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Map and Vector Field Transformations

The following methods calculate invariant foliations and invariant manifolds of maps and vectopr fields.

Let us consider the following four-dimensional map

function trivialMAP(z)
    om1 = 0.12
    om2 = 1
    dm1 = 0.002
    dm2 = 0.003
    return [(exp(dm1)*z[4]*(z[1]^3*z[2] + z[4]^2 + z[4]^4) + 2*z[1]*cos(om1) - 2*z[2]*sin(om1))/(2.0*exp(dm1)),
            -0.5*z[2]^3 - z[1]^2*z[2]^2*z[4] - (3*z[2]*z[3]*z[4])/4. + (z[2]*cos(om1) + z[1]*sin(om1))/exp(dm1),
            (-3*z[1]*z[2]^3*z[3])/4.0 + z[3]^5 - (3*z[1]^3*z[2]*z[4])/4.0 + (z[3]*cos(om2) - z[4]*sin(om2))/exp(dm2),
            (z[2]*z[3]^4)/4.0 + (z[2]*z[3]^3*z[4])/2.0 - z[4]^5 + (z[4]*cos(om2) + z[3]*sin(om2))/exp(dm2)]
end

To calculate the invariant foliation corresponding to a two-simensional invariant subspace we use

using FoliationsManifoldsAutoencoders

MF = DensePolyManifold(4, 4, 5)
XF = fromFunction(MF, trivial)

MWt, XWt, MS, XS = iFoliationMAP(MF, XF, [3, 4], [])

To calculate the invariant manifold corresponding to the same subspace we use

MWt, XWt, MR, XR = iManifoldMAP(MF, XF, [3, 4], [])

One can also calculate the frequency and damping curves using

That, Rhat_r = MAPFrequencyDamping(MWt, XWt, MS, XS, amp_max)
r = range(0, domain(That).right, length=1000)
frequency = abs.(That.(r))
damping = -log.(abs.(Rhat_r.(r))) ./ abs.(That.(r))

The relevant functions are the following:

FoliationsManifoldsAutoencoders.iFoliationMAP — Function
MUr, XUr, MSr, XSr, MU, XU, MS, XS = iFoliationMAP(MF::DensePolyManifold, XF, vars, par; order = PolyOrder(MF))

Calculates the invariant foliation of nonlinear map $\boldsymbol{F}$ with respect to selected eigenvectors of the Jacobian $D\boldsymbol{F}(\boldsymbol{0})$.

We solve the invariance equation

\[\boldsymbol{U} \circ \boldsymbol{F} = \boldsymbol{S} \circ \boldsymbol{U}\]

for $\boldsymbol{U}$ and $\boldsymbol{S}$ using polynomial expansion.

The input are

  • MF, XF represent the dense polynomial expansion of function \boldsymbol{F}`.
  • vars is a vector of integers, selecting the eigenvectors of $D\boldsymbol{F}(\boldsymbol{0})$.
  • par are the indices of input variable of $\boldsymbol{F}$, which do not have dynamics and can be treated as parameters.
  • order is the polynomial order for which the calculation is carried out. The default is the same order as the order of MF, XF, but it can be greater for increased accuracy.

The output is

  • MUr, XUr represent $\boldsymbol{U}$ in real coordinates
  • MSr, XSr represent $\boldsymbol{S}$ in real coordinates
  • MU, XU represent $\boldsymbol{U}$ in complex coordinates
  • MS, XS represent $\boldsymbol{S}$ in complex coordinates

Note that it calculates the complex result first, which has the minimal number of parameters, then it transforms the result into real form.

source
FoliationsManifoldsAutoencoders.iFoliationVF — Function
MUr, XUr, MSr, XSr, MU, XU, MS, XS = iFoliationVF(MF::DensePolyManifold, XF, vars, par; order = PolyOrder(MF))

Calculates the invariant foliation of nonlinear vector field $\boldsymbol{F}$ with respect to selected eigenvectors of the Jacobian $D\boldsymbol{F}(\boldsymbol{0})$.

We solve the invariance equation

\[D \boldsymbol{U} \boldsymbol{F} = \boldsymbol{S} \circ \boldsymbol{U}\]

for $\boldsymbol{U}$ and $\boldsymbol{S}$ using polynomial expansion.

