Analytic enclosure of the fundamental matrix solution

Castelli, Roberto; Lessard, Jean‐Philippe; James, J. D. Mireles

doi:10.1007/s10492-015-0114-6

Cited by 5 publications

(3 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the a-posteriori theory of [YGdlL] requires bounds on solutions of some variational equations associated with the periodic orbit, and solutions of the variational equation are not periodic. This problem could be addressed in Fourier space using the Floquet methods developed [CLMJ15]. However, in the present work we found it expedient to solve the variational equations directly using a Chebyshev scheme.…”

Section: Introductionmentioning

confidence: 84%

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

Gimeno¹,

Lessard²,

James³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

A computer-assisted argument is given, which provides existence proofs for periodic orbits in state-dependent delayed perturbations of ordinary differential equations (ODEs). Assuming that the unperturbed ODE has an isolated periodic orbit, we introduce a set of polynomial inequalities whose successful verification leads to the existence of periodic orbits in the perturbed delay equation. We present a general algorithm, which describes a way of computing the coefficients of the polynomials and optimizing their variables so that the polynomial inequalities are satisfied. The algorithm uses the tools of validated numerics together with Chebyshev series expansion to obtain the periodic orbit of the ODE as well as the solution of the variational equations, which are both used to compute rigorously the coefficients of the polynomials. We apply our algorithm to prove the existence of periodic orbits in a state-dependent delayed perturbation of the van der Pol equation.

show abstract

Section: Introductionmentioning

confidence: 84%

Persistence of Periodic Orbits under State-dependent Delayed Perturbations: Computer-assisted Proofs

Gimeno¹,

Lessard²,

James³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In practice, the unknowns are the periodic functions a α (θ) or, equivalently, the sequence of Fourier coefficients {a α,k } k of any α. We insert (11) into the invariance equation and expand f (P(θ, σ)) as its Taylor series with respect to σ at σ = 0. We obtain…”

Section: 3mentioning

confidence: 99%

“…Lastly, we remark one more time that the parameterisation method, in particular in the form of a Fourier-Taylor parameterisation, is well suited for rigorous computational analysis. The aposteriori error evaluation together with a contraction argument on the N -tail space may lead to a computer assisted enclosure of the local invariant manifold, see [10,11]. Thus, a projected boundary value problem can be set up to prove existence of connecting orbits.…”

Section: Polynomial Vector Field Formulationmentioning

confidence: 99%

Efficient representation of invariant manifolds of periodic orbits in the CRTBP

Castelli¹

2019

Discrete &Amp; Continuous Dynamical Systems - B

Self Cite

View full text Add to dashboard Cite

This paper deals with a methodology for defining and computing an analytical Fourier-Taylor series parameterisation of local invariant manifolds associated to periodic orbits of polynomial vector fields. Following the Parameterisation Method, the functions involved in the series result by solving some linear non autonomous differential equations. Exploiting the Floquet normal form decomposition, the time dependency is removed and the differential problem is rephrased as an algebraic system to be solved for the Fourier coefficients of the unknown periodic functions. The procedure leads to an efficient and fast computational algorithm. Motivated by mission design purposes, the technique is applied in the framework of the Circular Restricted Three Body problem and the parameterisation of local invariant manifolds for several halo orbits is computed and discussed.

show abstract