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:

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

References

[1]	Kanamaru (2007) Van der Pol oscillator Scholarpedia 2(1): 2202.

DOI: 10.4249/scholarpedia.2202.

Van der Pol oscillator dynamics

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 code: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)

As expected, we found harmonical oscillations for . Now, we will repeat the procedure for .

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

# set mu to 0
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 gallery section Model analysis.