Sakarya¶

Systems / ODEs / Coupled systems

A six-term Lorenz-family flow, realised as an electronic circuit, whose y*z and x*z nonlinearities produce a butterfly-like attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= a x + h y + s y z \\ \dot{y} &= - b y - p x + q x z \\ \dot{z} &= c z - r x y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`-1`	self-feedback gain on the x channel
`b`	`1`	self-feedback gain on the y channel
`c`	`1`	self-feedback gain on the z channel
`h`	`1`	x–y linear cross-coupling gain
`p`	`1`	x–y linear cross-coupling gain
`q`	`0.4`	x*z quadratic term strength
`r`	`0.3`	x*y quadratic term strength
`s`	`1`	y*z quadratic term strength

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a - b + c$
constant

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Sakarya()
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¶

Li et al. (2015), IEICE Electron. Express 12(4), 20141116
doi:10.1587/elex.12.20141116

BibTeX

@misc{sakarya,
  title = {Sakarya system},
  note = {Li et al. (2015), IEICE Electron. Express 12(4), 20141116},
  doi = {10.1587/elex.12.20141116}
}