1. 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)

2. 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 arc integrable (regular dynamics)

3. Poincaré sections: integrable cases SU(3) limit 2 independent integrals of motion I i restrict the motion to surfaces of topological tori points lie on “circles” - sections of the tori torus characterised by two winding frequencies ω i x pxpx y x E = E min /2

4. Poincaré sections: integrable cases 2 independent integrals of motion I i restrict the motion to surfaces of topological tori points lie on “circles” - sections of the tori torus characterised by two winding frequencies ω i O(6)-U(5) transition x E = 0 pxpx x y

5. no integral of motion besides energy E points ergodically fill the accessible phase space tori completely destroyed Poincaré sections: chaotic cases triangle interior y E = E min /2 x pxpx x

6. 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: distinct changes of dynamics in this region of the triangle pxpx x y x semiregular arc

7. 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: E=0 Fractions of regular area S reg in Poincare sections and of regular trajectories N reg in a random sample (dashed: N reg /N tot, full: S reg /S tot ) Method: Ch. Skokos, JPA: Math. Gen. 34, 10029 (2001), P. Stránský, M. Kurian, P. Cejnar, PRC 74, 014306 (2006) semiregular arc

8. 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

9. E5 E4 E3 E2 E1 |chi|>|chi reg | chi=chi reg |chi|<|chi reg | Energy dependence of regularity at both sides of the semiregular Arc ( eta = 0.5 ) |chi|>|chi reg | chi=chi reg |chi|<|chi reg | E10 E9 E8 E7 E6 10 equidistant energy values E i between E min and E lim

10. Crossover of two types of regular trajectories (2 a and 2 b ) Seen for in the regular arc... Coexistence of two species of regular trajectories (“knees and spectacles”) sligthly above E = 0 Increasing the energy, one of them prevails.. E 13 E 14

11. 0 + states of 40 bosons along the Arcs with k=1..5 by Stefan Heinze Quantum features: Level Bunching in the semiregular Arc η = 0.35 η = 0.5 η = 0.65 Cosine of action S along the primitive orbits of types 1, 2 a, 2 b. The shaded region corresponds to the “gap” in the spectrum at k=3.