Dynamical Systems

A dynamical system is a set of states S together with an iterator function f:S\rightarrow S which is used to determine the next state of a system in terms of the previous state. For example, if x_0\in S is the initial state, the subsequent states are given by:

x_1=f(x_0),

x_2=f(x_1)=f(f(x_0))=(f\circ f)(x_0)=:f^2(x_0)

x_3=f(x_2)=f(f^2(x_0))=f^3(x_0),

and in general, the next state is got by applying the iterator function:

x_{i}=f(x_{i-1})=f^i(x_0).

The sequence of states

\{x_0,x_1,x_2,\dots\}

is known as the orbit of x_0 and the x_i are known as the iterates.

Such dynamical systems are completely deterministic: if you know the state at any time you know it at all subsequent times. Also, if a state is repeated, for example:

\text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_4=x_2,x_5\dots,\}

then the orbit is destined to repeated forever because

x_5=f(x_4)=f(x_2)=x_3,

x_6=f(x_5)=f(x_3)=x_4=x_2, etc:

\Rightarrow \text{orb}(x_0)=\{x_0,x_1,x_2,x_3,x_2,x_3,x_2,\dots\}

Example: Savings

Suppose you save in a bank, where monthly you receive 0.1\%=0.001 interest and you throw in 50 per month, starting on the day you open the account.

This can be modeled as a dynamical system.

Let S=\mathbb{R} be the set of euro amounts. The initial amount of savings is x_0=50. After one month you get interest on this: 0.001\times50, you still have your original 50 and you are depositing a further €50, so the state of your savings, after one month, is given by:

x_1=50+0.001\times 50+50=(1+0.001)50+50.

Now, in the second month, there is interest on all this:

interest in second month 0.001\times((1+0.001)50+50)=0.001x_1,

we also have the x_1=(1+0.001)50+50 from the previous month and we are throwing in an extra €50 so now the state of your savings, after two months, is:

x_2=x_1+0.001x_1+50=(1+0.001)x_1+50,

and it shouldn’t be too difficult to see that how you get from x_i\longrightarrow x_{i+1} is by applying the function:

f(x)=(1+0.001)x+50.

Exercise

Use geometric series to find a formula for x_n.

Weather

If quantum effects are neglected, then weather is a deterministic system. This means that if we know the exact state of the weather at a certain instant (we can even think of the state of the universe – variations in the sun affecting the weather, etc), then we can calculate the state of the weather at all subsequent times.

This means that if we know everything about the state of the weather today at 12 noon, then we know what the weather will be at 12 noon tomorrow…

However this isn’t what we tend to experience… instead the weather seems to be very unpredictable.

However this is directly at odds with the contention that the weather is deterministic, which is an assumption underlying how we forecast weather. How do we explain this apparent contradiction?

Sensitivity to Initial Conditions

The first thing we need to understand is sensitivity to initial conditions, more commonly known as Butterfly Effect. What this says is that two initial states, arbitrarily close together, can have very different orbits, and their iterates diverge: that is we can have two initial states x_0\approx y_0 that are very close together but for a large enough n, x_n is very different to y_n.

To explain this, suppose for arguments sake that the temperature at a particular point on earth at 12 noon depends only on the temperature at 12 noon on the previous day. The temperature can be modeled as a dynamical system, with some iterator function giving the temperature now in terms of the temperature on the previous day:

T_i=f(T_{i-1}).

Here T_i is the temperature at 12 noon some i days after some given day. If the system displays sensitivity to initial conditions, then two different initial states, say T_0=15.5 and \tilde{T}_0=15.6, can, after a number of days, display wildly different behaviour, looking something like, for example:

graph15

Here we orbits of T_0=15.5 (red) and \tilde{T}_0=15.6 (green) in the presence of sensitivity to initial conditions. The behaviour is quite similar up about ten days, with very little between T_n and \tilde{T}_n. After this however, the iterates diverge.

Now this isn’t necessarily a big deal. We can still predict the temperature on subsequent days if we know T_0=15.5… or should I say T_0=15.500000\cdots…

Where do we get T_0 but from a measurement… and a measurement always comes with an error. For example, if the instrument used to measure T_0 is only accurate to the nearest decimal place, then when we measure T_0 and get T_0=15.5 in fact T_0 could be anything from 15.45 to 15.55:

15.45 \longleftarrow T_0\longrightarrow 15.55

This measurement challenge is always present for real-life weather forecasters… and sensitivity to initial conditions means we have the following principle:

Any limitation in measuring the weather conditions translates into a limitation of weather forecasting.

