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Symmetry triangle
O(6)SU(3) SU(3)*
U(5) vibrator
rotorγ softrotor
χ
η
0
1
-√7 ⁄ 2 √7 ⁄ 2
Spherical shape
Oblate shape
Prolate shape
Interacting Boson Model 1 (IBM1)
Regularity / Chaos in IBM1
• Complete integrability at dynamical symmetries due to Cassimir invariants
• Also at O(6)-U(5) transition due to underlying O(5) symmetry
• What about the triangle interior ?
varying degree of chaos initially studied by Alhassid and
Whelan quasiregular arc2
7
2
17
integrable(regular dynamics)
Poincaré sections: integrable cases
SU(3) limit
• 2 independent integrals of motion Ii restrict the motion to surfaces of topological tori
• points lie on “circles” - sections of the tori
• torus characterised by two winding frequencies ωi2
70
x
px y
xE = Emin /2
Poincaré sections: integrable cases
• 2 independent integrals of motion Ii restrict the motion to surfaces of topological tori
• points lie on “circles” - sections of the tori
• torus characterised by two winding frequencies ωi03.0
O(6)-U(5) transition
xE = 0
px
x
y
• no integral of motion besides energy E
• points ergodically fill the accessible phase space
• tori completely destroyed
Poincaré sections: chaotic cases
36.07.0
triangle interior
y
E = Emin /2x
px
x
Poincaré sections: semiregular arc
36.07.0
• semiregular Arc found by Alhassid and Whelan [Y.Alhassid,N.Whelan, PRL 67 (1991) 816 ]
• not connected to any known dynamical symmetry – partial dynamical symmetries possible
• linear fit: 27
217
reg
distinct changes of dynamics in this region of the triangle
px
x
y
x
91.05.0
semiregular arc
Poincaré sections: semiregular arc• semiregular Arc found by Alhassid and Whelan [Y.Alhassid,N.Whelan, PRL 67 (1991) 816 ]
• not connected to any known dynamical symmetry – partial dynamical symmetries possible
• linear fit: 27
217
reg
E=0
Fractions of regular area Sreg inPoincare sections and of regular trajectories Nreg in a random sample(dashed: Nreg/Ntot, full: Sreg/Stot)
Method: Ch. Skokos, JPA: Math. Gen. 34, 10029 (2001), P. Stránský, M. Kurian, P. Cejnar, PRC 74, 014306 (2006)
semiregular arc
91.05.0
• Phase space structure of mixed regular-chaotic systems is rather complicated – periodic trajectories crucial
As the strength of perturbation to an integrable system increases, the tori start to desintegrate but nevertheless, some survive (KAM – Kolmogorov-Arnold-Moser theorem).
Rational tori (i.e. those with periodic trajectories) are the most prone to decay, leaving behind alternating chains of stable and unstable fixed points in Poincaré section (Poincaré-Birkhoff theorem).
Digression: mixed dynamics
E5
E4
E3
E2
E1
|chi|>|chireg| chi=chireg |chi|<|chireg|
Energy dependence of regularity at both sides of the semiregular Arc (eta = 0.5)
|chi|>|chireg| chi=chireg |chi|<|chireg|
E10
E9
E8
E7
E6
10 equidistant energy values Ei between Emin and Elim
Crossover of two types of regular trajectories (2a and 2b)
Seen for in the regular arc...65.0,35.0
Coexistence of two species of regular trajectories (“knees and spectacles”) sligthly above E = 0
Increasing the energy, one of them prevails..
E13 E14
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