gemseo / problems / ode

# 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.

The adjoint of the problem relative to the design variables.

The adjoint of the problem relative to the state.

initial_state: NumberArray

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

integration_interval: tuple[float, float]

The interval of integration.

jac: Callable[[float, NumberArray], NumberArray] | NumberArray | None

The function to compute the Jacobian of $$f$$.

If Callable, the Jacobian is assumed to be dependent on time and state. It will be called as jac(time, state) as necessary. If NumberArray, the Jacobian is assumed to be constant. If None, the Jacobian will be approximated by finite differences.

jac_desvar: Callable[[float, NumberArray], NumberArray]

The function to compute the Jacobian of $$f$$ relative to the design variables.

result: ODEResult

The result of the ODE problem.

rhs_function: Callable[[float, NumberArray], NumberArray]

The function $$f$$.

state_vect: NDArray[float]

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

## Examples using VanDerPol¶

Solve an ODE: the Van der Pol problem

Solve an ODE: the Van der Pol problem