Chaos in open Hamiltonian systems
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1 Chaos in open Hamiltonian systems Tamás Kovács 5th Austrian Hungarian Workshop in Vienna April Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
2 What s going on? Introduction - phase space of conservative dynamics Open Hamiltonian system Long-lived chaotic transients in celestial mechanics Outlook Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
3 Phase space structure in conservative systems H(I, Θ) = H 0 (I) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
4 Phase space structure in conservative systems H(I, Θ) = H 0 (I) H(I, Θ) = H 0 (I)+ɛH 1 (I, Θ) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
5 Phase space structure in conservative systems H(I, Θ) = H 0 (I) H(I, Θ) = H 0 (I)+ɛH 1 (I, Θ) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
6 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
7 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
8 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
9 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Dvorak et al. Planetary ans Space Science (1993) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
10 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
11 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
12 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
13 Open systems and escape The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Kovács, T., Érdi, B., Astron. Nach. 104, 801 (2007) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
14 Transient chaos (chaotic scattering) Hsu, G.-H., Ott, E. and Grebogi, C., Phys. Lett. A127, 199 (1988) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
15 Transient chaos (chaotic scattering) double Cantor fractal structure Lebesgue measure is zero relation between geometry and dynamics: ( D = 2 1 κ ) λ Ott, E., Chaos in dynamical systemcs, Cambridge Univ. Press (1993) Kantz, H., and Grassberger, P., Physica D17, 75 (1985) The union of all the hyperbolic periodic orbits on the saddle and of all the homoclinic and heteroclinic points formed among their manifold (or the common part of the stable and unstable manifolds of all the infinitely many periodic orbits on the saddle). Tél, T, and Gruiz, M., Chaotic dynamics, Cambridge Univ. Press (2006) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
16 Transient chaos (chaotic scattering) Average lifetime of chaos: We are interested in non-escaping trajectories! N(t) N(0)e κt τ = 1 κ κ escape rate τ average escape time Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
17 Transient chaos (chaotic scattering) Average lifetime of chaos: We are interested in non-escaping trajectories! N(t) N(0)e κt τ = 1 κ κ escape rate τ average escape time Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
18 Transient chaos (chaotic scattering) Average lifetime of chaos: We are interested in non-escaping trajectories! N(t) N(0)e κt τ = 1 κ κ escape rate τ average escape time Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
19 Transient chaos (chaotic scattering) Average lifetime of chaos: We are interested in non-escaping trajectories! N(t) N(0)e κt τ = 1 κ Kantz, H., and Grassberger, P., Physica D17, 75 (1985) κ escape rate τ average escape time Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
20 Examples in astrodynamics The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
21 Examples in astrodynamics The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
22 Examples in astrodynamics The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
23 Examples in astrodynamics The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 κ = τ = 7.88 (e = 0.57) Kovács, T., Érdi, B., Cel. Mech. and Dyn. Astron. 104, 237 (2009) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
24 Examples in astrodynamics The Sitnikov problem: H(z, ż, t) = ż 2 2 z 2 +r(t) 2 κ = τ = 7.88 (e = 0.57) Kovács, T., Érdi, B., Cel. Mech. and Dyn. Astron. 104, 237 (2009) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
25 Examples in astrodynamics Hénon-Heiles system: H(x, y, ẋ, ẏ) = 1 2 (ẋ 2 + ẏ 2 ) (x 2 + y 2 ) + x 2 y 1 3 y 3 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
26 Examples in astrodynamics Hénon-Heiles system: H(x, y, ẋ, ẏ) = 1 2 (ẋ 2 + ẏ 2 ) (x 2 + y 2 ) + x 2 y 1 3 y 3 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
27 Examples in astrodynamics Hill s problem: H(Q 1, Q 2, Q 1, Q 2 ) = 1 2 ( Q Q Q2 1 + Q 2 2) 6(Q Q 2 2)(Q 2 1 Q 2 2) 2 Simó, C., Stuchi, T.J., Physica D (2000) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
28 Examples in astrodynamics Hill s problem: H(Q 1, Q 2, Q 1, Q 2 ) = 1 2 ( Q Q Q2 1 + Q 2 2) 6(Q Q 2 2)(Q 2 1 Q 2 2) 2 Simó, C., Stuchi, T.J., Physica D (2000) Kovács, T., unpublished Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
29 Examples in astrodynamics Hill s problem: H(Q 1, Q 2, Q 1, Q 2 ) = 1 2 ( Q Q Q2 1 + Q 2 2) 6(Q Q 2 2)(Q 2 1 Q 2 2) 2 Kovács, T., unpublished Kovács, T., unpublished Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
30 Outlook to atomic physics Ionization of hydrogen atom in crossed electric and magnetic field: H(u, v, P u, P v ) = 1 2 (P2 u + P 2 v ) (u2 + v 2 ) ɛ 2Ω (u4 v 4 ) + 1 4Ω 2 (u2 + v 2 )(up v vp u ) Ω 4 (u2 + v 2 ) 3 Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
31 Outlook to atomic physics Ionization of hydrogen atom in crossed electric and magnetic field: H(u, v, P u, P v ) = 1 2 (P2 u + P 2 v ) (u2 + v 2 ) ɛ 2Ω (u4 v 4 ) + 1 4Ω 2 (u2 + v 2 )(up v vp u ) Ω 4 (u2 + v 2 ) 3 Uzer, T., Farrelly, D., PRA 52, 2501 (1995) Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
32 Outlook to atomic physics Ionization of hydrogen atom in crossed electric and magnetic field: H(u, v, P u, P v ) = 1 2 (P2 u + P 2 v ) (u2 + v 2 ) ɛ 2Ω (u4 v 4 ) + 1 4Ω 2 (u2 + v 2 )(up v vp u ) Ω 4 (u2 + v 2 ) 3 Uzer, T., Farrelly, D., PRA 52, 2501 (1995) Kovács, T., unpublished Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
33 Outlook to atomic physics Ionization of hydrogen atom in crossed electric and magnetic field: H(u, v, P u, P v ) = 1 2 (P2 u + P 2 v ) (u2 + v 2 ) ɛ 2Ω (u4 v 4 ) + 1 4Ω 2 (u2 + v 2 )(up v vp u ) Ω 4 (u2 + v 2 ) 3 Kovács, T., unpublished Kovács, T., unpublished Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
34 What have we learned? Beside the permanent chaos exist a finite time chaotic behavior - chaotic scattering - in open Hamiltonian systems The main concept is the escape rate i.e. the exponencial decay of the non-escaped trajectories There is a well-defined chaotic set - the chaotic saddle - in the phase space responsible for transient chaos......which saddle is more unstable and more extended in size as the chaotic bands Finite time chaotic transients may appear in suitable dynamical systems from atomic physics to celestial mechanics Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
35 What have we learned? Beside the permanent chaos exist a finite time chaotic behavior - chaotic scattering - in open Hamiltonian systems The main concept is the escape rate i.e. the exponencial decay of the non-escaped trajectories There is a well-defined chaotic set - the chaotic saddle - in the phase space responsible for transient chaos......which saddle is more unstable and more extended in size as the chaotic bands Finite time chaotic transients may appear in suitable dynamical systems from atomic physics to celestial mechanics Tamás Kovács (MPI PKS, Dresden) Chaos in open Hamiltonian systems April / 13
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