gemseo / problems / ode

Show inherited members

van_der_pol module

The Van der Pol (VDP) problem describing an oscillator with non-linear damping.

Van der Pol, B. & Van Der Mark, J. Frequency Demultiplication. Nature 120, 363–364 (1927).

The Van der Pol problem is written as follows:

\[\frac{d^2 x(t)}{dt^2} - \mu (1-x(t)^2) \frac{dx(t)}{dt} + x = 0\]

where \(x(t)\) is the position coordinate as a function of time, and \(\mu\) is a scalar parameter indicating the stiffness.

This problem can be rewrittent in a 2-dimensional form with only first-order derivatives. Let \(y = \frac{dx}{dt}\) and \(s = \begin{pmatrix}x\\y\end{pmatrix}\). Then the Van der Pol problem is:

\[\frac{ds}{dt} = f(s, t)\]

with

\[\begin{split}f : s = \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \begin{pmatrix} y \\ \mu (1-x^2) y - x \end{pmatrix}\end{split}\]

The jacobian of this function can be expressed analytically:

\[\begin{split}\mathrm{Jac}\, f = \begin{pmatrix} 0 & 1 \\ -2\mu xy - 1 & \mu (1 - x^2) \end{pmatrix}\end{split}\]

There is no exact solution to the Van der Pol oscillator problem in terms of known tabulated functions (see Panayotounakos et al. « On the Lack of Analytic Solutions of the Van Der Pol Oscillator ». ZAMM 83, nᵒ 9 (1 septembre 2003)).

class gemseo.problems.ode.van_der_pol.VanDerPol(initial_time=0, final_time=0.5, mu=1000.0, use_jacobian=True, state_vector=None)[source]

Bases: ODEProblem

Representation of an oscillator with non-linear damping.

Parameters:

  • mu (float) –

    The stiffness parameter.

    By default it is set to 1000.0.

  • initial_time (float) –

    The start of the integration interval.

    By default it is set to 0.

  • final_time (float) –

    The end of the integration interval.

    By default it is set to 0.5.

  • use_jacobian (bool) –

    Whether to use the analytical expression of the Jacobian. If false, use finite differences to estimate the Jacobian.

    By default it is set to True.

  • state_vector (NDArray[float]) – The state vector of the system.

func: Callable[[NDArray[float], NDArray[float]], NDArray[float]]

The right-hand side of the ODE.

initial_state: NDArray[float]

The initial conditions \((t_0,s_0)\) of the ODE.

integration_interval: tuple[float, float]

The interval of integration.

jac: Callable[[NDArray[float], NDArray[float]], NDArray[float]]

The Jacobian function of the right-hand side of the ODE.

result: ODEResult

The result of the ODE problem.

state_vect: NDArray[float]

State vector \(s=(x, \dot{x})\) of the system.

time_vector: NDArray[float]

The times at which the solution should be evaluated.

Examples using VanDerPol

Solve an ODE: the Van der Pol problem

Solve an ODE: the Van der Pol problem