NuclearQuadrupole

Systems / ODEs / Exotic systems

A two-degree-of-freedom Hamiltonian for collective quadrupole vibrations of the nuclear surface, with a mixed regular–chaotic phase space that grows chaotic with energy.

continuous · ODE4 dimensionsconservative

projection (q1, q2, p1)

projection (q1, q2, p2)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{q1} &= a p_{1} \\ \dot{q2} &= a p_{2} \\ \dot{p1} &= - a q_{1} + \frac{3 \sqrt{2} b q_{1}^{2}}{2} - \frac{3 \sqrt{2} b q_{2}^{2}}{2} - d q_{1}^{3} - d q_{1} q_{2}^{2} \\ \dot{p2} &= - a q_{2} - 3 \sqrt{2} b q_{1} q_{2} - d q_{1}^{2} q_{2} - d q_{2}^{3} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	coefficient of the harmonic (quadratic) part
b	0.55	strength of the cubic anharmonic coupling between the two modes
d	0.4	strength of the quartic anharmonic terms

State variables: q1, q2, p1, p2

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = 0$
constant

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.NuclearQuadrupole()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference

Baran & Raduta (1998), Int. J. Mod. Phys. E
doi:10.1142/s0218301398000282

BibTeX

@misc{nuclearquadrupole,
  title = {NuclearQuadrupole system},
  note = {Baran & Raduta (1998), Int. J. Mod. Phys. E},
  doi = {10.1142/s0218301398000282}
}