CHΑΟS and (un-) predictability Simos Ichtiaroglou Section of Astrophysics, Astronomy and Mechanics Physics Department University of Thessaloniki

CHAOS Sensitivity 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 activity Population dynamics

One-dimensional maps x f ( x ) C

Fixed point of f Orbit of point x Periodic orbit of period k

The set S is an invariant set of f if The map f is topologically transitive in the compact invariant set S if for any intervals there is an n such that 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

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 1. The map is ordered. Fixed point: x = 0 All initial conditions tend to infinity. An initial uncertainty Δx 0 increases exponentially 0 x 2x 4x x 2x 4x but there is no mixing of states.

2. All points in the map are periodic. If α = p/q then There is no topological transitivity or sensitive dependence on initial conditions

3. In the map ebery orbit is dense in S 1. The map is topologically transitive but has no periodic points nor sensitive dependence on initial conditions and the sequence of points divides the circle in arcs of length less than ε

The Renyi map φ 2φ 4φ 8φ The map doubles the arc length

It is irreversible and every point has two preimages, e.g. point φ 1 =0 has the preimages φ 0 =0 and φ 0 =1/2 Since every point corresponds to the binary expression where

The map shifts the decimal point one place to the right and drops the integer part. If then The preimage of φ is

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 since e.g …. = ….

Orbits of the map 1. The point φ 0 = …. ή φ 0 = …. is a fixed point 2. Rationals of the form φ = (2m+1)/2 k or end up at point φ 0 after a finite number of iterations

3. Rationals represented by periodic sequences of k digits correspond to periodic orbits with period k 4. Rationals ending up to a periodic orbit after a finite number of iterations, e.g. 5. Non-periodic sequences correspond to irrationals, so the non- periodic orbits of f are non-countable

We define the distance as follows or d corresponds to the length of the smaller arc (φ,φ΄). If φ ΄ belongs to the ε – neighborhood of φ.

Let be sufficiently large so that If the binary representations of φ,φ ΄ are identical in their first k digits, i.e. φ ΄ belongs to the ε – neighborhood of φ since

The Renyi map has a dense set of periodic points If we consider the ε – neighborhood U of point with k sufficiently large so that then the point is a periodic point and moreover

The Renyi map is topologically transitive Consider the ε – neighborhood U of point and the ε΄ – neighborhood V of point Τhen so that

The Renyi map has sensitive dependence on the initial conditions Consider the point In every ε – neighborhood U of φ belong all points with binary representation Consider the point with

Obviously so that Moreover so that

Chaos and differential equations Consider the system and define the Poincarè map 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.