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.

We aim at finding periodic orbits for the Cauchy problem

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

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

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 $G(x_1,\cdots,x_M,T)=0$

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

where the cyclic matrix $\mathcal J_c$ is

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

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 $\Sigma_i=\{x\ / \ n_i\cdot(x-x^c_{i})=0\}$ 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.

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

where $k_i=argmax_p |n_{i,p}|$. The inverse operator is defined as (where $\bar x:=R_i(x)$)

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:

Computation of Floquet exponents

The (non trivial) Floquet exponents are eigenvalues of the Poincare 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

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

These are the eigenvalues of $M_M\cdots M_1$.

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

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 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.

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:

newton(prob:: AbstractShootingProblem, orbitguess, par0, options::NewtonPar; kwargs...)

and

newton(prob:: AbstractShootingProblem, orbitguess, par0, options::NewtonPar, defOp::DeflationOperator; kwargs...)

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.