Geometric Phase Bifurcation 
(taken as a model to simulate nuclear instability)

 Bernd Binder, 29.8.2002, 18.11.2006 (formula and text added), 16.9.2007 MAP  Z = Mg

Magic Angle Precession "MAP" is based on
  
      cos(JZpa) = Ma .

The MAP chaotic precession angle q =pa can be found from a recursive iteration subject to a variable quantum spin unit J and charge/curvature Z showing critical bifurcation points (at Feigenbaum distance). The underlying chaotic relation has been taken to model the fine-structure spin-orbit coupling (JZ=1, two charges with J=0.5 and M=137, where a=1/137.0360094..). Here we model/simulate nuclear instability at high charge densities. In the bifurcation diagram below the "charge" Z is the horizontal variable running from 1 to M, the precession angle q =pa is the vertical variable (range 0....p). At low charge values (conic rotation frequency, explanation below) the dynamics is stabile, but after the first bifurcation point there are two precession angles (chaotic attractor). Bifurcation are shown sometimes by a blue vertical line with Z -value displayed as the third number. The precession quantum number M (slider on the left) and quantum spin unit J (slider on the right in 0.5 steps) can be changed. The most interesting case is at: 
M=137, J/a=137.0360094.., J=0.5, with first bifurcation (coupling instability) near "critical charge" Z=115. Press "logistic" to see the logistic bifurcation.

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Background: precession and spin-orbit coupling can be modeled by rolling cones, where nonlinear dynamics is induced by curvature "contracting" both, precession and orbital motion (Berry phase, Thomas precession, gyroscopic magic angle precession). The curvature contraction (angular defect) is multiplied by "charge" Z (multiplying both curvature and cone angular velocity) and quantum spin unit J. In flat space the precession angle would be q =p/M. But due to spin and "charge" providing for curvature affecting ("contracting") both, orbital and precession (azimuthal) angular ranges we have instabilites due to "charge overload".

More can be found in Geometric Phase Locked in the Nucleus (pdf 279kb, 26.8.2002),

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(c) Bernd Binder 2002-2008