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
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:
When μ = 0, i.e. there is no damping function, the equation becomes:
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.
The Van der Pol oscillator does not have an exact, analytic solution. However, 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
Randomly chosen initial conditions are attracted to a stable orbit.
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
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 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:
Mary Cartwright, British mathematician, one of the first to study the theory of deterministic chaos, particularly as applied to this oscillator.
The quantum van der Pol oscillator, which is the quantum version of the classical van der Pol oscillator, has been proposed using a Lindblad equation 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 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.
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