# Van der Pol oscillator

The **van der Pol oscillator** is an oscillator
with nonlinear damping governed by the second-order differential equation
\[\tag{1}
\ddot x - \epsilon (1-x^2) \dot x + x = 0
\ ,\]

where \(x\) is the dynamical variable and \(\epsilon>0\) a parameter. This model was proposed by Balthasar van der Pol (1889-1959) in 1920 when he was an engineer working for Philips Company (in the Netherlands).

## Analysis

When \(x\) is small, the quadratic term \(x^2\) is negligible and the system becomes a linear differential equation with a negative damping \(-\epsilon \dot{x}\ .\) Thus, the fixed point \((x=0,\dot{x}=0)\) is unstable (an unstable focus when \(0 < \epsilon < 2\) and an unstable node, otherwise). On the other hand, when \(x\) is large, the term \(x^2\) becomes dominant and the damping becomes positive. Therefore, the dynamics of the system is expected to be restricted in some area around the fixed point. Actually, the van der Pol system (1) satisfies the Liénard's theorem ensuring that there is a stable limit cycle in the phase space.The van der Pol system is therefore a Liénard system.

Using the **Liénard's transformation**
\(y = x - x^3/3 - \dot{x}/\epsilon\ ,\)
equation (1) can be rewritten as
\[\tag{2}
\dot x = \epsilon \left( x - \frac{1}{3} x^3 - y \right)
\]

\[\tag{3} \dot y = \frac{x}{\epsilon} \]

which can be regarded as a special case of the FitzHugh-Nagumo model (also known as Bonhoeffer-van der Pol model).

### Small Damping

When \(\epsilon\) << 1, it is convenient to rewrite equation (1) as \[\tag{4} \dot x = \epsilon \left( x - \frac{1}{3} x^3\right) - y \]

\[\tag{5} \dot y = x \]

where the transformation \(y = \epsilon (x - x^3/3) - \dot{x}\)
was used.
When \(\epsilon = 0\ ,\) the system preserves the energy and
has the solution \(x=A\cos(t+\phi)\) and \(y=A\sin(t+\phi)\ .\)
To obtain the approximated solution for small \(\epsilon\ ,\)
new variables \((u,v)\) which rotate with the unperturbed solution,
*i.e.*,
\[
u = x \cos t + y \sin t
\]
\[
v = -x \sin t + y \cos t
\]
are considered.
By substituting them into equations (4) and (5), we obtain
\[\tag{6}
\dot{u} = \epsilon \left[ u \cos t - v \sin t - \frac{1}{3}( u \cos t - v \sin t)^3 \right] \cos t
\]

\[\tag{7} \dot{v} = - \epsilon \left[ u \cos t - v \sin t - \frac{1}{3}( u \cos t - v \sin t)^3 \right] \sin t \, . \]

Because \(\dot{u}\) and \(\dot{v}\) are \(O(\epsilon)\ ,\) the varying speed of \(u\) and \(v\) is much slower than \(\cos t\) and \(\sin t\ .\) Therefore, the averaging theory can be applied to equations (6) and (7). Integrating the righthand sides of equations (6) and (7) with respect to \(t\) from \(0\) to \(T=2\pi\ ,\) keeping \(u\) and \(v\) fixed, \[ \dot{u} = \frac{\epsilon}{8}\,u\left[ 4 - ( u^2+ v^2) \right] \] \[ \dot{v} = \frac{\epsilon}{8}\,v \left[ 4 - ( u^2+ v^2) \right] \] are obtained. Introducing \(r=\sqrt{u^2+v^2}\ ,\) a differential equation \[\tag{8} \dot{r} = \frac{\epsilon}{8}\, r\, ( 4 - r^2 ) \]

which has a stable equilibrium with \(r=2\) is obtained. Therefore, the original system (4) and (5) has a stable limit cycle with \(r=2\) for small \(\epsilon\ .\)

### Large Damping

When \(\epsilon\) >> 1, it is convenient to use equations (2) and (3). When the system is away from the curve \(y=x-x^3/3\ ,\) a relation \(|\dot{x}|\) >> \(|\dot{y}|=O(1/\epsilon)\) is obtained from equations (2) and (3). Therefore, the system moves quickly in the horizontal direction. When the system enters the region where \(|x-x^3/3-y| = O(1/\epsilon^2)\ ,\) \(\dot{x}\) and \(\dot{y}\) are comparable because both of them are \(O(1/\epsilon)\ .\) Then the system goes slowly along the curve, and eventually exits from this region. Such a situation is shown in Figure 3. It can be observed that the system has a stable limit cycle.

