BeerRNN

Systems / ODEs / Exotic systems

A three-neuron continuous-time recurrent neural network (CTRNN) whose fixed weights place it on a chaotic attractor.

continuous · ODE3 dimensionschaotic

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Definition

\[ \begin{aligned} \dot{x1} &= - 1.0 x_{1} + \frac{2.75}{1 + 3.04652013527946 e^{- x_{3}}} - \frac{0.018}{1 + 16.2322499337625 e^{- x_{2}}} + \frac{5.422}{1 + 60.8249459466964 e^{- x_{1}}} \\ \dot{x2} &= - 0.4 x_{2} + \frac{0.484}{1 + 3.04652013527946 e^{- x_{3}}} + \frac{1.836}{1 + 16.2322499337625 e^{- x_{2}}} - \frac{0.096}{1 + 60.8249459466964 e^{- x_{1}}} \\ \dot{x3} &= - 1.0 x_{3} + \frac{3.885}{1 + 3.04652013527946 e^{- x_{3}}} - \frac{2.25}{1 + 16.2322499337625 e^{- x_{2}}} + \frac{0.535}{1 + 60.8249459466964 e^{- x_{1}}} \end{aligned} \]

Properties

Lyapunov spectrum

$+0.0161,\; +0.005874,\; -0.4918$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.045$

Divergence ∇·f

$\nabla\!\cdot f = -2.4 + \frac{11.8357307255607 e^{- x_{3}}}{1 + 6.09304027055891 e^{- x_{3}} + 9.28128493466316 e^{- 2 x_{3}}} + \frac{29.8024108783879 e^{- x_{2}}}{1 + 32.464499867525 e^{- x_{2}} + 263.485937912133 e^{- 2 x_{2}}} + \frac{329.792856922988 e^{- x_{1}}}{1 + 121.649891893393 e^{- x_{1}} + 3699.67404941854 e^{- 2 x_{1}}}$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Beer (1995), Adapt. Behav. 3, 469-509

BibTeX

@misc{beerrnn,
  title = {BeerRNN system},
  note = {Beer (1995), Adapt. Behav. 3, 469-509}
}