On the Multiple Shooting Continuation of Periodic Orbits by Newton–krylov Methods

Sánchez, J.; Net, Marta

doi:10.1142/s0218127410025399

Cited by 36 publications

(29 citation statements)

References 39 publications

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“…These solutions might be relevant in organizing the global dynamics [18]. To find them and provide a deeper description of the phase space, continuation methods [19][20][21][22] must be used.…”

Section: Introductionmentioning

confidence: 99%

Continuation and stability of convective modulated rotating waves in spherical shells

2016

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Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E = 10 −4 , and a low Prandtl number fluid Pr = 0.1 in a moderately thick shell of radius ratio η = 0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined.

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Section: Introductionmentioning

confidence: 99%

Continuation and stability of convective modulated rotating waves in spherical shells

2016

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“…This value was used in [36,40,37] to provide examples of the continuation of periodic orbits by multiple shooting, and of invariant tori with two different algorithms, so the main branch of periodic orbits and that of invariant tori were already known. The Euclidean norm of the vector U = (ψ ij , Θ ij , η ij ), containing the values of the three functions ψ, Θ, and η at the mesh of inner collocation points, is plotted versus Ra.…”

Section: Resultsmentioning

confidence: 99%

“…NewtonPicard algorithms were used in [25] and implemented in the package PDECONT, a limited memory Broyden method was applied in [46], and Newton-Krylov techniques were used in [38]. Nontrivial extensions to parallel-shooting [36] and to the calculation of the coefficients of a normal form at a multicritical periodic orbit [41] were later implemented. The continuation of invariant tori was considered first in [40] and then improved with a parallel method in [37].…”

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confidence: 99%

Continuation of Bifurcations of Periodic Orbits for Large-Scale Systems

Net¹,

Sánchez²

2015

SIAM J. Appl. Dyn. Syst.

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Abstract.A methodology to track bifurcations of periodic orbits in large-scale dissipative systems depending on two parameters is presented. It is based on the application of iterative Newton-Krylov techniques to extended systems. To evaluate the action of the Jacobian it is necessary to integrate variational equations up to second order. It is shown that this is possible by integrating systems of dimension at most four times that of the original equations. In order to check the robustness of the method, the thermal convection of a mixture of two fluids in a rectangular domain has been used as a test problem. 1. Introduction. The study of dynamical systems involves, in addition to pure numerical simulations, the computation of invariant manifolds (fixed points, periodic orbits, invariant tori, etc.) and the connections among them (homo-and heteroclinic orbits and heteroclinic chains), the investigation of the stability of these objects, and the examination of their bifurcations when the parameters present in the system are varied. These essential tools help to understand the full dynamics of the system and its dependence on the parameters. The existence of robust continuation and bifurcation packages such as AUTO [10], CONTENT [22], MATCONT [7], etc., allows many of these calculations to be almost routinely performed for moderate-dimensional systems of ordinary differential equations (ODEs). These packages include the continuation of codimension-one bifurcations of fixed points, and even of periodic orbits, and some also include the detection and analysis of codimension-two points. They implement direct solvers for the linear systems involved in the computations, and find the full spectrum when solving eigenvalue problems to study of the stability of the invariant objects.The theory of the extended systems used in these packages to follow bifurcations of fixed points is well developed and can be found in, among others, [42,26,18,53,29,5,16]. The bordered systems for periodic orbits, based on boundary value problems, are analyzed in [11]. In this latter case piecewise collocation in time is used instead of shooting methods.The difficulties in the application of these methodologies to high-dimensional systems, obtained in most cases by discretizing systems of partial differential equations (PDEs), come

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“…1. The multiple shooting version using parallelism was studied in [20]. The extension is not trivial if some speedup is expected.…”

Section: Poincaré Maps and Its Derivativesmentioning

confidence: 99%

“…Newton-Picard algorithms [17] were implemented in the package PDECONT, Broyden method were used in [18], and Newton-Krylov techniques in [19,20]. The computation of two-dimensional unstable manifolds of periodic orbits was developed in [21].…”

Section: Introductionmentioning

confidence: 99%

Numerical continuation methods for large-scale dissipative dynamical systems

Umbría

Net

2016

Eur. Phys. J. Spec. Top.

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Abstract. A tutorial on continuation and bifurcation methods for the analysis of truncated dissipative partial differential equations is presented. It focuses on the computation of equilibria, periodic orbits, their loci of codimension-one bifurcations, and invariant tori. To make it more self-contained, it includes some definitions of basic concepts of dynamical systems, and some preliminaries on the general underlying techniques used to solve non-linear systems of equations by inexact Newton methods, and eigenvalue problems by means of subspace or Arnoldi iterations.

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On the Multiple Shooting Continuation of Periodic Orbits by Newton–krylov Methods

Cited by 36 publications

References 39 publications

Continuation and stability of convective modulated rotating waves in spherical shells

Continuation and stability of convective modulated rotating waves in spherical shells

Continuation of Bifurcations of Periodic Orbits for Large-Scale Systems

Numerical continuation methods for large-scale dissipative dynamical systems

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