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:

${\displaystyle {d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0,}$

Phase portrait of the unforced Van der Pol oscillator, showing a limit cycle and the direction field

Evolution of the limit cycle in the phase plane. The limit cycle begins as circle and, with varying μ, become increasingly sharp. An example of a Relaxation oscillator.

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.

History

The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips.[1] Van der Pol found stable oscillations,[2] which he subsequently called relaxation-oscillations[3] and are now known as a type of limit cycle 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[4] that at certain drive frequencies an irregular noise was heard, which was later found to be the result of deterministic chaos.[5]

The Van der Pol equation has a long history of being used in both the physical and biological sciences. For instance, in biology, Fitzhugh[6] and Nagumo[7] 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,[8] and in studies of phonation to model the right and left vocal fold oscillators.[9]

Two-dimensional form

Liénard's theorem can be used to prove that the system has a limit cycle. Applying the Liénard transformation ${\displaystyle y=x-x^{3}/3-{\dot {x}}/\mu }$, where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:[10]

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

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

Another commonly used form based on the transformation ${\displaystyle y={\dot {x}}}$ leads to:

${\displaystyle {\dot {x}}=y}$

${\displaystyle {\dot {y}}=\mu (1-x^{2})y-x}$.

Results for the unforced oscillator

Relaxation oscillation in the Van der Pol oscillator without external forcing. The nonlinear damping parameter is equal to μ = 5.

Two interesting regimes for the characteristics of the unforced oscillator are:[11]

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

${\displaystyle {d^{2}x \over dt^{2}}+x=0.}$

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. Near the origin x = dx/dt = 0, the system is unstable, and far from the origin, the system is damped.
• Van der Pol oscillator doesn't have an exact, analytic solution.[12] Such a solution does exist for the limit cycle if f(x) in the Lienard equation is a constant piece-wise function.

Hamiltonian for Van der Pol oscillator

One can also write a time-independent Hamiltonian formalism for the Van der Pol oscillator by augmenting it to a four-dimensional autonomous dynamical system using an auxiliary second-order nonlinear differential equation as follows:

${\displaystyle {\ddot {x}}-\mu (1-x^{2}){\dot {x}}+x=0,}$

${\displaystyle {\ddot {y}}+\mu (1-x^{2}){\dot {y}}+y=0.}$

Note that the dynamics of the original Van der Pol oscillator is not affected due to the one-way coupling between the time-evolutions of x and y variables. A Hamiltonian H for this system of equations can be shown to be[13]

${\displaystyle H(x,y,p_{x},p_{y})=p_{x}p_{y}+xy-\mu (1-x^{2})yp_{y},}$

where ${\displaystyle p_{x}={\dot {y}}+\mu (1-x^{2})y}$ and ${\displaystyle p_{y}={\dot {x}}}$ are the conjugate momenta corresponding to x and y, respectively. This may, in principle, lead to quantization of the Van der Pol oscillator. Such a Hamiltonian also connects[14] the geometric phase of the limit cycle system having time dependent parameters with the Hannay angle of the corresponding Hamiltonian system.

Forced Van der Pol oscillator

Chaotic behaviour in the Van der Pol oscillator with sinusoidal forcing. The nonlinear damping parameter is equal to μ = 8.53, while the forcing has amplitude A = 1.2 and angular frequency ω = 2π / 10.

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:

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

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

Popular culture

Electrical circuit involving a triode, resulting in a forced Van der Pol oscillator.[15] The circuit contains: a triode, a resistor R, a capacitor C, a coupled inductor-set with self inductance L and mutual inductance M. In the serial RLC circuit there is a current i, and towards the triode anode ("plate") a current i_a, while there is a voltage u_g on the triode control grid. The Van der Pol oscillator is forced by an AC voltage source E_s.

Author James Gleick described a vacuum tube Van der Pol oscillator in his book from 1987 Chaos: Making a New Science.[16] According to a New York Times article,[17] Gleick received a modern electronic Van der Pol oscillator from a reader in 1988.

