In the study of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, the van der Pol oscillator (named for Dutch physicist
Balthasar van der Pol
Balthasar van der Pol (27 January 1889 – 6 October 1959) was a Dutch physicist.
Life and work
Van der Pol began his studies of physics in Utrecht in 1911. J. A. Fleming offered Van der Pol the use of the Pender Electrical Laboratory at U ...
) is a non-
conservative
Conservatism is a cultural, social, and political philosophy and ideology that seeks to promote and preserve traditional institutions, customs, and values. The central tenets of conservatism may vary in relation to the culture and civiliza ...
,
oscillating
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
system with non-linear
damping
In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include ...
. It evolves in time according to the
second-order differential equation
where is the position
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
—which is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-orie ...
of the time —and is a
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
parameter indicating the
nonlinearity
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
and the strength of the damping.
History
The Van der Pol oscillator was originally proposed by the Dutch
electrical engineer
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
and physicist
Balthasar van der Pol
Balthasar van der Pol (27 January 1889 – 6 October 1959) was a Dutch physicist.
Life and work
Van der Pol began his studies of physics in Utrecht in 1911. J. A. Fleming offered Van der Pol the use of the Pender Electrical Laboratory at U ...
while he was working at
Philips
Koninklijke Philips N.V. (), simply branded Philips, is a Dutch multinational health technology company that was founded in Eindhoven in 1891. Since 1997, its world headquarters have been situated in Amsterdam, though the Benelux headquarter ...
.
Van der Pol found stable oscillations, which he subsequently called
relaxation-oscillations and are now known as a type of
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
, in
electrical circuit
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
s employing
vacuum tube
A vacuum tube, electron tube, thermionic valve (British usage), or tube (North America) is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied. It ...
s. When these circuits are driven near the
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
, they become
entrained, i.e. the driving
signal
A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology.
In ...
pulls the current along with it. Van der Pol and his colleague, van der Mark, reported in the September 1927 issue of ''
Nature
Nature is an inherent character or constitution, particularly of the Ecosphere (planetary), ecosphere or the universe as a whole. In this general sense nature refers to the Scientific law, laws, elements and phenomenon, phenomena of the physic ...
'' that at certain drive
frequencies
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
an irregular
noise
Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
was heard,
which was later found to be the result of
deterministic chaos
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to h ...
.
The Van der Pol equation has a long history of being used in both the
physical
Physical may refer to:
*Physical examination
In a physical examination, medical examination, clinical examination, or medical checkup, a medical practitioner examines a patient for any possible medical signs or symptoms of a Disease, medical co ...
and
biological
Biology is the scientific study of life and living organisms. It is a broad natural science that encompasses a wide range of fields and unifying principles that explain the structure, function, growth, origin, evolution, and distribution of ...
science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
s. For instance, in biology, Fitzhugh
and Nagumo
extended the equation in a
planar field as a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
for
action potential
An action potential (also known as a nerve impulse or "spike" when in a neuron) is a series of quick changes in voltage across a cell membrane. An action potential occurs when the membrane potential of a specific Cell (biology), cell rapidly ri ...
s of
neurons
A neuron (American English), neurone (British English), or nerve cell, is an membrane potential#Cell excitability, excitable cell (biology), cell that fires electric signals called action potentials across a neural network (biology), neural net ...
. The equation has also been utilised in
seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic ...
to model the two plates in a
geological fault
Geology (). is a branch of natural science concerned with the Earth and other astronomical objects, the rocks of which they are composed, and the processes by which they change over time. Modern geology significantly overlaps all other Earth ...
, and in studies of
phonation
The term phonation has slightly different meanings depending on the subfield of phonetics. Among some phoneticians, ''phonation'' is the process by which the vocal folds produce certain sounds through quasi-periodic vibration. This is the defi ...
to model the right and left
vocal fold
In humans, the vocal cords, also known as vocal folds, are folds of throat tissues that are key in creating sounds through Speech, vocalization. The length of the vocal cords affects the pitch of voice, similar to a violin string. Open when brea ...
oscillators.
Two-dimensional form
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:
:
:
.
Another commonly used form based on the transformation
leads to:
:
:
.
Results for the unforced oscillator
* When , i.e. there is no damping function, the equation becomes
This is a form of the
simple harmonic oscillator
In mechanics and physics, simple harmonic motion (sometimes abbreviated as ) is a special type of periodic function, periodic motion an object experiences by means of a restoring force whose magnitude is directly proportionality (mathematics), ...
