Sensitivity and Unpredictability in Chaos Theory: A Study in Dynamics

1. CHΑΟSand (un-) predictabilitySimos IchtiaroglouSection of Astrophysics, Astronomy and MechanicsPhysics DepartmentUniversity of Thessaloniki

3. CHAOS Sensitivity in very small variations of the initial state There is no possibility for predictions after a certain finite time interval

4. Τhe Butterfly Effect The weather in a city of USA depends on the flight of a butterfly in China

5. Solar system Fluid dynamics Population dynamics Solar flares Brain activity

6. f ( x ) x C One-dimensional maps

7. Orbitofpointx Fixed pointoff Periodic orbitof periodk

8. The mapfistopologically transitivein thecompact invariant setS if foranyintervals there is ann such that The set Sis an invariant set of f if The mapfhassensitive dependence on initial conditionsonthe invariant setSif there is a δ > 0such that for any pointxand every intervalUofxthere is another point x΄  Uandn Zsuch that

9. Properties of chaos The map fischaoticonthe compact invariant set Sif • There is a dense set of periodic points • It istopologically transitive • It hassensitive dependence on the initial conditions The definition has been given byDevaney (1989) These three properties are not independent. The third property can be proven by the first two, seeBanks et al. (1992), Glasner & Weiss (1993)

10. Dense set of periodic points: These points are all unstableand act as repellors Topological transitivity: Relates to theergodicity of the map Sensitive dependence on the initial conditions: Relates to the unpredictability after a definite time interval

11. 0 4x x 4x 2x x 2x Counterexamples 1. The map isordered. Fixed point: x = 0 All initial conditions tend to infinity. An initial uncertainty Δx0increases exponentially but there is nomixingof states.

12. 2. All points in the map areperiodic. Ifα= p/q then There is no topological transitivity or sensitive dependence on initial conditions

13. 3. In the map ebery orbit is denseinS1. and the sequence of points divides the circle in arcs of length less than ε The map is topologically transitive but has no periodic points nor sensitive dependence on initial conditions

14. 4φ 8φ 2φ φ TheRenyi map The map doubles the arc length

15. It isirreversibleand every point has two preimages, e.g. pointφ1=0 has the preimages φ0=0 andφ0=1/2 Since every point corresponds to the binary expression where

16. The map shifts the decimal point one place to the right and drops the integer part. If then The preimage ofφis

17. ... ή . 0 1 0 0 1 1 1 0 ... The values ofφcorrespond 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. .100000…. =.0111111….

18. Orbits of the map 1.The point φ0=.00000…. ή φ0=.111111…. is afixed point 2.Rationals of the formφ = (2m+1)/2kor end up at pointφ0after a finite number of iterations

19. 3.Rationals represented byperiodic sequencesofkdigits correspond toperiodic orbits with periodk 4.Rationals ending up to aperiodic orbitafter a finite number of iterations, e.g. 5.Non-periodic sequencescorrespond to irrationals, so thenon-periodic orbitsoffarenon-countable

20. We define the distance as follows or dcorresponds to the length of the smaller arc (φ,φ΄).If φ΄belongs to theε – neighborhood ofφ.

21. Letbe sufficiently large so that If the binary representations ofφ,φ΄are identical in their firstkdigits, i.e. φ΄belongs to theε – neighborhood ofφ since

22. TheRenyi map has a dense set of periodic points If we consider the ε – neighborhoodUof point withksufficiently large so that then the point is aperiodic pointand moreover

23. TheRenyi map is topologically transitive Consider theε – neighborhoodUof point and theε΄– neighborhood Vof point Τhen so that

24. TheRenyi map has sensitive dependence on the initial conditions Consider the point In everyε – neighborhoodUofφ belongallpoints with binary representation Consider the point with

25. Obviouslyso that Moreover so that .00... .01... .10... .11...

26. Chaos and differential equations Consider the system and define thePoincarè map Theasymptotic manifolds of a hyperbolic fixed point intersect generically transversally and transverse homoclinic points appear

29. Smale’s theorem • There is a suitably defined compact invariant set in the neighborhood of the hyperbolic fixed pointwhere Poincarè map ischaotic, i.e. it possesses: • A dense set of periodic points • Topological transitivity • Sensitive dependence on the initial conditions

30. 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. Conclusions