Aizawa¶

Systems / ODEs / Oscillatory systems

An ornamental forced attractor with a pinched-torus geometry — the TSDynamics signature shape.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - b x - d y + x z \\ \dot{y} &= - b y + d x + y z \\ \dot{z} &= a z + c - e x^{2} z - e y^{2} z + f x^{3} z - x^{2} - y^{2} - \frac{z^{3}}{3} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`0.95`	shape parameter of the attractor
`b`	`0.7`	shape parameter of the attractor
`c`	`0.6`	shape parameter of the attractor
`d`	`3.5`	shape parameter of the attractor
`e`	`0.25`	shape parameter of the attractor
`f`	`0.1`	shape parameter of the attractor

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a - 2 b - e x^{2} - e y^{2} + f x^{3} - z^{2} + 2 z$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Aizawa()
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¶

Aizawa & Uezu (1982), Prog. Theor. Phys. 67, 982-985
doi:10.1143/PTP.67.982

BibTeX

@misc{aizawa,
  title = {Aizawa system},
  note = {Aizawa & Uezu (1982), Prog. Theor. Phys. 67, 982-985},
  doi = {10.1143/PTP.67.982}
}