A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

Berg, Jan Bouwe van den; Queirolo, Elena

doi:10.3934/jcd.2021004

Cited by 10 publications

(4 citation statements)

References 26 publications

(71 reference statements)

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“…On the other hand, mathematically rigorous computer-assisted methods for studying continuous branches of periodic orbits are by now quite advanced. This is true even in the case of DDEs, and we refer for example to the work of [41,42,43,44,45,46]. Moreover, methods of computer-assisted proof have been developed for proving the existence of, and continuing through, a number of infinite dimensional bifurcations.…”

Section: Discussionmentioning

confidence: 99%

From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

Calleja,

García-Azpeitia,

Lessard

et al. 2020

Preprint

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The present work studies the continuation class of the regular n-gon solution of the n-body problem. For odd numbers of bodies between n = 3 and n = 15 we apply one parameter numerical continuation algorithms to the energy/frequency variable, and find that the figure eight choreography can be reached starting from the regular n-gon. The continuation leaves the plane of the n-gon, and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the n-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions, and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the n-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here we conjecture that the n-gon and the eight are in the same continuation class for all odd numbers of bodies.Keywords n-body problem • choreographies • numerical continuation • delay differential equations

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Section: Discussionmentioning

confidence: 99%

From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

Calleja,

García-Azpeitia,

Lessard

et al. 2020

Preprint

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“…Orbits can be proven in the vicinity of (and at) Hopf bifurcations, whether these are non-degenerate or degenerate. The first major release of the library BiValVe (Bifurcation Validation Venture, [4]) is being made in conjunction with the present work, and builds on an earlier version of the code associated to the work [29,30]. Some non-polynomial delay differential equations can be handled using the polynomial embedding technique.…”

Section: Contributions and Applicationsmentioning

confidence: 99%

“…• in the 1 ν (C n+m ) components: convolution polynomials of the form (29) involving Fourier components coming from the set…”

Section: The Bound Z 21mentioning

confidence: 99%

Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

Church¹,

Queirolo²

2022

Preprint

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“…The first extension is to let the vector field explicitly depend on a parameter and to perform rigorous (pseudo-arclength) continuation of connecting orbits. This involves a relatively straightforward application of the uniform contraction principle and a slight modification of the estimates developed in this paper (see [7,25,28,31] for instance). Furthermore, in order to carry out continuation efficiently, we need to develop algorithms (heuristics) which automatically determine near-optimal parameter values for the validation of the charts on the local (un)stable manifolds and the connecting orbit.…”

Section: Extensions and Future Workmentioning

confidence: 99%

Validated computations for connecting orbits in polynomial vector fields

Berg

Sheombarsing

2020

Indagationes Mathematicae

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In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simultaneously solves for the local (un)stable manifolds and the orbit which connects these. We obtain explicit rigorous control on the distance between the numerical approximation and the heteroclinic orbit. Transversality of the stable and unstable manifolds is also proven.

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A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

Cited by 10 publications

References 26 publications

From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

From the Lagrange polygon to the figure eight I: Numerical evidence extending a conjecture of Marchal

Computer-assisted proofs of Hopf bubbles and degenerate Hopf bifurcations

Validated computations for connecting orbits in polynomial vector fields

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