Note
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Brusselator
In a well-mixed reactor with b = 3 > 1 + a^2, the Brusselator leaves its unstable fixed point and settles on a limit
cycle: the concentrations u and v oscillate out of phase with a fixed amplitude, whatever the initial state.
The model is created by name from the catalog with a = 1 and b = 3, starting at u = v = 1, and runs for about
seven periods. The default blueprint shows both concentrations in one time series view.
import matplotlib.pyplot as plt
from skcomplex.engine.runtime import Simulation
from skcomplex.engine.viz import RerunObserver
from skcomplex.models import create
MODEL = 'brusselator_ode'
system = create(MODEL)
observer = RerunObserver(MODEL, save_path=f'{MODEL}.rrd', blueprint='default', interval=2)
simulation = Simulation(system, dt=0.1, observers=[observer]).check().prepare().run(until=50)
Open the recording in the Rerun viewer
One limit cycle from every initial state
The fixed point (a, b / a) = (1, 3) is unstable for b > 1 + a^2. Runs started inside and outside the cycle
should all wind onto the same closed orbit, the one the run above reaches.
fig, ax = plt.subplots(figsize=(6, 5))
for u0, v0 in [(1.0, 3.2), (0.5, 0.5), (4.0, 1.0), (3.0, 5.0)]:
run = Simulation(create(MODEL, u=u0, v=v0), dt=0.05).check().prepare().run(until=100).trajectory_
ax.plot(run.values('reactor.u'), run.values('reactor.v'), linewidth=0.8, alpha=0.7)
ax.plot(u0, v0, 'o', color='gray', markersize=4)
trajectory = simulation.trajectory_
late = trajectory.times > 30
ax.plot(
trajectory.values('reactor.u')[late],
trajectory.values('reactor.v')[late],
color='black',
linewidth=2,
label='limit cycle',
)
ax.plot(1.0, 3.0, 'x', color='red', markersize=10, label='unstable fixed point')
ax.set_xlabel('u')
ax.set_ylabel('v')
ax.set_title('All initial states converge to the limit cycle')
ax.legend()
plt.show()
About the model
The well-mixed Brusselator, which oscillates on a limit cycle when b > 1 + a**2.
References
- Prigogine and Lefever 1968
- Datseris and Parlitz 2022
- Trzaska 2023
Total running time of the script: ( 0 minutes 1.755 seconds)
Download Python source code: plot_brusselator_ode.py