Periodic orbits based on the shooting method

A set of shooting algorithms is provided which are called either Simple Shooting (SS) if a single shooting is used and Multiple Shooting (MS) otherwise. For the exposition, we follow the PhD thesis[Lust]

We aim at finding periodic orbits for the Cauchy problem

\[\tag{1} \frac{d x}{d t}=f(x)\]

and we write $\phi^t(x_0)$ the associated flow (or semigroup of solutions).

Tip about convenience functions

For convenience, we provide some functions plotPeriodicShooting for plotting, getAmplitude (resp. getMaximum) for getting the amplitude (resp. maximum) of the solution encoded by a shooting problem. See the tutorials for examples of use.

Standard Shooting

Simple shooting

A periodic orbit is found when we have a couple $(x, T)$ such that $\phi^T(x) = x$ and the trajectory is non constant. Therefore, we want to solve the equations $G(x,T)=0$ given by

\[\tag{SS} \begin{array}{l}{\phi^T(x)-x=0} \\ {s(x,T)=0}\end{array}.\]

The section $s(x,T)=0$ is a phase condition to remove the indeterminacy of the point on the limit cycle.

Multiple shooting

This case is similar to the previous one but more sections are used. To this end, we partition the unit interval with $m+1$ points $0=s_{0}<s_{1}<\cdots<s_{m-1}<s_{m}=1$ and consider the equations $G(x_1,\cdots,x_m,T)=0$

\[\begin{aligned} \phi^{\delta s_1T}(x_{1})-x_{2} &=0 \\ \phi^{\delta s_2T}(x_{2})-x_{3} &=0 \\ & \vdots \\ \phi^{\delta s_{m-1}T}(x_{m-1})-x_{m} &=0 \\ \phi^{\delta s_mT}(x_{m})-x_{1} &=0 \\ s(x_{1}, T) &=0. \end{aligned}\tag{MS}\]

where $\delta s_i:=s_{i+1}-s_i$. The Jacobian of the system of equations w.r.t. $(x,T)$ is given by

\[\mathcal{J}=\left(\begin{array}{cc}{\mathcal J_c} & {\partial_TG} \\ {\star} & {d}\end{array}\right)\]

where the cyclic matrix $\mathcal J_c$ is

\[\mathcal J_c := \left(\begin{array}{ccccc} {M_{1}} & {-I} & {} & {} \\ {} & {M_{2}} & {-I} & {}\\ {} & {} & {\ddots} & {-I}\\ {-I} & {} & {} & {M_{m}}\\ \end{array}\right)\]

and $M_i=\partial_x\phi^{\delta s_i T}(x_i)$.

Section

The periodic orbits solutions of (SS) or (MS) are not uniquely defined because of the phase invariance. A section $s(x,T)=0$ (resp. $s(x_1,T)=0$) for (SS) (resp. (MS)) must be provided. The default is the same for both $ s(x,T) = T\cdot \langle x-x_\pi, \phi\rangle.$

Encoding of the functional

The functional is encoded in the composite type ShootingProblem. In particular, the user can pass its own time stepper or one can use the different ODE solvers in DifferentialEquations.jl which makes it very easy to choose a solver tailored for the a specific problem. See the link ShootingProblem for more information ; for example on how to access the underlying functional, its jacobian...

Poincaré shooting

The algorithm is based on the one described in Newton–Krylov Continuation of Periodic Orbits for Navier–Stokes Flows., Sánchez, J., M. Net, B. Garcı́a-Archilla, and C. Simó (2004) and Matrix-Free Continuation of Limit Cycles for Bifurcation Analysis of Large Thermoacoustic Systems. Waugh, Iain, Simon Illingworth, and Matthew Juniper (2013).

We look for periodic orbits solutions of (1) using the hyperplanes $\Sigma_i=\{x\ / \ \langle x-x^c_{I}, n_i\rangle=0\}$ for $i=1,\cdots,M$, centered on $x^c_i$, which intersect transversally an initial periodic orbit guess. We write $\Pi_i:\Sigma_i\to\Sigma_{mod(i+1,M)}$, the Poincaré return map to $\Sigma_{mod(i+1,M)}$. The main idea of the algorithm is to use the fact that the problem is $(N-1)\cdot M$ dimensional if $x_i\in\mathbb R^N$ because each $x_i$ lives in $\Sigma_i$. Hence, one has to constrain the unknowns to these hyperplanes otherwise the Newton algorithm does not converge well.

