Liénard's theorem can be used to prove that the system has a limit cycle. Applying the Liénard transformation , where the dot indicates the time derivative, the Van der Pol oscillator can be written in its two-dimensional form:[11]
.
Another commonly used form based on the transformation leads to:
When μ > 0, all initial conditions converge to a globally unique limit cycle. Near the origin the system is unstable, and far from the origin, the system is damped.
The Van der Pol oscillator does not have an exact, analytic solution.[13] However, such a solution does exist for the limit cycle if f(x) in the
Lienard equation is a constant piece-wise function.
The period at small μ has serial expansion See
Poincaré–Lindstedt method for a derivation to order 2. See chapter 10 of [14] for a derivation up to order 3, and [15] for a numerical derivation up to order 164.
For large μ, the behavior of the oscillator has a slow buildup, fast release cycle (a cycle of building up the tension and releasing the tension, thus a relaxation oscillation). This is most easily seen in the form In this form, the oscillator completes one cycle as follows:
Slowly ascending the right branch of the cubic curve from (2, –2/3) to (1, 2/3).
Rapidly moving to the left branch of the cubic curve, from (1, 2/3) to (–2, 2/3).
Repeat the two steps on the left branch.
The leading term in the period of the cycle is due to the slow ascending and descending, which can be computed as: Higher orders of the period of the cycle is where α ≈ 2.338 is the smallest root of Ai(–α) = 0, where Ai is the
Airy function. (Section 9.7 [16]) ([17] contains a derivation, but has a misprint of 3α to 2α.) This was derived by
Anatoly Dorodnitsyn.[18][19]
As μ moves from less than zero to more than zero, the spiral sink at origin becomes a spiral source, and a limit cycle appears "out of the blue" with radius two. This is because the transition is not generic: when ε = 0, both the differential equation becomes linear, and the origin becomes a circular node.
Knowing that in a
Hopf bifurcation, the limit cycle should have size we may attempt to convert this to a Hopf bifurcation by using the change of variables which givesThis indeed is a Hopf bifurcation.[21]
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:
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[22]
where and 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[23] the
geometric phase of the limit cycle system having time dependent parameters with the
Hannay angle of the corresponding Hamiltonian system.
Quantum oscillator
The quantum van der Pol oscillator, which is the
quantum mechanics version of the classical van der Pol oscillator, has been proposed using a
Lindblad equation to study its quantum dynamics and
quantum synchronization.[24] Note the above Hamiltonian approach with an auxiliary second-order equation produces unbounded phase-space trajectories and hence cannot be used to quantize the van der Pol oscillator. In the limit of weak nonlinearity (i.e. μ→0) the van der Pol oscillator reduces to the
Stuart–Landau equation. The Stuart–Landau equation in fact describes an entire class of limit-cycle oscillators in the weakly-nonlinear limit. The form of the classical Stuart–Landau equation is much simpler, and perhaps not surprisingly, can be quantized by a Lindblad equation which is also simpler than the Lindblad equation for the van der Pol oscillator. The quantum Stuart–Landau model has played an important role in the study of quantum synchronisation[25][26] (where it has often been called a van der Pol oscillator although it cannot be uniquely associated with the van der Pol oscillator). The relationship between the classical Stuart–Landau model (μ→0) and more general limit-cycle oscillators (arbitrary μ) has also been demonstrated numerically in the corresponding quantum models.[24]
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:
^B. van der Pol: "A theory of the amplitude of free and forced triode vibrations", Radio Review (later Wireless World) 1 701–710 (1920)
^van der Pol, Balth. (1926). "On "relaxation-oscillations"". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 2 (11). Informa UK Limited: 978–992.
doi:
10.1080/14786442608564127.
ISSN1941-5982.
^Nagumo, J.; Arimoto, S.; Yoshizawa, S. (1962). "An Active Pulse Transmission Line Simulating Nerve Axon". Proceedings of the IRE. 50 (10). Institute of Electrical and Electronics Engineers (IEEE): 2061–2070.
doi:
10.1109/jrproc.1962.288235.
ISSN0096-8390.
S2CID51648050.
^K. Tomita (1986): "Periodically forced nonlinear oscillators". In: Chaos, Ed. Arun V. Holden. Manchester University Press,
ISBN0719018110, pp. 213–214.
^Gleick, James (1987). Chaos: Making a New Science. New York: Penguin Books. pp. 41–43.
ISBN0-14-009250-1.