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

Τhe Butterfly Effect

The weather in a city of USA depends on the flight of a butterfly in China

Fluid dynamics

Solar flares

Solar system

Brain activityPopulation dynamics

One-dimensional maps

1or

:

( ) ( )n n

f C C

x f x x f x

x f ( x )

C

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

The Renyi map

12 mod1 S

φ2φ4φ

8φ

The map doubles the arc length

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.