Thus we need to parametrize these hyperplanes.

To this end, we introduce the projection operator $R_i:\mathbb R^N\to \mathbb R^{N-1}$ such that

\[R_{i}\left(x_{1}, x_{2}, \ldots, x_{k_i-1}, x_{k_i}, x_{k_i+1}, \ldots, x_{N}\right)=\left(x_{1}, x_{2}, \ldots, x_{k_i-1}, x_{k_i+1}, \ldots, x_{N}\right)\]

where $k_i=argmax_p |n_{i,p}|$. The inverse operator $E_i:\mathbb R^{N-1}\to\Sigma_i$ is defined as (where $\bar x:=R_i(x)$)

\[E_{i}(\bar x) := E_{i}\left(x_{1}, x_{2}, \ldots, x_{k_i-1}, x_{k_i+1}, \ldots, x_{N}\right)= \left(x_{1}, x_{2}, \ldots, x_{k_i-1}, x^c_{i,k_i}-\frac{\bar{n}_i \cdot\left(\overline{x}-\overline{x}^c_{i}\right)}{n_{i,k_i}}, x_{k_i+1}, \ldots, x_{N}\right).\]

We note that $R_i\circ E_i = I_{\mathbb R^{N-1}}$ and $E_i\circ R_i = I_{\mathbb R^{N}}$.

We then look for solutions of the following problem:

\[\begin{aligned} \bar x_1 - R_M\Pi_M(E_M(\bar x_M))&=0 \\ \bar x_2 - R_1\Pi_1(E_i(\bar x_1))&=0 \\ & \vdots \\ \bar x_M - R_{M-1}\Pi_{M-1}(E_{M-1}(\bar x_{M-1}))&=0. \end{aligned}\]

Encoding of the functional

The functional is encoded in the composite type PoincareShootingProblem. In particular, the user can pass their own time stepper or he can use the different ODE solvers in DifferentialEquations.jl which makes it very easy to choose a tailored solver: the partial Poincaré return maps are implemented using callbacks. See the link PoincareShootingProblem for more information, in particular on how to access the underlying functional, its jacobian...

Floquet multipliers computation

Standard shooting

The Floquet multipliers are computed as the eigenvalues of the monodromy matrix $M=M_M\cdots M_1$.

Unlike the case with Finite differences, the matrices $M_i$ are not sparse.

Poincaré shooting

The (non trivial) Floquet exponents are eigenvalues of the Poincaré return map $\Pi:\Sigma_1\to\Sigma_1$. We have $\Pi = \Pi_M\circ\Pi_{M-1}\circ\cdots\circ\Pi_2\circ\Pi_1$. Its differential is thus

\[d\Pi(x)\cdot h = d\Pi_M(x_{M})d\Pi_{M-1}(x_{M-1})\cdots d\Pi_1(x_1)\cdot h\]

Numerical method

A not very precise algorithm for computing the Floquet multipliers is provided. The method, dubbed Quick and Dirty (QaD), is not numerically very precise for large / small Floquet exponents.

It amounts to computing the eigenvalues of $M=M_M\cdots M_1$ (resp. $d\Pi$) for the Standard (resp. Poinncaré) Shooting.

The method allows, nevertheless, to detect bifurcations of periodic orbits. It seems to work reasonably well for the tutorials considered here. For more information, have a look at FloquetQaD.

Algorithm

A more precise algorithm, based on the periodic Schur decomposition will be provided in the future.

Computation with newton

We provide a simplified call to newton to locate the periodic orbit. Have a look at the tutorial Continuation of periodic orbits (Standard Shooting) for a simple example on how to use the above methods.

The docs for this specific newton are located at newton.

