10. Symplectic Maps¶
The celmech.maps module implements a couple of area-preserving maps,
StandardMap and EncounterMap.
10.1. Chirikov Standard Map¶
The Chirikov standard map is a symplectic map that depends on a single parameter, \(K\), defined by the equations
10.2. Encounter Map¶
The encounter map is a symplectic map that approximates the dynamics of a pair of closely-spaced planets. The map depends on three parameters , \(\epsilon,y\), and \(J\). The map is defined by the equations
where
and the \(s_k(y)\) are the resonance amplitude functions given by the
sk() function.
10.3. API¶
- class celmech.maps.EncounterMap(m, J, y0, Nmax=7, mod=True)[source]¶
A class representing the encounter map. The map depends on three parameters, \(\epsilon,y\), and \(J\). The map is defined by the equations
\[\begin{split}\begin{align} x' &= x + \epsilon f(\theta;y) \\ \theta' &= \theta + 2\pi(J-x') \end{align}\end{split}\]By default, the map is defined on the cylinder with the \(\theta\) coordinate taken mod \(2\pi\). The parameter mod_p=True can be set to take the \(p\) coordinate modulo \(2\pi\) as well.
- Parameters
m (float) – Planet-star mass ratio
y (float) – The eccentricity divided by the orbit-crossing eccentricity.
J (float) – Center the map on the \(J\):\(J-1\) MMR. For integer J, the map is centered on a first order MMR. For rational \(J=p/q\), the map is centered on a :math:`q`th order MMR
mod_theta (bool, optional) – If True, the \(\theta\) coordinate is taken modulo \(2\pi\). Default is True
mod_p (bool, optional) – If True, the \(p\) coordinate is taken modulo \(2\pi\). Default is False.
- class celmech.maps.StandardMap(K, mod_theta=True, mod_p=False)[source]¶
A class representing the Chirikov standard map. The map depends on a single parameter, \(K\) and is defind by
\[\begin{split}\begin{align} p' &=& p + K \sin\theta\\ \theta' &=& \theta + p' \end{align}\end{split}\]By default, the map is defined on the cylinder with the \(\theta\) coordinate taken mod \(2\pi\). The parameter mod_p=True can be set to take the \(p\) coordinate modulo \(2\pi\) as well.
- Parameters
- action(pt)[source]¶
Evaluate The action zero-form,
\[\]lambda(theta,w) = 2pileft(frac{w’^2}{2}- frac{epsilon}{2pi} F_beta(theta)right)~,
where \(w' = w - \epsilon \partial_\theta F_\beta(\theta)\). The action zero-form satisfies \(T^*(w d\theta) - w d\theta = d\lambda\) where \(T^*\) is the pullback of the map.
- Parameters
pt (array-like) – the point \((\theta,w)\) at which to evlauate the action.
- Returns
The value of the action zero-form, \(\lambda(\theta,w)\)
- Return type
- inv(x)[source]¶
The inverse mapping
\[\begin{split}\begin{align} \theta &=& p' - \theta' \\ p &=& p' - K \sin\theta \end{align}\end{split}\]- Parameters
x (array-like) – The point \((\theta',p')\)
- Returns
The point \((\theta,p)\)
- Return type
array-like
- inv_partial_derivs(x, Nmax)[source]¶
Get the partial derivatives of the inverse map evaluated at the point x0 up to order Nmax.
- Parameters
x (array-like) – The point at which derivatives are to be evaluated
Nmax (int) – Maximum order of the partial derivatives
- Returns
T – The partial derivatives of the map. Writing the value of the map at a point \((x_1,x_2)\) as \(T(x_1,x_2) = (T_1(x_1,x_2),T_2(x_1,x_2))\), the entry T[i,n,m] stores
\[\frac{\partial^{(n+m)}}{\partial x_1^n \partial x_2^m} T_i\]Note that T[:,0,0] give the value of the map.
- Return type
array, shape (2,Nmax+1,Nmax+1)
- jac(x)[source]¶
Evaluate the Jacobian map at \(x=(\theta,p)\), given by
\[\begin{split}DT(x) = \begin{pmatrix} 1 + K\cos\theta & 1 \\ K\cos\theta & 1 \end{pmatrix}\end{split}\]10. Aruments¶
- xarray
Point(s) at which to evaluate the Jacobian.
- returns
DT – Value of the Jacobain at point x.
- rtype
array
- property mod_theta¶
Is the coordinate \(\theta\) calculated modulo \(2\pi\)?
- partial_derivs(x, Nmax)[source]¶
Get the partial derivatives of the map evaluated at the point x0 up to order Nmax.
- Parameters
x (array-like) – The point at which derivatives are to be evaluated
Nmax (int) – Maximum order of the partial derivatives to return
- Returns
T – The partial derivatives of the map. Writing the value of the map at a point \((x_1,x_2)\) as \(T(x_1,x_2) = (T_1(x_1,x_2),T_2(x_1,x_2))\), the entry T[i,n,m] stores .. math:
\frac{\partial^{(n+m)}}{\partial x_1^n \partial x_2^m} T_i
Note that
T[:,0,0]give the value of the map.- Return type
array, shape (2,Nmax+1,Nmax+1)
- symmetry_lines()[source]¶
Return the symmetry lines of the map.
- Returns
Tuple containing three functions that parameterize the symmetry lines of the map.
- Return type
- with_variational(X, dX)[source]¶
Apply the map along with the tangent map to point plus variationals. In particular,
\[\begin{split}\begin{align} (\theta', w') &=& T(\theta, w) \\ (\delta \theta',\delta w') &=& DT(\theta, w) \cdot (\delta \theta,\delta w)\end{split}\]where \(T\) is the usual map and \(DT\) is the Jacobian of the map.
- Parameters
X (array-like) – The point \(X = (\theta,w)\)
dX (array-like) – The variational vector \((\delta\theta,\delta w)\)
- Returns
X’ (array-like) – The new point
dX’ (array-lke) – The new variationl vector