the National Institutes of Health, study whether small, carefully timed pulses can steer a chaotic rhythm back to a regular one, using the sensitivity that makes chaos unpredictable as a lever to control it.
Chaotic motion looks random: a chaotic pendulum's swings seem to follow no pattern, and its long-run statistics resemble noise. But nothing random enters its equations. Start it in exactly the same state twice and it does exactly the same thing. What makes it unpredictable is that no measurement is exact, and in a chaotic system the unavoidable error grows exponentially.
A second error is to think chaos needs a complicated system. A single pendulum with a periodic push, or one line of arithmetic, is enough. What it needs is nonlinearity, and, for a continuous system, at least three dimensions of state space, which the drive's phase supplies.
Take $r = 2.5$ and $x_0 = 0.2$. Apply the map once.
$x_1 = 2.5 \times 0.2 \times 0.8 = 0.4$
$rx(1 - x)$.
Apply it again.
$x_2 = 2.5 \times 0.4 \times 0.6 = 0.6$
The output is the next input.
Apply it a third time.
$x_3 = 2.5 \times 0.6 \times 0.4 = 0.6$
It stays put.
Find the fixed point from the formula.
$x^* = 1 - \dfrac{1}{2.5} = 0.6$
The iterates reached it exactly.
Check its stability.
$f'(x^*) = 2 - 2.5 = -0.5, \quad |f'| < 1$
Stable: nearby starts are drawn in, alternating sides.
Take $r = 3.2$. Find the fixed point.
$x^* = 1 - \dfrac{1}{3.2} = 0.6875$
$1 - 1/r$.
Find the slope there.
$f'(x^*) = 2 - 3.2 = -1.2$
Larger than one in size.
Conclude about the fixed point.
$|f'| > 1 \Rightarrow \text{unstable}$
Deviations grow by $1.2$ each step.
Write the condition for a two-cycle.
$f(f(x)) = x, \quad f(x) \ne x$
A point that returns after two steps.
Solve it for $r = 3.2$.
$x = \dfrac{r + 1 \pm \sqrt{(r + 1)(r - 3)}}{2r} = 0.513 \text{ or } 0.799$
Divide the quartic $f(f(x)) - x$ by the fixed-point factors.
Check that they map into each other.
$f(0.513) = 3.2 \times 0.513 \times 0.487 = 0.799$
A genuine two-cycle, which is stable just past $r = 3$.
A chaotic pendulum has $\lambda = 1.0$ per drive period. Two runs start $10^{-6}$ rad apart. Write the separation.
$\delta(n) = 10^{-6}e^{1.0n}$
Exponential growth with $n$ drive periods.
Set the separation equal to $1$ rad, where predictions fail.
$10^{-6}e^{n} = 1$
The size of the motion.
Solve for the number of periods.
$n = \ln 10^{6} = 6 \times 2.303 = 13.8$
Take the natural log.
Improve the start a thousandfold.
$10^{-9}e^{n} = 1 \Rightarrow n = 9 \times 2.303 = 20.7$
A thousand times better measurement.
Compare the horizons.
$20.7 - 13.8 = 6.9\ \text{periods gained}$
A thousandfold improvement buys only seven periods.
State the general rule.
$n = \dfrac{1}{\lambda}\ln\dfrac{\delta_{\max}}{\delta_0}$
Logarithmic in the starting error.
Find the fixed point.
$x^* = 1 - \dfrac{1}{2} = 0.5$
$1 - 1/r$.
Find the slope there.
$f'(x^*) = 2 - 2 = 0$
$2 - r$.
Decide the stability.
The logistic map $x_{n+1} = 2x_n(1 - x_n)$ is started at $x_0 = 0.1$. What does $x_n$ do after many steps?
Complete the worked solution: for the logistic map $x_{n+1} = 2.5x_n(1 - x_n)$, find the nonzero fixed point, the slope of the map there, and $x_1$ starting from $x_0 = 0.5$.
Subtract the reciprocal of the parameter from one.
$x^* = 1 - \dfrac{1}{r} =$ x
From $x = rx(1 - x)$ with $x \ne 0$.
Evaluate the derivative at the fixed point.
$f'(x^) = r(1 - 2x^) = 2 - r =$ d
Its size decides whether nearby points are drawn in or pushed out.
Apply the map once from one half.
$x_1 = r \times 0.5 \times 0.5 =$ q
The largest value the map can produce.
Decide the fixed point's stability from its slope.
$\text{stable when the slope lies between } -1 \text{ and } 1$
Beyond that, a two-cycle takes over: the first period doubling.
Match each idea of nonlinear dynamics to its meaning.
| $f(x^*) = x^*$ | a cycle splitting into one twice as long | rate of exponential separation | $4.669$ | |
|---|---|---|---|---|
| fixed point | ||||
| period doubling | ||||
| Lyapunov exponent | ||||
| Feigenbaum constant |
Iterate the logistic map $x_{n+1} = 4x_n(1 - x_n)$ from $x_0 = 0.7$. Fill in $x_1$, $x_2$, and the map's nonzero fixed point.
| value | |
|---|---|
| $x_1$ | |
| $x_2$ | |
| fixed point |
Two chaotic pendulums start with angles that differ by $4$ microradians, and the system's Lyapunov exponent is $2$ s⁻¹. Write their separation, in microradians, as a formula in the time $t$ in seconds, while it is still small.
Answer:
For the logistic map, the first three period doublings happen at $3.0$, $3.4495$ and $3.5441$. Using the Feigenbaum constant $\delta = 4.669$, predict where the fourth happens.
Answer: fourth doubling threshold
A forecaster at the National Weather Service treats the atmosphere as chaotic: small errors in its starting state double every $2$ days. A forecast starts with a temperature uncertainty of $0.1$ °C and is useless once the uncertainty reaches $6.4$ °C. How many days ahead is it useful?
Answer: days of useful forecast
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Two chaotic pendulums start with angles that differ by $4$ microradians, and the system's Lyapunov exponent is $1$ s⁻¹. Write their separation, in microradians, as a formula in the time $t$ in seconds, while it is still small.
Answer:
You can analyze nonlinear systems and chaos. Explain to someone why doubling the accuracy of today's weather observations does not double how far ahead we can forecast.
18. Your turn: find the fixed point of the logistic map with $r = 2$, and decide its stability., step 3
$|f'| = 0 < 1 \Rightarrow \text{stable}$
Superstable: deviations shrink very fast.