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 (x, y, z)

projection (x, y, w)

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= a y_{2} \\ \dot{y_{1}} &= a y_{3} \\ \dot{y_{2}} &= - a y_{0} + \frac{3 \sqrt{2} b y_{0}^{2}}{2} - \frac{3 \sqrt{2} b y_{1}^{2}}{2} - d y_{0}^{3} - d y_{0} y_{1}^{2} \\ \dot{y_{3}} &= - a y_{1} - 3 \sqrt{2} b y_{0} y_{1} - d y_{0}^{2} y_{1} - d y_{1}^{3} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.2996,\; +0.01191,\; -0.01192,\; -0.2996$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 4$

Divergence ∇·f

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

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.NuclearQuadrupole()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Baran & Raduta (1998), Int. J. Mod. Phys. E

BibTeX

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