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).
Contents
Previous: Chaos in simple maps
Next: Tent map updated 12 July 2006
A much more important factor in the social movement than those already mentioned was the ever-increasing influence of women. This probably stood at the lowest point to which it has ever fallen, during the classic age of Greek life and thought. In the history of Thucydides, so far as it forms a connected series of events, four times only during a period of nearly seventy years does a woman cross the scene. In each instance her apparition only lasts for a moment. In three of the four instances she is a queen or a princess, and belongs either to the half-barbarous kingdoms of northern Hellas or to wholly barbarous Thrace. In the one remaining instance208— that of the woman who helps some of the trapped Thebans to make their escape from Plataea—while her deed of mercy will live for ever, her name is for ever lost.319 But no sooner did philosophy abandon physics for ethics and religion than the importance of those subjects to women was perceived, first by Socrates, and after him by Xenophon and Plato. Women are said to have attended Plato’s lectures disguised as men. Women formed part of the circle which gathered round Epicurus in his suburban retreat. Others aspired not only to learn but to teach. Arêtê, the daughter of Aristippus, handed on the Cyrenaic doctrine to her son, the younger Aristippus. Hipparchia, the wife of Crates the Cynic, earned a place among the representatives of his school. But all these were exceptions; some of them belonged to the class of Hetaerae; and philosophy, although it might address itself to them, remained unaffected by their influence. The case was widely different in Rome, where women were far more highly honoured than in Greece;320 and even if the prominent part assigned to them in the legendary history of the city be a proof, among others, of its untrustworthiness, still that such stories should be thought worth inventing and preserving is an indirect proof of the extent to which feminine influence prevailed. With the loss of political liberty, their importance, as always happens at such a conjuncture, was considerably increased. Under a personal government there is far more scope for intrigue than where law is king; and as intriguers women are at least the209 equals of men. Moreover, they profited fully by the levelling tendencies of the age. One great service of the imperial jurisconsults was to remove some of the disabilities under which women formerly suffered. According to the old law, they were placed under male guardianship through their whole life, but this restraint was first reduced to a legal fiction by compelling the guardian to do what they wished, and at last it was entirely abolished. Their powers both of inheritance and bequest were extended; they frequently possessed immense wealth; and their wealth was sometimes expended for purposes of public munificence. Their social freedom seems to have been unlimited, and they formed combinations among themselves which probably served to increase their general influence.321 The old religions of Greece and Italy were essentially oracular. While inculcating the existence of supernatural beings, and prescribing the modes according to which such beings were to be worshipped, they paid most attention to the interpretation of the signs by which either future events in general, or the consequences of particular actions, were supposed to be divinely revealed. 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Men who had learned to study the constant sequences of Nature for themselves, and to shape their conduct according to fixed principles of prudence or of justice, either thought it irreverent to trouble the god about questions on which they were competent to form an opinion for themselves, or did not choose to place a well-considered scheme at the mercy of his possibly interested responses. That such a revolution occurred about the middle of the fifth century B.C., seems proved by the great change of tone in reference to this subject which one perceives on passing from Aeschylus to Sophocles. That anyone should question the veracity of an oracle is a supposition which never crosses the mind of the elder dramatist. A knowledge of augury counts among the greatest benefits222 conferred by Prometheus on mankind, and the Titan brings Zeus himself to terms by his acquaintance with the secrets of destiny. Sophocles, on the other hand, evidently has to deal with a sceptical generation, despising prophecies and needing to be warned of the fearful consequences brought about by neglecting their injunctions. The stranger had a pleasant, round face, with eyes that twinkled in spite of the creases around them that showed worry. No wonder he was worried, Sandy thought: having deserted the craft they had foiled in its attempt to get the gems, the man had returned from some short foray to discover his craft replaced by another. “Thanks,” Dick retorted, without smiling. When they reached him, in the dying glow of the flashlight Dick trained on a body lying in a heap, they identified the man who had been warned by his gypsy fortune teller to “look out for a hidden enemy.” He was lying at full length in the mould and leaves. "But that is sport," she answered carelessly. On the retirement of Townshend, Walpole reigned supreme and without a rival in the Cabinet. Henry Pelham was made Secretary at War; Compton Earl of Wilmington Privy Seal. He left foreign affairs chiefly to Stanhope, now Lord Harrington, and to the Duke of Newcastle, impressing on them by all means to avoid quarrels with foreign Powers, and maintain the blessings of peace. With all the faults of Walpole, this was the praise of his political system, which system, on the meeting of Parliament in the spring of 1731, was violently attacked by Wyndham and Pulteney, on the plea that we were making ruinous treaties, and sacrificing British interests, in order to benefit Hanover, the eternal millstone round the neck of England. Pulteney and Bolingbroke carried the same attack into the pages of The Craftsman, but they failed to move Walpole, or to shake his power. The English Government, instead of treating Wilkes with a dignified indifference, was weak enough to show how deeply it was touched by him, dismissed him from his commission of Colonel of the Buckinghamshire Militia, and treated Lord Temple as an abettor of his, by depriving him of the Lord-Lieutenancy of the same county, and striking his name from the list of Privy Councillors, giving the Lord-Lieutenancy to Dashwood, now Lord Le Despencer. "I tell you what I'll do," said the Deacon, after a little consideration. "I feel as if both Si and you kin stand a little more'n you had yesterday. I'll cook two to-day. We'll send a big cupful over to Capt. McGillicuddy. That'll leave us two for to-morrer. After that we'll have to trust to Providence." "Indeed you won't," said the Surgeon decisively. "You'll go straight home, and stay there until you are well. You won't be fit for duty for at least a month yet, if then. If you went out into camp now you would have a relapse, and be dead inside of a week. The country between here and Chattanooga is dotted with the graves of men who have been sent back to the front too soon." "Adone do wud that—though you sound more as if you wur in a black temper wud me than as if you pitied me." "Wot about this gal he's married?" "Don't come any further." "Davy, it 'ud be cruel of us to go and leave him." "Insolent priest!" interrupted De Boteler, "do you dare to justify what you have done? Now, by my faith, if you had with proper humility acknowledged your fault and sued for pardon—pardon you should have had. But now, you leave this castle instantly. I will teach you that De Boteler will yet be master of his own house, and his own vassals. And here I swear (and the baron of Sudley uttered an imprecation) that, for your meddling knavery, no priest or monk shall ever again abide here. If the varlets want to shrieve, they can go to the Abbey; and if they want to hear mass, a priest can come from Winchcombe. But never shall another of your meddling fraternity abide at Sudley while Roland de Boteler is its lord." "My lord," said Edith, in her defence, "this woman has sworn falsely. The medicine I gave was a sovereign remedy, if given as I ordered. Ten drops would have saved the child's life; but the contents of the phial destroyed it. The words I uttered were prayers for the life of the child. My children, and all who know me, can bear witness that I have a custom of asking His blessing upon all I take in hand. I raised my eyes towards heaven, and muttered words; but, my lord, they were words of prayer—and I looked up as I prayed, to the footstool of the Lord. 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The sawtooth map is determined as
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)