See also

• Mary Cartwright, British mathematician, one of the first to study the theory of deterministic chaos, particularly as applied to this oscillator.[18]
• Quantum Van der Pol oscillator, which is the quantum version of the classical Van der Pol oscillator has been proposed [19][20] using Lindblad equation formalism to study quantum synchronization.

References

1. Cartwright, M.L., "Balthazar van der Pol", J. London Math. Soc., 35, 367–376, (1960).
2. B. van der Pol: "A theory of the amplitude of free and forced triode vibrations", Radio Review (later Wireless World) 1 701–710 (1920)
3. Van der Pol, B., "On relaxation-oscillations", The London, Edinburgh and Dublin Phil. Mag. & J. of Sci., 2(7), 978–992 (1926).
4. Van der Pol, B. and Van der Mark, J., “Frequency demultiplication”, Nature, 120, 363–364, (1927).
5. Kanamaru, T., "Van der Pol oscillator", Scholarpedia, 2(1), 2202, (2007).
6. FitzHugh, R., “Impulses and physiological states in theoretical models of nerve membranes”, Biophysics J, 1, 445–466, (1961).
7. Nagumo, J., Arimoto, S. and Yoshizawa, S. "An active pulse transmission line simulating nerve axon", Proc. IRE, 50, 2061–2070, (1962).
8. 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).
9. Lucero, Jorge C.; Schoentgen, Jean (2013). "Modeling vocal fold asymmetries with coupled van der Pol oscillators". Proceedings of Meetings on Acoustics. 19 (1): 060165. doi:10.1121/1.4798467. ISSN 1939-800X.
10. Kaplan, D. and Glass, L., Understanding Nonlinear Dynamics, Springer, 240–244, (1995).
11. Grimshaw, R., Nonlinear ordinary differential equations, CRC Press, 153–163, (1993), ISBN 0-8493-8607-1.
12. Panayotounakos, D. E., Panayotounakou, N. D., & Vakakis, A. F. (2003). On the lack of analytic solutions of the Van der Pol oscillator. ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik: Applied Mathematics and Mechanics, 83(9), 611–615.
13. Shah, Tirth; Chattopadhyay, Rohitashwa; Vaidya, Kedar; Chakraborty, Sagar (2015). "Conservative perturbation theory for nonconservative systems". Physical Review E. 92 (6): 062927. arXiv:1512.06758. Bibcode:2015PhRvE..92f2927S. doi:10.1103/physreve.92.062927.
14. Chattopadhyay, Rohitashwa; Shah, Tirth; Chakraborty, Sagar (2018). "Finding the Hannay angle in dissipative oscillatory systems via conservative perturbation theory". Physical Review E. 97 (6): 062209. arXiv:1610.05218. doi:10.1103/PhysRevE.97.062209.
15. K. Tomita (1986): "Periodically forced nonlinear oscillators". In: Chaos, Ed. Arun V. Holden. Manchester University Press, ISBN 0719018110, pp. 213–214.
16. Gleick, James (1987). Chaos: Making a New Science. New York: Penguin Books. pp. 41–43. ISBN 0-14-009250-1.
17. Colman, David (11 July 2011). "There's No Quiet Without Noise". New York Times. Retrieved 11 July 2011.
18. Mary Cartwright and J. E. Littlewood (1945) "On Non-linear Differential Equations of the Second Order", Journal of the London Mathematical Society 20: 180 doi:10.1112/jlms/s1-20.3.180
19. Stefan Walter, Andreas Nunnenkamp, and Christoph Bruder (2014). Quantum Synchronization of a Driven Self-Sustained Oscillator. Physical Review Letters, 112(9), 094102. doi:10.1103/PhysRevLett.112.094102
20. T E Lee, HR Sadeghpour (2013). Quantum synchronization of quantum van der Pol oscillators with trapped ions. Physical Review Letters, 111(23), 234101. doi:10.1103/PhysRevLett.111.234101

External links

• "Van der Pol equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Van der Pol oscillator on Scholarpedia
• Van Der Pol Oscillator Interactive Demonstrations