def seir(timespan, initrv, params=(0.3, 0.3, 0.1)): r"""Initial value problem (IVP) based on the SEIR model. The SEIR model with no vital dynamics is defined through .. math:: f(t, y) = \begin{pmatrix} \frac{-\beta y_1 y_3}{N} \\ \frac{\beta y_1 y_3}{N} - \alpha y_2 \\ \alpha y_2 - \gamma y_3 \\ \gamma y_3 \end{pmatrix} for some parameters :math:`(\alpha, \beta, \gamma)` and population count :math:`N`. Without taking vital dynamics into consideration, :math:`N` is constant such that for every time point :math:`t` .. math:: S(t) + E(t) + I(t) + R(t) = N holds. Default parameters are :math:`(\alpha, \beta, \gamma)=(0.3, 0.3, 0.1)`. The population count is computed from the (mean of the) initial value random variable. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(4, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`4`-dimensional mean vector and :math:`4 \times 4`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float, float, float), optional Parameters :math:`(\alpha, \beta, \gamma)` for the SEIR model IVP. Default is :math:`(\alpha, \beta, \gamma)=(0.3, 0.3, 0.1)`. Returns ------- IVP IVP object describing the SEIR model IVP with the prescribed configuration. """ population_count = np.sum(initrv.mean) params_and_population_count = (*params, population_count) def rhs(t, y): return seir_rhs(t, y, params_and_population_count) def jac(t, y): return seir_jac(t, y, params_and_population_count) return IVP(timespan, initrv, rhs, jac=jac)
def threebody(timespan, initrv, params=0.012277471): r"""Initial value problem (IVP) based on a three-body problem. Let the initial conditions be :math:`y = (y_1, y_2, \dot{y}_1, \dot{y}_2)^T`. This function implements the second-order three-body problem as a system of first-order ODEs, which is defined as follows: [1]_ .. math:: f(t, y) = \begin{pmatrix} \dot{y_1} \\ \dot{y_2} \\ y_1 + 2 \dot{y}_2 - \frac{(1 - \mu) (y_1 + \mu)}{d_1} - \frac{\mu (y_1 - (1 - \mu))}{d_2} \\ y_2 - 2 \dot{y}_1 - \frac{(1 - \mu) y_2}{d_1} - \frac{\mu y_2}{d_2} \end{pmatrix} with .. math:: d_1 &= ((y_1 + \mu)^2 + y_2^2)^{\frac{3}{2}} \\ d_2 &= ((y_1 - (1 - \mu))^2 + y_2^2)^{\frac{3}{2}} and a constant parameter :math:`\mu` denoting the standardized moon mass. Default is :math:`\mu = 0.012277471`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(4, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`4`-dimensional mean vector and :math:`4 \times 4`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float), optional Parameter :math:`\mu` for the three-body problem Default is :math:`\mu = 0.012277471`. Returns ------- IVP IVP object describing a three-body problem IVP with the prescribed configuration. References ---------- .. [1] Hairer, E., Norsett, S. and Wanner, G.. Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics, 1993. """ def rhs(t, y): return threebody_rhs(t, y, params) return IVP(timespan, initrv, rhs)
def logistic(timespan, initrv, params=(3.0, 1.0)): r"""Initial value problem (IVP) based on the logistic ODE. The logistic ODE is defined through .. math:: f(t, y) = a y \left( 1 - \frac{y}{b} \right) for some parameters :math:`(a, b)`. Default is :math:`(a, b)=(3.0, 1.0)`. This implementation includes the Jacobian :math:`J_f` of :math:`f` as well as a closed form solution given by .. math:: f(t) = \frac{b y_0 \exp(a t)}{b + y_0 \left[ \exp(at) - 1 \right]} where :math:`y_0= y(t_0)` is the initial value. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=())* -- Scalar-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free or Normal (no Random Variable isy) with scalar mean and scalar variance. To replicate "classical" initial values use the Constant distribution. params : (float, float), optional Parameters :math:`(a, b)` for the logistic IVP. Default is :math:`(a, b) = (3.0, 1.0)`. Returns ------- IVP IVP object describing the logistic IVP with the prescribed configuration. """ def rhs(t, y): return log_rhs(t, y, params) def jac(t, y): return log_jac(t, y, params) def hess(t, y): return log_hess(t, y, params) def sol(t): return log_sol(t, params, initrv.mean) return IVP(timespan, initrv, rhs, jac, hess, sol)
