chaos in open hamiltonian systems - max planck …systems from atomic physics to celestial mechanics...
TRANSCRIPT
![Page 1: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/1.jpg)
Chaos in open Hamiltonian systems
Tamas Kovacs
5th Austrian Hungarian Workshop in Vienna
April 9. 2010
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 1 / 13
![Page 2: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/2.jpg)
What’s going on?
Introduction - phase space of conservative dynamics
Open Hamiltonian system
Long-lived chaotic transients in celestial mechanics
Outlook
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 2 / 13
![Page 3: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/3.jpg)
Phase space structure in conservative systems
H(I,Θ) = H0(I)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 3 / 13
![Page 4: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/4.jpg)
Phase space structure in conservative systems
H(I,Θ) = H0(I) H(I,Θ) = H0(I)+εH1(I,Θ)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 3 / 13
![Page 5: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/5.jpg)
Phase space structure in conservative systems
H(I,Θ) = H0(I) H(I,Θ) = H0(I)+εH1(I,Θ)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 3 / 13
![Page 6: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/6.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 7: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/7.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 8: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/8.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 9: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/9.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Dvorak et al. Planetary ans Space Science 46 1567 (1993)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 10: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/10.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 11: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/11.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 12: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/12.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 13: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/13.jpg)
Open systems and escape
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Kovacs, T., Erdi, B., Astron. Nach. 104, 801 (2007)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 4 / 13
![Page 14: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/14.jpg)
Transient chaos (chaotic scattering)
Hsu, G.-H., Ott, E. and Grebogi, C., Phys. Lett. A127, 199 (1988)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 5 / 13
![Page 15: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/15.jpg)
Transient chaos (chaotic scattering)
double Cantor fractal structure
Lebesgue measure is zero
relation between geometry anddynamics:
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 allthe homoclinic and heteroclinic points formed among their manifold(or the common part of the stable and unstable manifolds of all theinfinitely many periodic orbits on the saddle).
Tel, T, and Gruiz, M., Chaotic dynamics, Cambridge Univ. Press (2006)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 6 / 13
![Page 16: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/16.jpg)
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 rateτ–average escape time
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 7 / 13
![Page 17: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/17.jpg)
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 rateτ–average escape time
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 7 / 13
![Page 18: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/18.jpg)
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 rateτ–average escape time
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 7 / 13
![Page 19: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/19.jpg)
Transient chaos (chaotic scattering)
Average lifetime of chaos:
Kantz, H., and Grassberger, P., Physica D17, 75 (1985)
We are interested in non-escapingtrajectories!
N(t) ≈ N(0)e−κt
τ =1
κ
κ– escape rateτ–average escape time
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 7 / 13
![Page 20: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/20.jpg)
Examples in astrodynamics
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 8 / 13
![Page 21: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/21.jpg)
Examples in astrodynamics
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 8 / 13
![Page 22: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/22.jpg)
Examples in astrodynamics
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 8 / 13
![Page 23: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/23.jpg)