The input are

  • MF, XF represent the dense polynomial expansion of function \boldsymbol{F}`.
  • vars is a vector of integers, selecting the eigenvectors of $D\boldsymbol{F}(\boldsymbol{0})$.
  • par are the indices of input variable of $\boldsymbol{F}$, which do not have dynamics and can be treated as parameters.
  • order is the polynomial order for which the calculation is carried out. The default is the same order as the order of MF, XF, but it can be greater for increased accuracy.

The output is

  • MUr, XUr represent $\boldsymbol{U}$ in real coordinates
  • MSr, XSr represent $\boldsymbol{S}$ in real coordinates
  • MU, XU represent $\boldsymbol{U}$ in complex coordinates
  • MS, XS represent $\boldsymbol{S}$ in complex coordinates

Note that it calculates the complex result first, which has the minimal number of parameters, then it transforms the result into real form.

source
FoliationsManifoldsAutoencoders.iManifoldMAP — Function
MWr, XWr, MRr, XRr, MW, XW, MR, XR = iManifoldMAP(MF0::DensePolyManifold, XF0, vars, par; order = PolyOrder(MF0))

Calculates the invariant manifold of nonlinear map $\boldsymbol{F}$ with respect to selected eigenvectors of the Jacobian $D\boldsymbol{F}(\boldsymbol{0})$.

We solve the invariance equation

\[\boldsymbol{W} \circ \boldsymbol{R} = \boldsymbol{F} \circ \boldsymbol{W}\]

for $\boldsymbol{U}$ and $\boldsymbol{S}$ using polynomial expansion.

The input are

  • MF, XF represent the dense polynomial expansion of function \boldsymbol{F}`.
  • vars is a vector of integers, selecting the eigenvectors of $D\boldsymbol{F}(\boldsymbol{0})$.
  • par are the indices of input variable of $\boldsymbol{F}$, which do not have dynamics and can be treated as parameters.
  • order is the polynomial order for which the calculation is carried out. The default is the same order as the order of MF, XF, but it can be greater for increased accuracy.

The output is

  • MWr, XUr represent $\boldsymbol{W}$ in real coordinates
  • MWr, XSr represent $\boldsymbol{R}$ in real coordinates
  • MW, XW represent $\boldsymbol{W}$ in complex coordinates
  • MR, XR represent $\boldsymbol{R}$ in complex coordinates

Note that it calculates the complex result first, which has the minimal number of parameters, then it transforms the result into real form.

source
FoliationsManifoldsAutoencoders.iManifoldVF — Function
MWr, XWr, MRr, XRr, MW, XW, MR, XR = iManifoldVF(MF0::DensePolyManifold, XF0, vars, par; order = PolyOrder(MF0))

Calculates the invariant manifold of nonlinear vector field $\boldsymbol{F}$ with respect to selected eigenvectors of the Jacobian $D\boldsymbol{F}(\boldsymbol{0})$.

We solve the invariance equation

\[D \boldsymbol{W} \boldsymbol{R} = \boldsymbol{F} \circ \boldsymbol{W}\]

for $\boldsymbol{U}$ and $\boldsymbol{S}$ using polynomial expansion.

The input are

  • MF, XF represent the dense polynomial expansion of function \boldsymbol{F}`.
  • vars is a vector of integers, selecting the eigenvectors of $D\boldsymbol{F}(\boldsymbol{0})$.
  • par are the indices of input variable of $\boldsymbol{F}$, which do not have dynamics and can be treated as parameters.
  • order is the polynomial order for which the calculation is carried out. The default is the same order as the order of MF, XF, but it can be greater for increased accuracy.

The output is

  • MWr, XUr represent $\boldsymbol{W}$ in real coordinates
  • MWr, XSr represent $\boldsymbol{R}$ in real coordinates
  • MW, XW represent $\boldsymbol{W}$ in complex coordinates
  • MR, XR represent $\boldsymbol{R}$ in complex coordinates

Note that it calculates the complex result first, which has the minimal number of parameters, then it transforms the result into real form.

source