It is also observed that the period of oscillation is determined mainly by the time during which the system stays around the cubic function where both \(\dot{x}\) and \(\dot{y}\) are \(O(1/\epsilon)\ .\) Thus, the period of oscillation is roughly estimated to be \(T\propto \epsilon\ .\)

When van der Pol (1927) realized equation (1)
with an electrical circuit composed of two resistances \(R\) and \(r\ ,\)
a capacitance \(C\ ,\) an inductance, and a tetrode, the period of oscillation
was determined by \(\epsilon = RC\) in his circuit.
Because \(RC\) is the time constant of relaxation
in RC circuit, he named this oscillation as **relaxation oscillation**.
The characteristics of the relaxation oscillation are
the slow asymptotic behavior and the sudden discontinuous jump to another
value.
Using few relaxation oscillations, van der Pol and van der Mark (1928) modeled the electric activity of the heart.

## Electrical Circuit

To make electrical circuits described by equation (1), active circuit elements with the cubic nonlinear property, \(i=\phi(v)= \gamma v^3 - \alpha v\ ,\) are required, where \(i\) and \(v\) are current and voltage, respectively. In the 1920s, van der Pol built the oscillator using the triode or tetrode. After Reona Esaki (1925-) invented the tunnel diode in 1957, making the van der Pol oscillator with electrical circuits became much simpler.

Using the tunnel diode with input-output relation \[ i=\phi_t(v) = \phi(v-E_0) + I_0 \] the equation for the circuit shown in Figure 5 is written as follows. \[ \dot{V} = \frac{1}{C}\left(- \phi(V) - W\right) \] \[ \dot{W} = \frac{1}{L}V \] This can be rewritten as \[\tag{9} \ddot{V} - \frac{1}{C} (\alpha - 3\gamma V^2) \dot{V} + \frac{1}{LC} V = 0 \]

Introducing new variables \(x = \sqrt{3\gamma/\alpha} V\ ,\) \(t' =t/\sqrt{LC}\ ,\) and \(\epsilon = \sqrt{L/C} \alpha\ ,\) equation (9) can be transformed into equation (1). As shown in the previous section, when \(\epsilon\) is large, the period of oscillation is proportional to \(\epsilon\ .\) Thus, the original system has a period \(T\propto \epsilon \sqrt{LC}= L\alpha\ .\) Because \(\alpha\) has an order of the reciprocal of resistance \(r\ ,\) \(T\propto L/r\) is obtained. \(L/R\) is the time constant of relaxation in LR circuit; therefore, the name of "relaxation oscillation" is justified.

The electrical circuit elements with the nonlinear property can also be realized using operational amplifiers. By this method, much research has been done to study the nonlinear dynamics in physical systems.

## Periodic Forcing and Deterministic Chaos

Van der Pol had already examined the response of the van der Pol oscillator to a periodic forcing in his paper in 1920, which can be formulated as \[ \ddot x - \epsilon (1-x^2) \dot x + x = F \cos\left( \frac{2 \pi t}{T_{in}}\right) \] There exist two frequencies in this system, namely, the frequency of self-oscillation determined by \(\epsilon\) and the frequency of the periodic forcing. The response of the system is shown in Figure 6 (upper) for \(T_{in}=10\) and \(F=1.2\ .\) It is observed that the mean period \(T_{out}\) of \(x\) often locks to \(mT_{in}/n\ ,\) where \(m\) and \(n\) are integers. It is also known that chaos can be found in the system when the nonlinearity of the system is sufficiently strong.Figure 6 (lower) shows the largest Lyapunov exponent, and it is observed that chaos takes place in the narrow ranges of \(\epsilon\ .\)

Van der Pol and van der Mark (1927) considered an electrical circuit composed of a resistance, a capacitance, and a Ne lamp, and they heard the response of the system by inserting the telephone receivers into their circuit. Besides the locking behaviors, they heard irregular noises before the period of the system jumps to the next value. They stated that this noise is a subsidiary phenomenon, but today it is thought that they heard the deterministic chaos in 1927 before Yoshisuke Ueda (1961) and Edward Lorenz (1963). Nevertheless, van der Pol did not identify the structure underlying a chaotic attractor in the phase space. Lorenz published a picture of a chaotic attractor in the phase space in the early 60s and Ueda did in the early 70s.

Typical sounds of the system can be heard in the following links (before clicking the link, please lower the volume of your speaker)

- (A) Media:vdP-Periodic1.mp3 (Periodic, \(\epsilon=6\))

- (B) Media:vdP-Chaotic.mp3 (Chaotic, \(\epsilon=8.53\))

- (C) Media:vdP-Periodic2.mp3 (Periodic, \(\epsilon=10\))

where (A), (B), and (C) correspond to the letters in Figure 6. A transformation of the timescale was applied so that the oscillation with \(T_{out}=10\) was transformed into the oscillation with 440 [Hz]. An irregular noise would be heard when chaos exists in the system.

