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

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

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

Properties

Lyapunov spectrum

$+0.1151,\; +0.000203,\; -0.3646$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.316$

Divergence ∇·f

$\nabla\!\cdot f = a - 2 b - e y_{0}^{2} - e y_{1}^{2} + f y_{0}^{3} - y_{2}^{2} + 2 y_{2}$
state-dependent

Equilibria

3 equilibria
1 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Aizawa & Uezu (1982), Prog. Theor. Phys. 67, 982-985

BibTeX

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