Van der Pol oscillator

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second order differential equation:

${d^2x \over dt^2}-\mu(1-x^2){dx \over dt}%2Bx= 0$

where x is the position coordinate — which is a function of the time t, and μ is a scalar parameter indicating the nonlinearity and the strength of the damping.

1 History
2 Two dimensional form
3 Results for the unforced oscillator
4 The forced Van der Pol oscillator
5 Popular Culture
6 References
7 External links

History

The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol whilst he was working at Philips.^[1] Van der Pol found stable oscillations, which he called relaxation-oscillations^[2] and are now known as limit cycles, in electrical circuits employing vacuum tubes. When these circuits were driven near the limit cycle they become entrained, i.e. the driving signal pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of Nature ^[3] that at certain drive frequencies an irregular noise was heard. This irregular noise was always heard near the natural entrainment frequencies. This was one of the first discovered instances of deterministic chaos.^[4]

The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh^[5] and Nagumo^[6] extended the equation in a planar field as a model for action potentials of neurons. The equation has also been utilised in seismology to model the two plates in a geological fault.^[7]

Two dimensional form

Liénard's Theorem can be used to prove that the system has a limit cycle. Applying the Liénard transformation $y = x - x^3/3 - \dot x/\mu$ , where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two dimensional form^[8]:

$\dot x = \mu \left(x-\frac{1}{3}x^3-y\right)$

$\dot y = \frac{1}{\mu} x.$

Results for the unforced oscillator

Two interesting regimes for the characteristics of the unforced oscillator are:^[9]

When μ = 0, i.e. there is no damping function, the equation becomes:

${d^2x \over dt^2}%2Bx= 0.$

This is a form of the simple harmonic oscillator and there is always conservation of energy.

When μ > 0, the system will enter a limit cycle, where energy continues to be conserved. Near the origin x = dx/dt = 0 the system is unstable, and far from the origin the system is damped. Energy can be lost or gained, and work is done, if the system does not enter a limit cycle immediately.

The forced Van der Pol oscillator

The forced, or driven, Van der Pol oscillator takes the 'original' function and adds a driving function Asin(ωt) to give a differential equation of the form:

${d^2x \over dt^2}-\mu(1-x^2){dx \over dt}%2Bx-A \sin(\omega t)= 0,$

where A is the amplitude, or displacement, of the wave function and ω is its angular velocity.

Popular Culture

Author James Gleick described a vacuum-tube Van der Pol oscillator in his book Chaos: Making a New Science ^[10]. According to a New York Times article^[11], Gleick received a modern electronic Van der Pol oscillator from a reader in 1988.

References

^ Cartwright, M.L., "Balthazar van der Pol", J. London Math. Soc., 35, 367-376, (1960).
^ Van der Pol, B., "On relaxation-oscillations", The London, Edinburgh and Dublin Phil. Mag. & J. of Sci., 2(7), 978-992 (1927).
^ Van der Pol, B. and Van der Mark, J., “Frequency demultiplication”, Nature, 120, 363-364, (1927).
^ Kanamaru, T., "Van der Pol oscillator", Scholarpedia, 2(1), 2202, (2007).
^ FitzHugh, R., “Impulses and physiological states in theoretical models of nerve membranes”, Biophysics J, 1, 445-466, (1961).
^ Nagumo, J., Arimoto, S. and Yoshizawa, S. "An active pulse transmission line simulating nerve axon", Proc. IRE, 50, 2061-2070, (1962).
^ Cartwright, J., Eguiluz, V., Hernandez-Garcia, E. and Piro, O., "Dynamics of elastic excitable media", Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9, 2197–2202, (1999).
^ Kaplan, D. and Glass, L., Understanding Nonlinear Dynamics, Springer, 240-244, (1995).
^ Grimshaw, R., Nonlinear ordinary differential equations, CRC Press, 153–163, (1993), ISBN 0849386071.
^ Gleick, James (1987). Chaos: Making a New Science. New York: Penguin Books. pp. 41–43. ISBN 0140092501.
^ Colman, David (11 July 2011). "There's No Quiet Without Noise". New York Times. http://www.nytimes.com/2011/07/10/fashion/a-machine-designed-to-add-chaos-to-order.html. Retrieved 11 July 2011.