The locking behaviors of the mean period can be understood using the circle map and related mappings. This was done in a series of papers by M.L. Cartwright and J.E. Littlewood (1945-1950) and in work on an important piece-wise linear approximation by N. Levinson (1949). Both of these investigations uncovered "random-like" dynamics. Levinson's analysis led to S. Smale's introduction of the horseshoe mapping, which was used by M. Levi (1981) to complete the picture of limit behavior of all solutions. van der Pol's model was simulated using high resolution computations by J.E. Flaherty and F.C. Hoppensteadt (1978) who identified overlapping regions in the parameter domain where phase locking occurs, similar to Arnold's tongues. That work motivated a successful investigation of phase-locking in neural tissue done by R. Guttman et al.(See Voltage-Controlled Oscillations in Neurons). As for chaos in the Arnold's tongues, please see Horita et al. (1988) and Ott (1993).

## References

- B. van der Pol, A theory of the amplitude of free and forced triode vibrations,
*Radio Review*,**1**, 701-710, 754-762, 1920.

- E. V. Appleton and B. van der Pol, On the form of free triode vibrations,
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.6*,**42**, 201-220, 1921.

- E. V. Appleton and B. van der Pol, On a type of oscillation-hysteresis in a simple triode generator,
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.6*,**43**, 177-193, 1922.

- B. van der Pol, On oscillation hysteresis in a triode generator with two degrees of freedom,
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.6*,**43**, 700-719, 1922.

- B. van der Pol, On "relaxation-oscillations",
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.7*,**2**, 978-992, 1926.

- B. van der Pol, Forced oscillations in a circuit with non-linear resistance (reception with reactive triode),
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.7*,**3**, 65-80, 1927.

- B. van der Pol and J. van der Mark, Frequency demultiplication,
*Nature*,**120**, 363-364, 1927

- B. van der Pol and J. van der Mark, The heartbeat considered as a relaxation oscillation, and an electrical model of the heart.
*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Ser.7*,**6**, 763-775, 1928.

- B. van der Pol, The nonlinear theory of electric oscillations,
*Proceedings of the Institute of Radio Engineers*,**22**, 1051-1086, 1934.

- M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order: I. The equation \(\ddot y - k(1 - y^2 )\dot y + y = b\lambda k cos (\lambda t + a); k\) large,
*Journal of the London Mathematical Society*,**20**, 180-189, 1945.

- N. Levinson, A second order differential equation with singular solutions,
*Ann. Math.*,**50**, No. 1, 127-153, 1949.

- M.L. Cartwright, Forced oscillations in nonlinear systems,
*Contrib. to theory of nonlinear oscillations*, Princeton University Press (Study 20) 149-241, 1950.

- R. FitzHugh, Impulses and physiological states in models of nerve membrane,
*Biophysical Journal*,**1**, 445-466, 1961.

- J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon,
*Proceedings of the Institute of Radio Engineers*,**50**, 2061-2070, 1962.

- J.E. Flaherty, F.C. Hoppensteadt, Frequency entrainment of a forced van der Pol oscillator,
*Studies in Appl. Math.*,**58**, 5-15, 1978.

- M. Levi, Qualitative analysis of the periodically forced relaxation oscillations,
*Memoirs of the Amer. Math. Soc.*,**32**, No. 244, 1981.

- J. Guckenheimer and P. Holmes,
*Nonlinear oscillations, dynamical systems, and bifurcations of vector fields*, Springer-Verlag, 1983.

- T. Horita, H. Hata, H. Mori, T. Morita, K. Tomita, S. Kuroki, and H. Okamoto, Local Structures of Chaotic Attractors and q-Phase Transitions at Attractor-Merging Crises in the Sine-Circle Maps,
*Progress of Theoretical Physics*,**80**, 793-808, 1988

- E. Ott,
*Chaos in Dynamical Systems*, Cambridgeã€€University Press, New York, 1993.

**Internal references**

- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Steve Smale and Michael Shub (2007) Smale horseshoe. Scholarpedia, 2(11):3012.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
- Frank Hoppensteadt (2006) Voltage-controlled oscillations in neurons. Scholarpedia, 1(11):1599.

## External Links

## See Also

Averaging, Chaos, FitzHugh-Nagumo Model, Periodic Orbit, Relaxation Oscillator, Stability, Voltage-Controlled Oscillations in Neurons