Periodic orbits based on orthogonal collocation

• Periodic orbits based on orthogonal collocation
• • Introduction
  • Mesh adaptation
  • Jacobians
  • Linear solvers
  • Floquet multipliers computation
  • References

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

The general method is explained in BifurcationKit.jl.

We recall the basics for completeness.

Introduction

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

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

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 $[-1,1]$:

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

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 $z_l$, $l=1,\cdots,m$:

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

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

Mesh adaptation

Mesh adaptation can be turned on like in the case of ODEs.

🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧

🚧 🚧 BELOW IS WORK IN PROGRESS 🚧 🚧

🚧 🚧 🚧 🚧 DO NOT USE YET 🚧 🚧 🚧 🚧

🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧🚧

Jacobians

All jacobians of the ODE case are available.

Linear solvers

All linear solvers of the ODE case are available.

Floquet multipliers computation

This part is specific to delay differential equations. Let's explain it in the simple case

\[\frac{dx(t)}{dt} = F(x(t), x(t-\tau)).\]

The variational equation around a periodic orbit reads

\[\frac{dz(t)}{dt} = A(t)\cdot x(t) +B(t) x(t-\tau).\]

where $A,B$ are periodic functions. The Floquet $\lambda$ exponents are solution to

\[\left(\lambda + \frac{d}{dt}\right) z(t) = A(t)\cdot x(t) + e^{-\lambda\tau} B(t) x(t-\tau)\tag{1}\]

We provide one method(s) to compute the Floquet coefficients.

1. The algorithm (Default) FloquetColl is based on ^[Lust] and it computes an approximation of the monodromy operator from the jacobian matrix of the functional.
2. The algorithm FloquetGEV boils down to solving a large generalized eigenvalue problem based on (1). There is clearly room for improvements here but this can be used to check the results of the previous method.

References