def get_index_from_key(key, j, n):
r"""
Return first occurrence of given key over a Kronecker power.
INPUT:
- ``key`` -- list or tuple, key corresponding to the exponent vector in the
Kronecker power
- ``j`` -- integer, order of the Kronecker power
- ``n`` -- integer, number of dimensions
NOTES:
- We assume `n \geq 2`. Notice that if `n=1`, we would return always that ``first_occurence = 0``.
EXAMPLES:
Take `x^{[2]}` for `x=(x_1, x_2)` and compute retrive the first ocurrence of
the given key::
sage: from carlin.transformation import get_index_from_key
sage: get_index_from_key([1, 1], 2, 2)
1
"""
x = polygens(QQ, ['x' + str(1 + k) for k in range(n)])
x_power_j = kron_power(x, j)
for i, monomial in enumerate(x_power_j):
if (list(monomial.dict().keys()[0]) == key):
first_occurence = i
break
return first_occurence
def cubic_scalar(a=1, b=1):
r"""
A scalar ODE with a cubic term.
It is defined as:
.. MATH::
x'(t) = ax(t) + bx(t)^3
where `a` and `b` are paremeters of the ODE.
EXAMPLES::
sage: from carlin.library import cubic_scalar
sage: C = cubic_scalar(-1, 1)
sage: C.funcs()
[x0^3 - x0]
Compute the Carleman embedding truncated at order `N=4`::
sage: from carlin.transformation import get_Fj_from_model, truncated_matrix
sage: Fj = get_Fj_from_model(C.funcs(), C.dim(), C.degree())
sage: matrix(truncated_matrix(4, *Fj, input_format="Fj_matrices").toarray())
[-1.0 0.0 1.0 0.0]
[ 0.0 -2.0 0.0 2.0]
[ 0.0 0.0 -3.0 0.0]
[ 0.0 0.0 0.0 -4.0]
"""
# define the vector of symbolic variables
x = polygens(QQ, ['x0'])
f = [None] * 1
f[0] = a * x[0] + b * x[0]**3
return PolynomialODE(f, n=1, k=3)
def get_key_from_index(i, j, n):
r"""
Return multi-index of Kronecker power given an index and the order.
INPUT:
- ``i`` -- integer, index in the canonial Kronecker power enumeration
- ``j`` -- integer, order of the Kronecker power
- ``n`` -- integer, number of dimensions
EXAMPLES:
Take `x^{[2]}` for `x=(x_1, x_2)` and compute the exponent vector of the element
in position `1`::
sage: from carlin.transformation import get_key_from_index
sage: get_key_from_index(1, 2, 2)
[1, 1]
"""
x = polygens(QQ, ['x' + str(1 + k) for k in range(n)])
x_power_j = kron_power(x, j)
d = x_power_j[i].dict()
return list(d.items()[0][0])
def chen_seven_dim(u=0):
r"""
This is a seven-dimensional nonlinear system of quadratic order.
It appears as ``'example_nonlinear_reach_04_sevenDim_nonConvexRepr.m'`` in
the tool CORA 2016, in the examples for continuous nonlinear systems.
.. NOTE:
There is an independent term, `u`, in the fourth equation with value `2.0` that has
been neglected here for convenience (hence we take `u=0` by default).
"""
# dimension of state-space
n = 7
# vector of variables
x = polygens(QQ, ['x' + str(i) for i in range(n)])
f = [None] * n
f[0] = 1.4 * x[2] - 0.9 * x[0]
f[1] = 2.5 * x[4] - 1.5 * x[1]
f[2] = 0.6 * x[6] - 0.8 * x[2] * x[1]
f[3] = -1.3 * x[3] * x[2] + u
f[4] = 0.7 * x[0] - 1.0 * x[3] * x[4]
f[5] = 0.3 * x[0] - 3.1 * x[5]
f[6] = 1.8 * x[5] - 1.5 * x[6] * x[1]
return PolynomialODE(f, n, k=2)
def biomodel_2():
r"""
This is a nine-dimensional polynomial ODE used as benchmark model in
`the Flow star tool <https://ths.rwth-aachen.de/research/projects/hypro/biological-model-ii/>`_.
