Van der Pol oscillator

In the study of

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

History

The Van der Pol oscillator was originally proposed by the Dutch

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:

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

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

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

^[12]

• When μ = 0, i.e. there is no damping function, the equation becomes
  $${\displaystyle {\frac {d^{2}x}{dt^{2}}}+x=0.}$$

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

• When μ > 0, all initial conditions converge to a globally unique limit cycle. Near the origin ${\displaystyle x={\tfrac {dx}{dt}}=0,}$ 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.
  Lienard equation
  is a constant piece-wise function.
• The period at small μ has serial expansion
  $${\displaystyle T={\frac {2\pi }{1-\mu ^{2}/16+17\mu ^{4}/3072+O(\mu ^{6})}}.}$$

  
  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
  $${\displaystyle {\dot {x}}=\mu \left(x-{\frac {1}{3}}x^{3}-y\right)\!,\quad {\dot {y}}={\frac {1}{\mu }}x.}$$

  
  In this form, the oscillator completes one cycle as follows:
  • Slowly ascending the right branch of the cubic curve ${\displaystyle y=x-{\tfrac {x^{3}}{3}},}$ 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:
  $${\displaystyle T=2\int dt=2\int \mu {\frac {dy}{x}}=2\mu \int _{2}^{1}{\frac {dy}{dx}}{\frac {dx}{x}}=(3-2\ln 2)\mu }$$

  
  Higher orders of the period of the cycle is
  $${\displaystyle T=(3-2\ln 2)\mu +3\alpha \mu ^{-1/3}-{\frac {23}{\mu ^{-1}}}\ln \mu +O(\ln \mu ^{-1}).}$$

  
  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.)
• The amplitude of the cycle is ^[18]
  $${\displaystyle 2+{\frac {\alpha }{3}}\mu ^{-4/3}-{\frac {16}{27}}\mu ^{-2}\ln \mu +O(\mu ^{-2})}$$

  

Hopf bifurcation

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 ${\displaystyle \propto \varepsilon ^{1/2},}$ we may attempt to convert this to a Hopf bifurcation by using the change of variables ${\displaystyle u=\varepsilon ^{1/2}x,}$ which gives

$${\displaystyle {\ddot {u}}+u+u^{2}{\dot {u}}-\varepsilon {\dot {u}}=0}$$

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^[20]

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

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^[21] 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

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

• Mary Cartwright, British mathematician, one of the first to study the theory of deterministic chaos, particularly as applied to this oscillator.^[28]

References