BelousovZhabotinsky¶
Systems / ODEs / Chemical & biological systems
The Györgyi–Field three-variable model of deterministic chaos in the oscillating Belousov–Zhabotinsky reaction.
Definition¶
\[
\begin{aligned}
\dot{x} &= t_{0} \left(\frac{c_{1} v yb_{1} z \max\left(0, x\right)}{y_{0} \left(kf + x yb_{2} + yb_{3}\right)} + \frac{c_{2} v yb_{1} z}{y_{0} \left(kf + x yb_{2} + yb_{3}\right)} + c_{3} \max\left(0, x\right)^{2} + c_{4} \left(ci - z z_{0}\right) \sqrt{\max\left(0, x\right)} + c_{5} z \max\left(0, x\right) - kf \max\left(0, x\right)\right) \\
\dot{z} &= t_{0} \left(\frac{c_{6} \left(ci - z z_{0}\right) \sqrt{\max\left(0, x\right)}}{z_{0}} + c_{7} z \max\left(0, x\right) + c_{8} v z + c_{9} z - kf z\right) \\
\dot{v} &= t_{0} \left(\frac{c_{10} v yb_{1} z \max\left(0, x\right)}{y_{0} \left(kf + x yb_{2} + yb_{3}\right)} + \frac{c_{11} v yb_{1} z}{y_{0} \left(kf + x yb_{2} + yb_{3}\right)} + c_{12} \max\left(0, x\right)^{2} + c_{13} v z - kf v\right)
\end{aligned}
\]
Parameters¶
| Symbol | Default | Role |
|---|---|---|
c1 |
-8.03474 |
|
c10 |
0.0223915 |
|
c11 |
7.53559e-05 |
|
c12 |
8.07384e-06 |
|
c13 |
-0.000499825 |
|
c2 |
0.05408 |
|
c3 |
-0.0115886 |
|
c4 |
832.587 |
|
c5 |
-0.029155 |
|
c6 |
0.00321617 |
|
c7 |
-0.01352 |
|
c8 |
-0.0831709 |
|
c9 |
-0.0199985 |
|
ci |
0.000833 |
bromate feed concentration |
kf |
0.00035 |
reactor in/out-flow rate |
t0 |
2308.62 |
overall time-scaling factor |
y0 |
7.72571e-06 |
|
yb1 |
6.92813e-07 |
|
yb2 |
2.00869 |
|
yb3 |
0.01352 |
|
z0 |
8.33e-06 |
State variables: x, z, v
Properties¶
Lyapunov spectrum
TODO — Lyapunov compute exceeded 20s (killed)
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
n/a — non-smooth right-hand side — divergence is piecewise (defined almost everywhere)
Equilibria
none found (no equilibria)
Define it in TSDynamics¶
import tsdynamics as ts
sys = ts.systems.BelousovZhabotinsky()
traj = sys.integrate(final_time=100.0, dt=0.01)
exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)
Reference¶
Györgyi & Field (1992), Nature 355, 808-810