Computation with newton and deflation

We also provide a simplified call to newton to locate the periodic orbit with a deflation operator:

BifurcationKit.newtonMethod
newton(prob, orbitguess, options; lens, kwargs...)

This is the Newton-Krylov Solver for computing a periodic orbit using (Standard / Poincaré) Shooting method. Note that the linear solver has to be apropriately set up in options.

Arguments

Similar to newton except that prob is either a ShootingProblem or a PoincareShootingProblem. These two problems have specific options to be tuned, we refer to their link for more information and to the tutorials.

  • prob a problem of type <: AbstractShootingProblem encoding the shooting functional G.
  • orbitguess a guess for the periodic orbit. See ShootingProblem and See PoincareShootingProblem for information regarding the shape of orbitguess.
  • par parameters to be passed to the functional
  • options same as for the regular newton method.

Optional argument

  • jacobian Specify the choice of the linear algorithm, which must belong to [:autodiffMF, :MatrixFree, :autodiffDense, :autodiffDenseAnalytical, :FiniteDifferences]. This is used to select a way of inverting the jacobian dG
    • For :MatrixFree, we use an iterative solver (e.g. GMRES) to solve the linear system. The jacobian was specified by the user in prob.
    • For :autodiffMF, we use iterative solver (e.g. GMRES) to solve the linear system. We use Automatic Differentiation to compute the (matrix-free) derivative of x -> prob(x, p).
    • For :autodiffDense. Same as for :autodiffMF but the jacobian is formed as a dense Matrix. You can use a direct solver or an iterative one using options.
    • For :FiniteDifferencesDense, same as for :autodiffDense but we use Finite Differences to compute the jacobian of x -> prob(x, p) using the δ = 1e-8 which can be passed as an argument.
    • For :autodiffDenseAnalytical. Same as for :autodiffDense but the jacobian is using a mix of AD and analytical formula.
    • For :FiniteDifferences, use Finite Differences to compute the jacobian of x -> prob(x, p) using the δ = 1e-8 which can be passed as an argument.

and

newton(prob::BifurcationKit.AbstractShootingProblem,
				orbitguess::vectype,
				defOp::DeflationOperator{Tp, Tdot, T, vectype},
				options::NewtonPar{T, S, E};
				lens::Union{Lens, Nothing} = nothing,
				kwargs...,
			) where {T, Tp, Tdot, vectype, S, E}

Continuation

Have a look at the Continuation of periodic orbits (Standard Shooting) example for the Brusselator.

The docs for this specific newton are located at continuation.

BifurcationKit.continuationMethod
continuation(probPO, orbitguess, alg, contParams, linearAlgo; δ, kwargs...)

This is the continuation method for computing a periodic orbit using a (Standard / Poincaré) Shooting method.

Arguments

Similar to continuation except that probPO is either a ShootingProblem or a PoincareShootingProblem. By default, it prints the period of the periodic orbit.

Optional argument

  • δ = 1e-8 used for finite differences
  • jacobian Specify the choice of the linear algorithm, which must belong to [:autodiffMF, :MatrixFree, :autodiffDense, :autodiffDenseAnalytical, :FiniteDifferences]. This is used to select a way of inverting the jacobian dG
    • For :MatrixFree, we use an iterative solver (e.g. GMRES) to solve the linear system. The jacobian was specified by the user in prob.
    • For :autodiffMF, we use iterative solver (e.g. GMRES) to solve the linear system. We use Automatic Differentiation to compute the (matrix-free) derivative of x -> prob(x, p).
    • For :autodiffDense. Same as for :autodiffMF but the jacobian is formed as a dense Matrix. You can use a direct solver or an iterative one using options.
    • For :FiniteDifferencesDense, same as for :autodiffDense but we use Finite Differences to compute the jacobian of x -> prob(x, p) using the δ = 1e-8 which can be passed as an argument.
    • For :autodiffDenseAnalytical. Same as for :autodiffDense but the jacobian is using a mix of AD and analytical formula.
    • For :FiniteDifferences, use Finite Differences to compute the jacobian of x -> prob(x, p) using the δ = 1e-8 which can be passed as an argument.

References

  • Lust

    Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations, Lust Kurt, 1997.