of a Van der Pol Oscillator

Ned J. Corron

In the study of nonlinear dynamics and applied mathematics, the Van der Pol oscillator is commonly used to illustrate various phenomena including stability, Hopf bifurcation, limit cycles, and relaxation oscillations. This mathematical model was originally developed for an electronic oscillator built using vacuum tubes; however, it is difficult today to realize this circuit in its original form due to the replacement of tubes with semiconductor technology. Thus, it is particularly instructive to have a modern circuit implementation for demonstrating these mathematical concepts in a physical device.

Mathematically, a general form of a Van der Pol oscillator is

_{} (1)

where *e* and *a* are constants, *u*(*t*) is the dependent
state that depends on time *t*, and a
dot denotes differentiation with respect to time. Equivalently, the oscillator (1) can be
written in system form as

_{} (2)

where _{} and _{}. This latter form is
convenient for directly interpreting the response of the system in *x*‑*y* phase space.

The
oscillator (1) is widely used to demonstrate nonlinear dynamics since it is
amenable to analysis. Specifically,
asymptotic techniques can be applied for the cases of both small and large *e*
to predict the amplitude response and waveform shape. For small *e*, it is a standard exercise to show that
*u* = 0 is stable for *a* < 0;
a Hopf bifurcation occurs at* a* = 0; and a stable limit
cycle exists for* a* > 0, for which

_{} (3)

where *f* is an arbitrary phase. For large *e*, the oscillator produces relaxation
oscillations, which approach a square wave.
Numerical simulations can be used to examine the continuum that bridges
these limits.

The
electronic circuit shown in Figure 1 is a Van der Pol oscillator; that is, the
circuit is appropriately modeled by equation (1), or equivalently by the system
(2). The voltages *V _{x}* and

Figure 1. Circuit for a Van der Pol oscillator.

Component |
Value or Device |

R |
10 kW |

R |
470 kW |

R |
100 kW potentiometer |

R |
1 kW (see text) |

C |
0.01 mF |

U |
TL082, ½ Dual BiFET Op Amp |

U |
AD633, Low Cost Analog Multiplier |

Table 1. Suggested component values and devices.

For the
component values suggested in Table 1, the parameter *e* is set to 0.1; however,
other values for *e* can be selected by changing R_{8}, where

_{}. (4)

The parameter *a* is set by adjusting the voltage divider
at R_{7},
where

_{}. (5)

Including R_{5} and R_{6} in the voltage
divider roughly compensates for the factor of 10 in (5), thereby providing the
useful range -15 < *a* < 15 using a ±15-volt
power source. The frequency at the Hopf
bifurcation is

_{} (6)

which can be adjusted by changing both C_{1}
and C_{2}
together. In operation, the voltage *V** _{a}* is monitored using a digital
voltmeter, and the voltages

Figure
2. Observed response viewed in the |

The
interesting nonlinear dynamics of the Van der Pol oscillator can be observed by
scanning *V** _{a}* from negative
to positive values. In Figure 2, the
system phase space trajectory observed for various

Figure 3.
Observed |

The
response described by (3) can be confirmed experimentally using the
circuit. In Figure 4, the theoretical
and observed amplitude responses for the oscillator are plotted as a function
of the control parameter *a*.
In this figure, two observed responses are shown. These responses are derived from the
peak-to-peak voltage observed for the voltages *V _{x}* and

Figure 4.
Observed amplitude response for the Van der Pol circuit as a function of
the control parameter *a*.

In conclusion, this brief tutorial presents a modern Van der Pol circuit that is capable of showing various nonlinear phenomena including stability, Hopf bifurcation, limit cycles, and relaxation oscillations. As a result, this simple experimental system provides a useful tool for exploring important concepts in nonlinear dynamics and serves as a starting point for further investigations in nonlinear and chaotic systems.