Perhaps at a later stage we will describe how real weather forecasters might get around this to a degree.

Chaotic Systems

Systems that display sensitivity to initial conditions are inherently difficult to predict (with the presence or rather inevitability of measurement error).

Topological Mixing

Another way that a system can be ‘chaotic’ is if orbits avoid any periodic pattern. For example, look at this plot of the price of IBM shares:

graph16

There doesn’t seem to be any rhyme nor reason: it just looks like… chaos. As written above, an aspect of dynamical systems:

initial state x_0, find next state by applying iterator function x_{i+1}=f(x_i),

is that if a state is ever repeated, then the system will fall into periodic behaviour. If we want a really chaotic orbit, that never repeats itself, then we must have a system with an orbit that never visits the same state twice.

If we have a chaotic orbit (never repeats) that actually gets close to every possible state, then we say we have a dense orbit. A periodic orbit can never be dense: it only contains finitely many distinct states (because it repeats itself) and so cannot get arbitrarily close to every single possible state.

In fact, we can go further and ask that most initial states are chaotic and have dense orbit. This is called topological mixing (basically most orbits never repeat and are… ‘all over the place’).

Density of Periodic Points

In the presence of sensitivity to initial conditions AND topological mixing (‘mad’ lookin’ orbits) there is one more thing that makes a system even harder to predict.

Sensitivity to initial conditions give quantitative differences between two initially close orbits. Quantitative because there is a number that describes that difference:

|x_n-y_n|.

In truly chaotic systems we want to add some qualitative differences too. We want it to be possible for the orbits of two close initial states x_0 and y_0 to have very different behaviour… for example,

\text{orb}(x_0) (eventually) periodic

\text{orb}(y_0) never periodic (‘chaotic’ and dense)

This is the final signifier of chaos… sort of like mega-sensitivity to initial conditions. For example, starting at a period-three temperature T_0\approx 19.0097 we get into a lovely periodic pattern, but starting at \tilde{T}_0=19 we get into a ‘chaotic’ and dense orbit:

graph16

There is an initial state T_0\approx 19.0097 whose orbit is periodic, repeating every three iterations, and never getting close to e.g. zero. In contrast, \tilde{T}_0=19 displays no such periodic behaviour and probably has a chaotic, dense orbit, eventually getting close to any possible temperature.

A system that exhibits all three of the following (or maybe just two: there are various definitions) is said to be a chaotic system:

  • Sensitivity to Initial Conditions
  • Topological Mixing
  • Density of Periodic Points

A double pendulum is an example of a chaotic system.

Chaos can exist in very simple systems. Here we show it is present in the following, very simple system.

Doubling Mapping

Let the set of states be given by [0,1) — the real numbers between zero and one, and consider the function D:[0,1)\rightarrow [0,1) given by:

D(x)=\begin{cases}2x & \text{ if }x< \frac12 \\ 2x-1 &\text{ if }x\geq \frac12 \end{cases}.

For example, as \frac13< \frac12:

D(1/3)=2(1/3)=2/3,

and as 0.8\geq 1/2:

D(0.8)=2(0.8)-1=1.6-1=0.6.

Model: Doubling Angle

Very similar to this, we have the dynamical system where the set of states is the set of points on the unit circle (given by the angle \theta made with the positive x-axis, so S=[0,2\pi))

D(\theta)=\begin{cases}2\theta & \text{ if }\theta< \pi=180^\circ \\ 2\theta-2\pi &\text{ if }\theta\geq \pi \end{cases}.

That is you get from one point to the next by doubling the angle… if you go over the 360^\circ=2\pi you start a new rotation, e.g.

2(200^\circ)=400^\circ \equiv 40^\circ.

Start at a given angle and keep doubling the angle basically.

Getting back to the Doubling Mapping (not points on a circle), we will show that this system is chaotic. First let us represent the states slightly differently.

Binary

We represent numbers between zero and one by decimals. How this works is as follows. Take the number 0.273. This is

\displaystyle 0.273=2\times 0.1+7\times 0.01+3\times 0.001=\frac{2}{10^1}+\frac{7}{10^2}+\frac{3}{10^3}.