Examples in astrodynamics
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
κ = 0.126⇒ τ = 7.88 (e = 0.57)
Kovacs, T., Erdi, B., Cel. Mech. and Dyn. Astron.104, 237 (2009)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 8 / 13
![Page 24: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/24.jpg)
Examples in astrodynamics
The Sitnikov problem: H(z , z , t) = z2 −
2√z2+r(t)2
κ = 0.126⇒ τ = 7.88 (e = 0.57)
Kovacs, T., Erdi, B., Cel. Mech. and Dyn. Astron.104, 237 (2009)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 8 / 13
![Page 25: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/25.jpg)
Examples in astrodynamics
Henon-Heiles system:
H(x , y , x , y) =1
2(x2 + y2)
+1
2(x2 + y2) + x2y − 1
3y3
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 9 / 13
![Page 26: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/26.jpg)
Examples in astrodynamics
Henon-Heiles system:
H(x , y , x , y) =1
2(x2 + y2)
+1
2(x2 + y2) + x2y − 1
3y3
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 9 / 13
![Page 27: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/27.jpg)
Examples in astrodynamics
Hill’s problem:
H(Q1,Q2, Q1, Q2) =1
2(Q2
1 + Q22 + Q2
1 + Q22 )− 6(Q2
1 + Q22 )(Q2
1 − Q22 )2
Simo, C., Stuchi, T.J., Physica D140 1-32 (2000)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 10 / 13
![Page 28: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/28.jpg)
Examples in astrodynamics
Hill’s problem:
H(Q1,Q2, Q1, Q2) =1
2(Q2
1 + Q22 + Q2
1 + Q22 )− 6(Q2
1 + Q22 )(Q2
1 − Q22 )2
Simo, C., Stuchi, T.J., Physica D140 1-32 (2000) Kovacs, T., unpublished
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 10 / 13
![Page 29: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/29.jpg)
Examples in astrodynamics
Hill’s problem:
H(Q1,Q2, Q1, Q2) =1
2(Q2
1 + Q22 + Q2
1 + Q22 )− 6(Q2
1 + Q22 )(Q2
1 − Q22 )2
Kovacs, T., unpublished Kovacs, T., unpublished
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 10 / 13
![Page 30: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/30.jpg)
Outlook to atomic physics
Ionization of hydrogen atom in crossed electric and magnetic field:
H(u, v ,Pu,Pv ) =1
2(P2
u + P2v ) +
1
2(u2 + v2)− ε
2Ω(u4 − v4)
+1
4Ω2(u2 + v2)(uPv − vPu) +
1
32Ω4(u2 + v2)3
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 11 / 13
![Page 31: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/31.jpg)
Outlook to atomic physics
Ionization of hydrogen atom in crossed electric and magnetic field:
H(u, v ,Pu,Pv ) =1
2(P2
u + P2v ) +
1
2(u2 + v2)− ε
2Ω(u4 − v4)
+1
4Ω2(u2 + v2)(uPv − vPu) +
1
32Ω4(u2 + v2)3
Uzer, T., Farrelly, D., PRA 52, 2501 (1995)
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 12 / 13
![Page 32: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/32.jpg)
Outlook to atomic physics
Ionization of hydrogen atom in crossed electric and magnetic field:
H(u, v ,Pu,Pv ) =1
2(P2
u + P2v ) +
1
2(u2 + v2)− ε
2Ω(u4 − v4)
+1
4Ω2(u2 + v2)(uPv − vPu) +
1
32Ω4(u2 + v2)3
Uzer, T., Farrelly, D., PRA 52, 2501 (1995) Kovacs, T., unpublished
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 12 / 13
![Page 33: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/33.jpg)
Outlook to atomic physics
Ionization of hydrogen atom in crossed electric and magnetic field:
H(u, v ,Pu,Pv ) =1
2(P2
u + P2v ) +
1
2(u2 + v2)− ε
2Ω(u4 − v4)
+1
4Ω2(u2 + v2)(uPv − vPu) +
1
32Ω4(u2 + v2)3
Kovacs, T., unpublished Kovacs, T., unpublished
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 12 / 13
![Page 34: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/34.jpg)
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 thenon-escaped trajectories
There is a well-defined chaotic set - the chaotic saddle - in the phasespace responsible for transient chaos...
...which saddle is more unstable and more extended in size as thechaotic bands
Finite time chaotic transients may appear in suitable dynamicalsystems from atomic physics to celestial mechanics
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 13 / 13
![Page 35: Chaos in open Hamiltonian systems - Max Planck …systems from atomic physics to celestial mechanics Tam as Kov acs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010](https://reader033.vdocuments.us/reader033/viewer/2022042121/5e9c2acc8ec96e2f080f153c/html5/thumbnails/35.jpg)
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 thenon-escaped trajectories
There is a well-defined chaotic set - the chaotic saddle - in the phasespace responsible for transient chaos...
...which saddle is more unstable and more extended in size as thechaotic bands
Finite time chaotic transients may appear in suitable dynamicalsystems from atomic physics to celestial mechanics
Tamas Kovacs (MPI PKS, Dresden) Chaos in open Hamiltonian systems April 9. 2010 13 / 13