def vanderpol(timespan, initrv, params=0.1): r"""Initial value problem (IVP) based on the Van der Pol Oscillator. This function implements the second-order Van-der-Pol Oscillator as a system of first-order ODEs. The Van der Pol Oscillator is defined as .. math:: f(t, y) = \begin{pmatrix} y_2 \\ \mu \cdot (1 - y_1^2)y_2 - y_1 \end{pmatrix} for a constant parameter :math:`\mu`. :math:`\mu` determines the stiffness of the problem, where the larger :math:`\mu` is chosen, the more stiff the problem becomes. Default is :math:`\mu = 0.1`. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`2`-dimensional mean vector and :math:`2 \times 2`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float), optional Parameter :math:`\mu` for the Van der Pol Equations Default is :math:`\mu=0.1`. Returns ------- IVP IVP object describing the Van der Pol Oscillator IVP with the prescribed configuration. """ def rhs(t, y): return vanderpol_rhs(t, y, params) def jac(t, y): return vanderpol_jac(t, y, params) return IVP(timespan, initrv, rhs, jac=jac)
def lorenz(timespan, initrv, params=(10.0, 28.0, 8.0 / 3.0)): r"""Initial value problem (IVP) based on the Lorenz system. The Lorenz system is defined through .. math:: f(t, y) = \begin{pmatrix} a(y_2 - y_1) \\ y_1(b-y_3) - y_2 \\ y_1y_2 - cy_3 \end{pmatrix} for some parameters :math:`(a, b, c)`. Default is :math:`(a, b, c)=(10, 28, 2.667)`. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(3, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`3`-dimensional mean vector and :math:`3 \times 3`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float, float, float, float), optional Parameters :math:`(a, b, c)` for the Lorenz system. Default is :math:`(a, b, c)=(10, 28, 2.667)`. Returns ------- IVP IVP object describing the Lorenz system IVP with the prescribed configuration. """ def rhs(t, y): return lor_rhs(t, y, params) def jac(t, y): return lor_jac(t, y, params) return IVP(timespan, initrv, rhs, jac)
def lotkavolterra(timespan, initrv, params=(0.5, 0.05, 0.5, 0.05)): r"""Initial value problem (IVP) based on the Lotka-Volterra model. The Lotka-Volterra (LV) model is defined through .. math:: f(t, y) = \begin{pmatrix} a y_1 - by_1y_2 \\ -c y_2 + d y_1 y_2 \end{pmatrix} for some parameters :math:`(a, b, c, d)`. Default is :math:`(a, b)=(0.5, 0.05, 0.5, 0.05)`. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`2`-dimensional mean vector and :math:`2 \times 2`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float, float, float, float), optional Parameters :math:`(a, b, c, d)` for the Lotka-Volterra IVP. Default is :math:`(a, b, c, d)=(0.5, 0.05, 0.5, 0.05)`. Returns ------- IVP IVP object describing the Lotka-Volterra IVP with the prescribed configuration. """ def rhs(t, y): return lv_rhs(t, y, params) def jac(t, y): return lv_jac(t, y, params) return IVP(timespan, initrv, rhs, jac)
def fitzhughnagumo(timespan, initrv, params=(0.0, 0.08, 0.07, 1.25)): r"""Initial value problem (IVP) based on the FitzHugh-Nagumo model. The FitzHugh-Nagumo (FHN) model is defined through .. math:: f(t, y) = \begin{pmatrix} y_1 - \frac{1}{3} y_1^3 - y_2 + a \\ \frac{1}{d} (y_1 + b - c y_2) \end{pmatrix} for some parameters :math:`(a, b, c, d)`. Default is :math:`(a, b)=(0.0, 0.08, 0.07, 1.25)`. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`2`-dimensional mean vector and :math:`2 \times 2`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. params : (float, float, float, float), optional Parameters :math:`(a, b, c, d)` for the FitzHugh-Nagumo IVP. Default is :math:`(a, b, c, d)=(0.0, 0.08, 0.07, 1.25)`. Returns ------- IVP IVP object describing the FitzHugh-Nagumo IVP with the prescribed configuration. """ def rhs(t, y): return fhn_rhs(t, y, params) def jac(t, y): return fhn_jac(t, y, params) return IVP(timespan, initrv, rhs, jac)
def rigidbody(timespan, initrv): r"""Initial value problem (IVP) for rigid body dynamics without external forces The rigid body dynamics without external forces is defined through .. math:: f(t, y) = \begin{pmatrix} y_2 y_3 \\ -y_1 y_3 \\ -0.51 \cdot y_1 y_2 \end{pmatrix} The ODE system has no parameters. This implementation includes the Jacobian :math:`J_f` of :math:`f`. Parameters ---------- timespan : (float, float) Time span of IVP. initrv : RandomVariable, *(shape=(3, ))* -- Vector-valued RandomVariable that describes the belief over the initial value. Usually it is a Constant (noise-free) or Normal (noisy) Random Variable with :math:`3`-dimensional mean vector and :math:`3 \times 3`-dimensional covariance matrix. To replicate "classical" initial values use the Constant distribution. Returns ------- IVP IVP object describing the rigid body dynamics IVP with the prescribed configuration. """ def rhs(t, y): return rigidbody_rhs(t, y) def jac(t, y): return rigidbody_jac(t, y) return IVP(timespan, initrv, rhs, jac=jac)