This is the decimal, or base-10, expansion.

We can also consider the binary, or base-two expansion. Note that all decimal digits are between zero and nine (=10-1). Similarly binary digits are between zero and one (=2-1).

So for example, consider the number between zero and one given in binary by 0.01011. We usually write 0.01011_2 to signify this is written in binary. Similarly to the decimal expansion above, this number represents

\displaystyle 0.01011_2=\frac{0}{2^1}+\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}=\frac{11}{32}.

The thing about the Doubling Mapping is that it is very to see what it does to a state when the state is written in binary.

Note first of all that any x< 1/2, written in binary, has a first binary digit of 0. So for example, all of the below are less than 1/2:

0.001,0.011,0.000001,0.0111.

Exercise

Use geometric series to show that:

0.111111\dots_2=1.

0.0111111\dots_2=\frac12.

Now take any state x<1/2, for example x_0=0.01011_2. As x_0< 1/2, D(x_0)=2x_0:

\displaystyle D(x_0)=2(0.01011_2)

\displaystyle =2\left(\frac{0}{2^1}+\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}+\frac{1}{2^5}\right),

noting that:

2\times \frac{a_i}{2^i}=\frac{a_i}{2^{i-1}}, and 0/2^i=0,

\displaystyle D(x_0)=\frac{1}{2^1}+\frac{0}{2^2}+\frac{1}{2^3}+\frac{1}{2^4}=0.1011_2,

that is

D(0.01011_2)=0.1011_2,

so all the iterator function D did was ‘chop’ off the first binary digit.

Exercise

Show that where a_i\in\{0,1\}

D(0.0a_2a_3a_4\dots_2)=0.a_2a_3a_4\dots.

Any x_0\geq 1/2 is of the form x_0=\frac{1}{2}+\underbrace{\varepsilon}_{\leq 1/2} so must, in binary, be of the form:

x_0=0.1_2+0.0a_2a_3\dots_2=0.1a_2a_3\dots,

that is the first binary digit must be one (with the exception of 0.0111\dots_2=0.1_2. Take say x_0=0.1101_2 and calculate D(x_0). Note in this region, x\geq 1/2, D(x)=2x-1:

\displaystyle D(x_0)=2(0.1101_2)-1=2\left(\frac12 +\frac{1}{2^2}+\frac{0}{2^3}+\frac{1}{2^4}\right)-1

\displaystyle =1+\frac12+\frac{0}{2^2}+\frac{1}{2^3}-1

\displaystyle =\frac{1}{2^1}+\frac{0}{2^2}+\frac{1}{2^3}=0.101_2,

so that

D(0.1101_2)=0.101_2,

so that, again, the iterator function D did was ‘chop’ off the first binary digit.

Exercises

Show that where a_i\in\{0,1\}

D(0.1a_2a_3a_4\dots_2)=0.a_2a_3a_4\dots.

Putting the two exercises together we see that:

D(0.a_1a_2a_3a_4\dots_2)=0.a_2a_3a_4\dots_2.

Sensitivity to Initial Conditions

Consider the state:

x_0=0.\overbrace{111\cdots 11}^{20 \text{ binary digits}}00_2

Consider also

x_0=0.\overbrace{1111\cdots 11}^{20 \text{ binary digits}}01_2.

Now the difference between these two is very small:

\displaystyle y_0-x_0=\frac{1}{2^{22}}\approx 10^{-7}=0.0000001.

Now apply the doubling mapping to both 20 times (i.e. chop off the first twenty ones:

D^{20}(x_0)=0 and D^{20}(y_0)=0.01_2=\frac14

D^{21}(x_0)=0 and D^{21}(y_0)=0.1_2=\frac12

This means that initially the distance between the states is small, \approx 10^{-7}, but after 21 iterations of D, the orbits are no longer close together:

graph18

The initial states x_0 and y_0 are close together but their orbits diverge. After twenty iterations, the distance between the orbits is 0.5 (whereas it was once \approx 10^{-7}.

Problems for Friday

  • Come up with a seed x_0 such which shows topological mixing (i.e. its orbit gets close to every state x\in [0,1]
  • Show that close to every point there is a periodic point.

If we can solve these two problems we have shown that the dynamical system ([0,1),D) is chaotic.

 

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