2.4. The Van der Pol Oscillator

Here, we will introduce the Van der Pol model, a widely studied non-linear oscillator model [1]. In its two-dimensional form, the Van der Pol oscillator is governed by the following non-linear ODEs:

\[ \begin{align}\begin{aligned}\dot x &= z,\\\dot z &= \mu (1-x^2) z - x,\end{aligned}\end{align} \]

with damping constant \(\mu\). Depending on the latter parameter, the periodic solutions of the Van der Pol oscillator change, with simple harmonic oscillations for \(\mu = 0\) and non-harmonic limit cycle oscillations for \(\mu > 0\). In this gallery, we will simulate and visualize the model dynamics for \(\mu = 0\) and \(\mu = 5\).

References

[1]

20. Kanamaru (2007) Van der Pol oscillator Scholarpedia 2(1): 2202. DOI: 10.4249/scholarpedia.2202.

2.4.1. Step 1: Numerical simulation of the model dynamics for \(\mu = 0\)

First, we will load a CircuitTemplate instance via the path to the model definition. We will the use the CircuitTemplate.run method to solve the initial value problem of the above defined differential equations for a time interval from 0 to the given simulation time. This solution will be calculated numerically by a differential equation solver in the backend, starting with a defined step-size. Here, we use the default backend and a Runge-Kutta 2(3) solver. Check out the arguments of the CircuitTemplate.run() method for a detailed explanation of the arguments that you can use to adjust this numerical procedure.

from pyrates import CircuitTemplate, clear

# define simulation time and input start and stop
T = 100.0
step_size = 1e-2

# load model template
model = CircuitTemplate.from_yaml("model_templates.oscillators.vanderpol.vdp")

# set mu to 0
model.update_var({'p/vdp_op/mu': 0.0})
results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'p/vdp_op/x'}, solver='scipy', method='RK23')

# plot resulting phases
import matplotlib.pyplot as plt
plt.plot(results)
plt.show()

# clear results
clear(model)

2.4.2. Step 2: Numerical simulation of the model dynamics for \(\mu = 5\)

As expected, we found harmonical oscillations for \(\mu = 0\). Now, we will repeat the procedure for \(\mu = 5\).

# load model template
model = CircuitTemplate.from_yaml("model_templates.oscillators.vanderpol.vdp")

# set mu to 5
model.update_var({'p/vdp_op/mu': 5.0})
results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'p/vdp_op/x'}, solver='scipy', method='RK23')

# plot resulting phases
plt.plot(results)
plt.show()

We find a much less harmonical waveform of the oscillations. For investigations of the Van der Pol oscillator driven by a periodic input signal, see the entrainment example in the use examples section Model analysis.