NoseHoover

Systems / ODEs / Chaotic attractors

A conservative thermostatted oscillator — chaos without dissipation, sampling a canonical ensemble.

continuous · ODE3 dimensionsconservative

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= y \\ \dot{y} &= - x + y z \\ \dot{z} &= a - y^{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a	1.5	target-temperature parameter of the thermostat

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = z$
state-dependent

Equilibria

none found (no equilibria)

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.NoseHoover()
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

Nosé (1984), J. Chem. Phys. 81, 511-519; Hoover (1985), Phys. Rev. A 31, 1695-1697
doi:10.1103/physreva.31.1695

BibTeX

@misc{nosehoover,
  title = {NoseHoover system},
  note = {Nosé (1984), J. Chem. Phys. 81, 511-519; Hoover (1985), Phys. Rev. A 31, 1695-1697},
  doi = {10.1103/physreva.31.1695}
}