diffeqzoo.ivps Package#

Initial value problem examples.

This module provides a number of example implementations of initial value problems (IVPs). IVPs are a combination of an ordinary differential equation

\[\dot u(t) = f(u(t), t, \theta)\]

and an initial condition \(u(0) = u_0\). The initial values \(u_0\) and the vector field \(f\) are known, the parameters \(\theta\) might be known, and \(u\) is unknown.

The functions in this module construct implementations of this kind of problem. They (loosely) follow the input/output rule

f, u0, (t0, tmax), param = constructor()

where the constructor is, e.g., lotka_volterra() or sir(). This API specification is only loose, because every problem is different. For example, second-order problems implement a second-order differential equation

\[\ddot u(t) = f(u(t), \dot u(t), t, \theta)\]

subject to the initial conditions \(u(0) = u_0\) and \(\dot u(0) = u_1\). For these problems (e.g., three_body_restricted() or van_der_pol()), there are two initial values:

f, (u0, u1), (t0, tmax), param = constructor()

We try to stick as closely as possible to the above signature, but if problem-specific issues arise, we allow ourselves to deviate from this specification. When in doubt, consult the documentation of the respect constructor function.

Functions#

affine_dependent(*[, initial_values, ...])

Construct an IVP with an affine vector field.

affine_independent(*[, initial_values, ...])

Construct an IVP with an affine vector field, where each dimension is treated independently.

fitzhugh_nagumo(*[, initial_values, ...])

Construct the FitzHugh-Nagumo model.

goodwin(*[, initial_values, time_span, r, ...])

Construct the Goodwin Oscillator dynamics.

heat_1d_dirichlet(*[, initial_values, ...])

Discretised heat equation in 1d with Dirichlet boundary.

henon_heiles(*[, initial_values, time_span, p])

Construct the Henon-Heiles problem.

henon_heiles_first_order(**kwargs)

Construct the Henon-Heiles problem as a first-order differential equation.

henon_heiles_with_unused_derivative_argument(...)

Construct the Henon-Heiles problem as \(\ddot u(t) = f(u(t), \dot u(t))\) (with an unused second argument).

hires(*[, initial_values, time_span])

Construct the High Irradiance Response (HIRES) problem.

logistic(*[, initial_value, time_span, ...])

Construct the logistic ODE model.

lorenz63(*[, initial_values, time_span, ...])

Construct the Lorenz63 model.

lorenz96(*[, initial_values, time_span, ...])

Construct the Lorenz96 model.

lotka_volterra(*[, initial_values, ...])

Construct the Lotka--Volterra / predator-prey model.

neural_ode_mlp(*[, initial_values, ...])

Construct an IVP with a neural ODE vector field.

nonlinear_chemical_reaction(*[, ...])

Construct the Nonlinear Chemical Reaction dynamics.

oregonator(*[, initial_values, time_span, ...])

Construct the scaled Oregonator Mass-Action dynamics in a well-stirred, homogeneous system.

pleiades(*[, initial_values, time_span])

Construct the Pleiades problem in its original, second-order form.

pleiades_first_order(**kwargs)

Construct the Pleiades problem as a first-order differential equation.

pleiades_with_unused_derivative_argument(...)

Construct the Pleiades problem as \(\ddot u(t) = f(u(t), \dot u(t))\) (with an unused second argument).

rigid_body(*[, time_span, initial_values, ...])

Construct the rigid body dynamics without external forces.

rober(*[, initial_values, time_span, k1, k2, k3])

Construct the ROBER problem due to Robertson (1966).

roessler(*[, initial_values, time_span, ...])

Construct the Roessler model.

seir(*[, initial_values, time_span, alpha, ...])

Construct the SEIR model.

sir(*[, initial_values, time_span, beta, gamma])

Construct the SIR model without vital dynamics.

sird(*[, initial_values, time_span, beta, ...])

Construct the SIRD model.

three_body_restricted(*[, initial_values, ...])

Construct the restricted three-body problem as a second-order differential equation.

three_body_restricted_first_order(**kwargs)

Construct the restricted three-body problem as a first-order differential equation.

van_der_pol(*[, stiffness_constant, ...])

Construct the Van-der-Pol system as a second-order differential equation.

van_der_pol_first_order(**kwargs)

Construct the Van-der-Pol system as a first-order differential equation.