Note

Click here to download the full example code

Finite-time blow-up

Hyperbolic growth x' = x² looks tame at first, slower than an exponential, but every doubling takes half the time of the previous one, and the state diverges at the finite time t* = 1 / x0: a singularity reached in finite time.

The run starts at x = 1 with p = 2, so the singularity is at t* = 1, and stops at t = 0.99, where the exact solution x = 1 / (1 - t) reaches 100. The default blueprint shows x as a time series.

import matplotlib.pyplot as plt

from skcomplex.engine.runtime import Simulation
from skcomplex.engine.viz import RerunObserver
from skcomplex.models import create

MODEL = 'finite_time_blow_up'

observer = RerunObserver(MODEL, save_path=f'{MODEL}.rrd', blueprint='default', interval=4)
simulation = Simulation(create(MODEL), dt=0.001, observers=[observer]).check().prepare().run(until=0.99)

Open the recording in the Rerun viewer

Approach to the singularity

The exact solution x = ((p - 1)(t* - t))^(-1 / (p - 1)) diverges as a power of the time left, with t* = 1 / (p - 1) for x0 = 1 and a = 1. Plotted against t* - t on log-log axes, runs for p = 1.5, 2, and 3 should follow straight lines of slopes -2, -1, and -1/2.

fig, ax = plt.subplots(figsize=(6, 4.5))
for p in (1.5, 2.0, 3.0):
    singular = 1 / (p - 1)
    trajectory = (
        Simulation(create(MODEL, p=p), dt=singular / 2000).check().prepare().run(until=0.999 * singular).trajectory_
    )
    remaining = singular - trajectory.times
    (line,) = ax.loglog(remaining, trajectory.values('growth.x'), label=f'p = {p}')
    ax.loglog(remaining, ((p - 1) * remaining) ** (-1 / (p - 1)), ':', color='black')
ax.set_xlabel(r'time left to the singularity $t^* - t$')
ax.set_ylabel('x')
ax.set_title('Power-law divergence (dotted: exact solutions)')
ax.legend()
plt.show()

Download Python source code: plot_finite_time_blow_up.py

Download Jupyter notebook: plot_finite_time_blow_up.ipynb

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About the model

Hyperbolic growth x' = a x^p, which diverges at the finite time x0^(1 - p) / (a (p - 1)) for p > 1.

References

• Forrest 2018