The model is adapted from E. Klipp, R. Herwig, A. Kowald, C. Wierling, H. Lehrach.
Systems Biology in Practice: Concepts, Implementation and Application. Wiley-Blackwell, 2005.
"""
# dimension of state-space
n = 9
# vector of variables
x = polygens(QQ, ['x' + str(i) for i in range(n)])
f = [None] * n
f[0] = 3 * x[2] - x[0] * x[5]
f[1] = x[3] - x[1] * x[5]
f[2] = x[0] * x[5] - 3 * x[2]
f[3] = x[1] * x[5] - x[3]
f[4] = 3 * x[2] + 5 * x[0] - x[4]
f[5] = 5 * x[4] + 3 * x[2] + x[3] - x[5] * (x[0] + x[1] + 2 * x[7] + 1)
f[6] = 5 * x[3] + x[1] - 0.5 * x[6]
f[7] = 5 * x[6] - 2 * x[5] * x[7] + x[8] - 0.2 * x[7]
f[8] = 2 * x[5] * x[7] - x[8]
return PolynomialODE(f, n, k=2)
def arrowsmith_and_place_fig_3_5e_page_79():
r"""
Nonlinear two-dimensional system with an hyperbolic fixed point.
It is defined as:
.. MATH::
\begin{aligned}
x' &= x^2+(x+y)/2 \\
y' &= (-x+3y)/2
\end{aligned}
Taken from p. 79 of the book by Arrowsmith and Place, Dynamical Systems:
Differential Equations, maps and chaotic behaviour.
"""
# dimension of state-space
n = 2
# vector of variables
x = polygens(QQ, ['x' + str(i) for i in range(n)])
# ODE and order k=2
f = [x[0]**2 + (x[0] + x[1]) / 2, (-x[0] + 3 * x[1]) / 2]
return PolynomialODE(f, n, k=2)
def quadratic_scalar(a=1, b=1):
r"""
A scalar ODE with a quadratic term.
It is defined as:
.. MATH::
x'(t) = ax(t) + bx(t)^2
where `a` and `b` are paremeters of the ODE.
EXAMPLES::
sage: from carlin.library import quadratic_scalar
sage: Q = quadratic_scalar(); Q
A Polynomial ODE in n = 1 variables
sage: Q.funcs()
[x0^2 + x0]
Compute the Carleman embedding truncated at order `N=4`::
sage: from carlin.transformation import get_Fj_from_model, truncated_matrix
sage: Fj = get_Fj_from_model(Q.funcs(), Q.dim(), Q.degree())
sage: matrix(truncated_matrix(4, *Fj, input_format="Fj_matrices").toarray())
[1.0 1.0 0.0 0.0]
[0.0 2.0 2.0 0.0]
[0.0 0.0 3.0 3.0]
[0.0 0.0 0.0 4.0]
"""
# define the vector of symbolic variables
x = polygens(QQ, ['x0'])
f = [None] * 1
f[0] = a * x[0] + b * x[0]**2
return PolynomialODE(f, 1, 2)
def vanderpol(mu=1, omega=1):
r"""
The Van der Pol oscillator is a non-conservative system with non-linear damping.
It is defined as:
.. MATH::
\begin{aligned}
x' &= y \\
y' &= -\omega^2 x - (x^2 - 1) \mu y
\end{aligned}
where `\omega` is the natural frequency and `\mu` is the damping parameter.
For additional information see the :wikipedia:`Van_der_Pol_oscillator`.
EXAMPLES::
sage: from carlin.library import vanderpol
sage: vanderpol(SR.var('mu'), SR.var('omega')).funcs()
[x1, -omega^2*x0 - (x0^2 - 1)*mu*x1]
"""
# dimension of state-space
n = 2
# define the vector of symbolic variables
x = polygens(QQ, ['x' + str(i) for i in range(n)])
# vector field (n-dimensional)
f = [None] * n
f[0] = x[1]
f[1] = -omega**2 * x[0] + mu * (1 - x[0]**2) * x[1]
# the order is k=3
return PolynomialODE(f, n, k=3)