Periodic orbits based on orthogonal collocation

We compute Ntst time slices of a periodic orbit using orthogonal collocation. This is implemented in the structure PeriodicOrbitOCollProblem.

Large scale

The current implementation is not yet optimized for large scale problems. This will be improved in the future.

The general method is very well exposed in [Dankowicz],[Doedel] and we adopt the notations of [Dankowicz]. However our implementation is based on [Doedel] because it is more economical (less equations) when it enforces the continuity of the solution.

We look for periodic orbits as solutions $(x(0), T)$ of

\[\dot x = T\cdot F(x),\ x(0)=x(1).\]

We consider a partition of the time domain

\[0=\tau_{1}<\cdots<\tau_{j}<\cdots<\tau_{N_{tst}+1}=1\]

where the points are referred to as mesh points. On each mesh interval $[\tau_j,\tau_{j+1}]$ for $j=1,\cdots,N_{tst}$, we define the affine transformation

\[\tau=\tau^{(j)}(\sigma):=\tau_{j}+\frac{(1+\sigma)}{2}\left(\tau_{j+1}-\tau_{j}\right), \sigma \in[-1,1].\]

The functions $x^{(j)}$ defined on $[-1,1]$ by $x^{(j)}(\sigma) \equiv x(\tau_j(\sigma))$ satisfies the following equation on $[-1,1]$:

\[\dot x^{(j)} = T\frac{\tau_{j+1}-\tau_j}{2}\cdot F(x^{(j)})\tag{$E_j$}\]

with the continuity equation $x^{(j+1)}(-1) = x^{(j)}(1)$.

We now aim at solving $(E_j)$ by using an approximation with a polynomial of degree $m$. Following [Dankowicz], we define a (uniform) partition:

\[-1=\sigma_{1}<\cdots<\sigma_{i}<\cdots<\sigma_{m+1}=1.\]

The points $\tau_{i,j} = \tau^{(i)}(\sigma_j)$ are called the base points: they serve as collocation points.

The associated $m+1$ Lagrange polynomials of degree $m$ are:

\[\mathcal{L}_{i}(\sigma):=\prod_{k=1, k \neq i}^{m+1} \frac{\sigma-\sigma_{k}}{\sigma_{i}-\sigma_{k}}, i=1, \ldots, m+1.\]

We then introduce the approximation $p_j$ of $x^{(j)}$:

\[\mathcal p_j(\sigma)\equiv \sum\limits_{k=1}^{m+1}\mathcal L_k(\sigma)x_{j,k}\]

and the problem to be solved at the nodes $z_l$, $l=1,\cdots,m$:

\[\forall 1\leq l\leq m,\quad 1\leq j\leq N_{tst},\quad \dot p_j(z_l) = T\frac{\tau_{j+1}-\tau_j}{2}\cdot F(p_j(z_l))\tag{$E_j^2$}.\]

The nodes $(z_l)$ are associated with a Gauss–Legendre quadrature.

In order to have a unique solution, we need to remove the phase freedom. This is done by imposing a phase condition.

Phase condition

To ensure uniqueness of the solution to the functional, we add the following phase condition

\[\frac{1}{T} \int_{0}^{T}\left\langle x(s), \dot x_0(s)\right\rangle d s \approx \sum_{j=1}^{N_{tst}}\sum_{i=1}^{m}\omega_i\left\langle x_{i,j}, \phi_{i,j}\right\rangle=0\]

During continuation at step $k$, we use $\frac{1}{T} \int_{0}^{T}\left\langle x(s), \dot x_{k-1}(s)\right\rangle d s$

Interpolation

BifurcationKit.POOCollSolutionType

Structure to encode the solution associated to a functional ::PeriodicOrbitOCollProblem. In particular, this allows to use the collocation polynomials to interpolate the solution. Hence, if sol::POOCollSolution, one can call

sol = BifurcationKit.POOCollSolution(prob_coll, x)
sol(t)

on any time t.

Mesh adaptation

Work in progress.

We are still working on this.

The goal of this method[Russell] is to adapt the mesh $\tau_i$ in order to minimize the error.

Encoding of the functional

The functional is encoded in the composite type PeriodicOrbitOCollProblem. See the link for more information, in particular on how to access the underlying functional, its jacobian...

Floquet multipliers computation

The algorithm is a simplified version of the procedure described in [Fairgrieve]. It boils down to solving a large generalized eigenvalue problem. There is clearly room for improvements here.

The method allows, nevertheless, to detect bifurcations of periodic orbits. It seems to work reasonably well for the tutorials considered here.

Computation with newton

We provide a simplified call to newton to locate the periodic orbits. Compared to the regular newton function, there is an additional option jacobianPO to select one of the many ways to deal with the above linear problem. The default solver jacobianPO is :autodiffDense.

The docs for this specific newton are located at newton.

BifurcationKit.newtonMethod
newton(probPO, orbitguess, options; kwargs...)

This is the Newton Solver for computing a periodic orbit using orthogonal collocation method. Note that the linear solver has to be apropriately set up in options.

Arguments

Similar to newton except that prob is a PeriodicOrbitOCollProblem.

  • prob a problem of type <: PeriodicOrbitOCollProblem encoding the shooting functional G.
  • orbitguess a guess for the periodic orbit.
  • options same as for the regular newton method.

Optional argument

  • jacobian Specify the choice of the linear algorithm, which must belong to (:autodiffDense, ). This is used to select a way of inverting the jacobian dG
    • For :autodiffDense. The jacobian is formed as a dense Matrix. You can use a direct solver or an iterative one using options. The jacobian is formed inplace.

Continuation

We refer to continuation for more information regarding the arguments.

References

  • Dankowicz

    Dankowicz, Harry, and Frank Schilder. Recipes for Continuation. Computational Science and Engineering Series. Philadelphia: Society for Industrial and Applied Mathematics, 2013.

  • Doedel

    Doedel, Eusebius, Herbert B. Keller, and Jean Pierre Kernevez. “NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS.” International Journal of Bifurcation and Chaos 01, no. 04 (December 1991): 745–72.

  • Fairgrieve

    Fairgrieve, Thomas F., and Allan D. Jepson. “O. K. Floquet Multipliers.” SIAM Journal on Numerical Analysis 28, no. 5 (October 1991): 1446–62. https://doi.org/10.1137/0728075.

  • Russell

    Russell, R. D., and J. Christiansen. “Adaptive Mesh Selection Strategies for Solving Boundary Value Problems.” SIAM Journal on Numerical Analysis 15, no. 1 (February 1978): 59–80. https://doi.org/10.1137/0715004.