CHΑΟSand (un-) predictability
Simos IchtiaroglouSection of Astrophysics, Astronomy and Mechanics
Physics Department
University of Thessaloniki
CHAOSSensitivity in very small variations of the initial state
There is no possibility for predictions after a certain finite time interval
Fixed point of f
Orbit of point x
2 1 2, ( ), , ( ), ( ) , ( ), ( ), , ( ),, kk x f x ff x f x xx xf f
0 0( )x f x
Periodic orbit of period k
0 0( )kkx f x x
The set S is an invariant set of f if
( )kx S f x S k
The map f is topologically transitive in the compact invariant
set S if for any intervals ,U V S there is an n such that
( )nf U V
The map f has sensitive dependence on initial conditions on the
invariant set S if there is a δ > 0 such that for any point x and
every interval U of x there is another point x΄ U and n Z such that
( ( ), ( ))n nd f x f x
Properties of chaos
The map f is chaotic on the compact invariant set S if
There is a dense set of periodic points
It is topologically transitive
It has sensitive dependence on the initial conditions
The definition has been given by Devaney (1989)
These three properties are not independent. The third property can be proven by the first two, see Banks et al. (1992), Glasner & Weiss (1993)
Dense set of periodic points: These points are all unstable and act as repellors
Topological transitivity: Relates to the ergodicity of the map
Sensitive dependence on the initial conditions: Relates to the unpredictability after a definite time interval
Counterexamples
2x x x 1. The map
is ordered. Fixed point: x = 0
All initial conditions tend to infinity. An initial uncertainty Δx0
increases exponentially
02kkx x
0
x 2x 4x x 2x 4x
but there is no mixing of states.
2. All points in the map
1mod1 ,a S a
( ) mod1qf qa p
are periodic.
If α = p/q then
There is no topological transitivity or sensitive dependence on initial conditions
( ) ( )f f
0 : | ( ) | mod1kk f
3. In the map
1mod1 , \a S a
ebery orbit is dense in S1.
The map is topologically transitive but has no periodic points nor sensitive dependence on initial conditions
and the sequence of points
2( ), ( ), , ( ),k k mkf f f
divides the circle in arcs of length less than ε
It is irreversible and every point has two preimages, e.g. point φ1=0 has the preimages φ0=0 and φ0=1/2
Since 0 1
every point corresponds to the binary expression
1 2 3. ks s s s where
{0,1}is
1 2 3. ks s s s
The map shifts the decimal point one place to the right and drops the integer part. If
then
2 3( ) 2 . kf s s s
The preimage of φ is
1 2 3 1 2 31 0. .k ks s s s s s s s or
...
. 0 1 0 0 1 1 1 0 ...
ή
The values of φ correspond to all possible infinite sequences of two symbols.
The correspondence is 1-1 with the exception of the rationals of the form
2 1
2k
m
since e.g. .100000…. =.0111111….
Orbits of the map
1 2 3
1 2 3
. 0000
. 1111k
k
s s s s
s s s s
1. The point
φ0 =.00000…. ή φ0 =.111111….
is a fixed point
2. Rationals of the form φ = (2m+1)/2k or
end up at point φ0 after a finite number of iterations
0
0
( ) .0000
( ) .1111
k
k
f
f
3. Rationals represented by periodic sequences of k digits
correspond to periodic orbits with period k
1 2 1 2 1 2
1 2 1 2
.
( ) .
k k k
kk k
s s s s s s s s s
f s s s s s s
4. Rationals ending up to a periodic orbit after a finite number of iterations, e.g.
1 2 1 2
1 2
1 2
1 2
.
( ) .
m m m k m m m k
mm m m m m k
m
k m
s s s s s s
f s s
s
s
s
s s
s
s
5. Non-periodic sequences correspond to irrationals, so the non-
periodic orbits of f are non-countable
We define the distance
1 2 3 1 2 3. , .k ks s s s s s s s
as follows
1
( , )2
k kk
k
s sd
1/ 2 if
or1
( , )2
d 1/ 2 if
d corresponds to the length of the smaller arc (φ,φ΄). If
( , )d
φ΄ belongs to the ε – neighborhood of φ.
Let be sufficiently large so thatk 2 k
If the binary representations of φ,φ΄ are identical in their first k digits, i.e.
0i is s i k
φ΄ belongs to the ε – neighborhood of φ since
( , ) 2 kd
The Renyi map has a dense set of periodic points
1 2 3 1. k ks s s s s
2 k
U
If we consider the ε – neighborhood U of point
with k sufficiently large so that
then the point
1 2 1 2 1 2. k k ks s s s s s s s s
is a periodic point and moreover
The Renyi map is topologically transitive
Consider the ε – neighborhood U of point
1 2 3 1. , : 2 kk ks s s s s k
and the ε΄– neighborhood V of point
1 2 3 1. , : 2 mm ms s s s s m
Τhen
1 2 3 1 2 3
1 2 3
.
( ) .
k m
km
s s s s s s s s U
f s s s s V
so that
( )kf U V
The Renyi map has sensitive dependence on the initial conditions
Consider the point
1 21 2 3 3. k kk kss s s sss
In every ε – neighborhood U of φ belong all points with binary representation
1 2. 2 kks s s where
Consider the point
1 12 323. k kk kss ss s s s with
1 1k ks s
( , ) 2 kd Obviously so that
UMoreover
1 2 13 32( ) . , ( ) .k kkkk k kks sf f ss ss
1( ( ), ( ))
4k kd f f
so that .00....01...
.10... .11...
Chaos and differential equations
Consider the system
( , ), ( , ) ( , )x f x t f x t T f x t
and define the Poincarè map
2 2( ) (( 1) )
( ) (( 1) )
x kT x k T
x kT x k T
The asymptotic manifolds of a hyperbolic fixed point intersect generically transversally and transverse homoclinic points appear
Smale’s theorem
There is a suitably defined compact invariant set in the neighborhood of the hyperbolic fixed point where Poincarè map is chaotic, i.e. it possesses:
• A dense set of periodic points• Topological transitivity• Sensitive dependence on the initial conditions
Conclusions
• 1. Chaos is a well defined property and its main characteristic is the sensitive dependence of the final state on the initial one, so that prediction for arbitrarily large time intervals is impossible.
• 2. Almost all systems are chaotic. Complexity is not necessary for the appearance of chaotic dynamics, which may appear in very simple systems.
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