, and there is always
conservation of energy
The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
.
* When , 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. However, such a solution does exist for the limit cycle if in the
Lienard equation is a constant piece-wise function.
* The period at small has serial expansion
See
Poincaré–Lindstedt method
In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method remove ...
for a derivation to order 2. See chapter 10 of
for a derivation up to order 3, and for a numerical derivation up to order 164.
* For large , 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 to .
** Rapidly moving to the left branch of the cubic curve, from to .
** 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 is the smallest root of , where is the
Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear in ...
. (Section 9.7
) ( contains a derivation, but has a misprint of to .) This was derived by
Anatoly Dorodnitsyn
Anatoly Alekseyevich Dorodnitsyn (Russian: Анатолий Алексеевич Дородницын) 19 November (per Julian Calendar), 2 December (per Gregorian Calendar), 1910 – 7 June 1994, Moscow) was a Russian mathematician who worked a ...
.
* The amplitude of the cycle is
Hopf bifurcation
As 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 , both the differential equation becomes linear, and the origin becomes a circular node.
Knowing that in a
Hopf bifurcation
In the mathematics of dynamical systems and differential equations, a Hopf bifurcation is said to occur when varying a parameter of the system causes the set of solutions (trajectories) to change from being attracted to (or repelled by) a fixed ...
, the limit cycle should have size
we may attempt to convert this to a Hopf bifurcation by using the change of variables
which gives
This indeed is a Hopf bifurcation.
Hamiltonian for Van der Pol oscillator

One can also write a time-independent
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified 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
:
where
and
are the
conjugate momenta
In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
corresponding to ''x'' and ''y'', respectively. This may, in principle, lead to quantization of the Van der Pol oscillator. Such a Hamiltonian also connects the
geometric phase
In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the ...
of the limit cycle system having time dependent parameters with the
Hannay angle
In classical mechanics, the Hannay angle is a mechanics analogue of the geometric phase (or Berry phase). It was named after John Hannay of the University of Bristol, UK. Hannay first described the angle in 1985, extending the ideas of the recently ...
of the corresponding Hamiltonian system.
Quantum oscillator
The quantum van der Pol oscillator, which is the
quantum mechanical
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of a ...
version of the classical van der Pol oscillator, has been proposed using a
Lindblad equation
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation (named after Vittorio Gorini, Andrzej Kossakowski, E. C. George Sudarshan, George Sudarshan and Göran Lindblad (physicist), Göran Lindblad), master equation in ...
to study its quantum dynamics and
quantum synchronization.
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 describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, named after John Trevor Stuart and Lev Landau. In 1944, Landau proposed an equation for the evolution of the magnitude of the disturban ...
. 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 (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.
Forced Van der Pol oscillator
The forced, or driven, Van der Pol oscillator takes the 'original' function and adds a driving function to give a differential equation of the form:
:
where is the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
, or
displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
, of the
wave function
In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
and is its
angular velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
.
Popular culture

Author
James Gleick
James Gleick (; born August 1, 1954) is an American author and historian of science whose work has chronicled the cultural impact of modern technology. Recognized for his writing about complex subjects through the techniques of narrative nonficti ...
described a
vacuum tube
A vacuum tube, electron tube, thermionic valve (British usage), or tube (North America) is a device that controls electric current flow in a high vacuum between electrodes to which an electric voltage, potential difference has been applied. It ...
Van der Pol oscillator in his book from 1987 ''
Chaos: Making a New Science''.
According to a ''
New York Times
''The New York Times'' (''NYT'') is an American daily newspaper based in New York City. ''The New York Times'' covers domestic, national, and international news, and publishes opinion pieces, investigative reports, and reviews. As one of ...
'' article,
Gleick received a modern electronic Van der Pol oscillator from a reader in 1988.
See also
*
Mary Cartwright
Dame Mary Lucy Cartwright (17 December 1900 – 3 April 1998) was a British mathematician. She was one of the pioneers of what would later become known as chaos theory. Along with J. E. Littlewood, Cartwright saw many solutions to a prob ...
, British mathematician, one of the first to study the theory of deterministic chaos, particularly as applied to this oscillator.
[
]
References
External links
*
Van der Pol oscillator on Scholarpedia
{{Chaos theory
Chaotic maps
Dutch inventions
Ordinary differential equations
Electronic oscillators
Dynamical systems