Chaotic 1D maps

Surprisingly very simple 1D maps yield good model of chaotic systems.

Sawtooth map and Bernoulli shifts

The sawtooth map is determined as
    xn+1 = 2xn (mod 1)
where x (mod 1) is the fractional part of x. In the binary number system multiplying by 2 corresponds to the left shift by one bit site and taking the fractional part corresponds to the upper bit truncation. Therefore xn+1 is the Bernoulli shift of xn
    xo = 0.01011 ...
    x1 = 0.1011 ...
    x2 = 0.011 ...
and so on... The sequence (xo , x1 ...) is called orbit of the point xo.

Symbolic dynamics and chaos

If the n-th digit after the binary point in xo is 0 (1) then xn lies in the left (right) half-interval of [0,1]. Thus for the map any orbit is determined uniquely by its (so , s1 ...) symbolic sequence σ of visits of these intervals. For a random symbolic sequence points of corresponding orbit will visit the left or right half-interval randomly. Existence of continuum of complex orbits is a sign of chaos.

For the continuous noninvertible tent map (to the left) for any xn one can find preceding xn-1 value lying in the left or in the right half-interval. Thus in this case it is possible also to make orbit for any symbolic sequence by reverse iterations of the map. In general case not all symbolic sequences are allowed. E.g. 11 subsequence is deprecated for the map in Fig.3 to the right.

Unstable orbits and Lyapunov exponent

If xo and yo have k equal first binary digits then for the sawtooth map while n < k
    yn - xn =2n (yo - xo ) = (yo - xo ) en log 2.
where Λ=log 2 is the Lyapunov exponent for the map. Thus the distance between two close orbits diverges exponentially with increasing n. This property is called sensitivity to initial conditions. It means that all periodic orbits are unstable too.

Unstable periodic orbits

For the sawtooth map an orbit with rational xo = p/q is periodic for odd q. E.g. 1/3 orbit has period 2
    1/3 → 2/3 → 1/3 ...
and 1/7 orbit has period 3
    1/7 → 2/7 → 4/7 → 1/7 ...
For even q an orbit becames periodic eventually, e.g.
    1/6 → 1/3 → 2/3 → 1/3 ...   or   1/8 → 1/4 → 1/2 → 1 → 1 ...
Therefore there is infinite (countable) set of unstable periodic orbits and these orbits are dence in [0,1].
In Fig.4 the second iteration of the map and two points of the period-2 orbit are shown. Period-2 orbit is obtaned from the symbolic sequence σ= (01) (see Appendix)
    x0 = 0.0101... = 0.(01) = 1 / 112 = 1/3,
    x1 = 0.1010... = 0.(10) = 102 / 112 = 2/3
Two more combinations x2 = 0.(00) = 0 and x3 = 0.(11) = 1 are fixed points of the map. Two period-3 orbits are
    x0 = 0.001001... = 0.(001) = 12 / 1112 = 1/7,
    x1 = 0.(010) = 2/7,     x2 = 0.(100) = 4/7
and   x3 = 0.(110) = 6/7,   x4 = 0.(101) = 5/7,   x5 = 0.(011) = 3/7.

Stretching and folding

We may consider the sawtooth map to represent two steps:
  (1) uniform stretching of the interval [0,1] to twice its original length,
  (2) left shift of its right half in original position.
The stretching property leads to exponential separation of the nearby points and hence, sensitive dependence on initial conditions. The shift property keeps the generated sequence bounded, but also causes the map to be noninvertible, since it causes two different xn points to be mapped into one xn+1 point.

Shadowing

The exponential growth of errors iterating a chaotic dynamical system implies that a computer generated trajectory for some initial condition will rapidly diverge from the true orbit due to roundoff errors, so that after a relatively short time the computer generated orbit (called the pseudo-trajectory) will have no correlation with the true orbit.
However for given xn of the pseudo-trajectory we can imagine iterating backwards to find preimage of this point. Since the map is contracting under inverse iterations, the error decays for backwards orbits, and the trajectorry remains close to the backwards iteration of the true trajectory. Existence of a true trajectory that remains close to the pseudo-trajectory is called shadowing.

Invariant densities

In physical and computer experiments we can set initial conditions only approximately. But for any finite accurancy of the initial data chaotic dynamics is predictable only up to a finite number of steps! For such "turbulent" motions a statistical description may be of more use then actual knowledge of the true orbits. Therefore we have to trace evolution of the density of representative points.

For the sawtooth map after every iteration distance between close points increases two times, thus a smooth density spreads uniformly two times too. As since all points lay in the bounded [0,1] interval, therefore we get uniform distribution of the points in the n → ∞ limit. This density is left unchanged by the sawtooth map (it is called stationary or invariant density). Note that points of an unstable periodic orbit make singular invariant density.

Ergodicity

If we take random xo = 0.a1a2a3... then for any s = 0.b1b2b3...bk we can always find somewhere in xo coincident subsequence, i.e. xn will go close to s and probability of this "crossing" does not depend on s. Thus every random orbit will go arbitrary close to any point in [0,1] and cover this interval uniformly (a funny proof based on mysterious properties of randomness :)
One can use this fact to substitute "time" average <A> by "ensemble" average (ergodicity)
    <A> = ∑n A(xn) = ∫ A(x) dx.
In general case for a chaotic map
    <A> = ∑n A(xn) = ∫ A(x) dμ = ∫ A(x) ρ(x) dx ,
where μ is invariant measure and ρ(x) is invariant density for the map.

The degenerate circle map

The degenerate circle map
    xn = xn-1+Δ (mod 1) = xo+ nΔ (mod 1)
"wraps" regularly xn points around [0,1] interval (if one joins 0 and 1 points to make a circle). For a rational Δ = p/q all orbits are periodic with period q. For an irrational Δ every orbit covers [0,1] interval uniformly and dence, therefore one can introduce constant invariant density again (to calculate averages).

Note, however that all points of the circle [0,1] are displaced by the map on the same distance Δ. Therefore the distance between two orbits is constant and density of any ensemble of points keeps its shape. We have uniform invariant density with no mixing!

Decay of correlations

Average correlation function C(m) for a sequence xk is
    C(m) = limn→∞ 1/n ∑k=1,n (xk - <x>)(xk+m - <x>) ,     <x> = limn→∞ 1/n ∑k xk .
If invariant measure for a map is known then
    C(m) = limn→∞ 1/n ∑k(xk - <x>)(f om(xk) - <x>) = ∫(x - <x>)(f om(x) - <x>) dμ
E.g. for the degenerate circle map we have
    C(m) = 1/12 - δ(1 - δ) / 2 ,
where δ = mΔ (mod 1). This correlation function oscillates with increasing of m.
For the sawtooth map correlation function is
    C(m) = 2-m / 12 .
Thus mixing leads to exponential decay of correlations for large m.

Appendix: Binary code conversion to rational fraction
Consider decimal fraction at first
    0.(3) = 0.333... = 3 (10 -1 + 10 -2 + 10 -3 + ...) = 3/[10(1 - 1/10)] = 3/9 = 1/3
(formula for the sum of geometric series is used). In a similar way for binary code 0.(101) = 1012 / 1112 = 5/7 (subscript 2 points the binary system).

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updated 12 July 2006