# Periodic orbits based on orthogonal collocation

- 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`

.

The current implementation is optimized for ODE and for large scale problems for which the jacobian is sparse.

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

\[\dot x = T\cdot F(x),\ x(0)=x(1)\in\mathbb R^n.\]

We focus on the differential equality and 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.

## Number of unknowns

Putting the period unknown aside, we have to find the $x_{j,k}$ which gives $n\times N_{tst}\times (m+1)$ unknowns.

The equations $E_j^2$ provides $n\times N_{tst}\times m$ plus the $(N_{tst}-1)\times n$ equations for the continuity equations. This makes a total of $(N_{tst}-1)\times m\times n+n\times N_{tst}\times m = n[N_{tst}(m+1)-1]$ equations to which we add the $n$ equations for the periodic boundary condition. In total, we have

\[n\times N_{tst}\times (m+1)\]

equations which matches the number of unknowns.

## Phase condition

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

\[\frac{1}{T} \int_{0}^{T}\left\langle x(s), \dot x_0(s)\right\rangle d s =0\]

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

## Discretization of the BVP and jacobian

We only focus on the differential part. Summing up, we obtained the following equations for the $x_{j,l}\in\mathbb R^n$:

\[\sum\limits_{k=1}^{m+1}\mathcal L_k'(z_l)x_{j,k} = F\left(\sum\limits_{k=1}^{m+1}\mathcal L_k(z_l)x_{j,k}\right)\]

The jacobian in the case $m=2$ is given by:

\[\begin{array}{llllllll} x_{0,0} & x_{0,1} & x_{1,0} & x_{1,1} & x_{2,0} & x_{2,1} & x_{3,0} &\quad \mathbf{T} \end{array} \\ \left(\begin{array}{llllllll} H_{0,0}^0 & H_{0,1}^0 & H_{1,0}^0 & & & & & * \\ H_{0,0}^1 & H_{0,1}^1 & H_{1,0}^1 & & & & & * \\ & & H_{1,0}^0 & H_{1,1}^0 & H_{2,0}^0 & & & * \\ & & H_{1,0}^1 & H_{1,1}^1 & H_{2,0}^1 & & & * \\ & & & & H_{2,0}^0 & H_{2,1}^0 & H_{3,0}^0 & * \\ & & & & H_{2,0}^1 & H_{2,1}^1 & H_{3,0}^1 & * \\ & & & & & & & * \\ I & & & & & & -I & * \\ * & * & * & * & * & * & * & * \end{array}\right)\]

where

\[H_{k,l}^{l_2} = \mathcal L'_{l_2,l}\cdot I_n - T\frac{\tau_{j+1}-\tau_j}{2}\cdot\mathcal L_{l_2,l}\cdot dF\left(x_{k,l}\right)\in\mathbb R^n.\]

## Interpolation

`BifurcationKit.POSolution`

— TypeStructure to encode the solution associated to a functional like `::PeriodicOrbitOCollProblem`

or `::ShootingProblem`

. In the particular case of `::PeriodicOrbitOCollProblem`

, this allows to use the collocation polynomials to interpolate the solution. Hence, if `sol::POSolution`

, one can call

```
sol = BifurcationKit.POSolution(prob_coll, x)
sol(t)
```

on any time `t`

.

## Mesh adaptation

The goal of this method^{[Russell]} is to adapt the mesh $\tau_i$ in order to minimize the error. It is particularly helpful near homoclinic solutions where the period diverge. It can also be useful in order to use a smaller $N_{tst}$.

## 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...

## Jacobian and linear solvers

We provide many different linear solvers to take advantage of the formulations or the dimensionality. These solvers are available through the argument `jacobian`

in the constructor of `PeriodicOrbitOCollProblem`

. For example, you can pass `jacobian = FullSparse()`

. Note that all the internal linear solvers and jacobians are set up automatically so you don't need to do anything. However, for the sake of explanation, we detail how this works.

### 1. `DenseAnalytical()`

The jacobian is computed with an analytical formula, works for dense matrices. This is the default algorithm.

### 2. `AutoDiffDense()`

The jacobian is computed with automatic differentiation, works for dense matrices. Can be used for debugging.

### 3. `FullSparse()`

The jacobian is computed with an analytical formula, works for sparse matrices.

### 3. `FullSparseInplace()`

The sparse jacobian is computed in place, limiting memory allocations, with an analytical formula when the sparsity of the jacobian of the vector field is constant. This is much faster than `FullSparse()`

.

## Floquet multipliers computation

We provide two methods to compute the Floquet coefficients.

- The algorithm (Default)
`FloquetColl`

is based on the condensation of parameters described in^{[Doedel]}. It is the fastest method. - The algorithm
`FloquetCollGEV`

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 but this can be used to check the results of the previous method.

These methods allow to detect bifurcations of periodic orbits. It seems to work reasonably well for the tutorials considered here. However they may be imprecise^{[Lust]}.

- The state of the art method is based on a Periodic Schur decomposition. It is available through the package PeriodicSchurBifurcationKit.jl. For more information, have a look at
`FloquetPQZ`

.

## Computation with `newton`

`BifurcationKit.newton`

— Method```
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. - For
`DenseAnalytical()`

Same as for`AutoDiffDense`

but the jacobian is formed using a mix of AD and analytical formula.

- For

We provide a simplified call to `newton`

to locate the periodic orbits. `newton`

will look for `prob.jacobian`

in order to select the requested way to compute the jacobian.

The docs for this specific `newton`

are located at `newton`

.

## Continuation

We refer to `continuation`

for more information regarding the arguments. `continuation`

will look for `prob.jacobian`

in order to select the requested way to compute the jacobian.

## 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.

- Lust
Lust, Kurt. “Improved Numerical Floquet Multipliers.” International Journal of Bifurcation and Chaos 11, no. 09 (September 2001): 2389–2410. https://doi.org/10.1142/S0218127401003486.