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¶

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} + h y_{1} + s y_{1} y_{2} \\ \dot{y_{1}} &= - b y_{1} - p y_{0} + q y_{0} y_{2} \\ \dot{y_{2}} &= c y_{2} - r y_{0} y_{1} \end{aligned} \]

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

Properties¶

Lyapunov spectrum

$+0.2234,\; +0.01567,\; -1.239$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.193$

Divergence ∇·f

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

Equilibria

5 equilibria
0 stable · 5 unstable

Define it in TSDynamics¶

import tsdynamics as ts

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

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

Reference¶

Li et al. (2015), IEICE Electron. Express 12(4), 20141116

BibTeX

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