Unstable manifold computations for the two-dimensional plane Poiseuille flow

Casas, Pablo S.; Jorba, Àngel

doi:10.1007/s00162-004-0138-0

Cited by 2 publications

(1 citation statement)

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“…The basic algorithm to compute periodic orbits [22], using Newton-Krylov methods, can be applied in the frame of reference at which the quasiperiodic modulation becomes a periodic orbit. This method, but using direct linear solvers, was used, for instance, in [51] for the plane Poiseuille flow (see [43] for more examples). In this case the size of the systems to solve is the same as for the calculation of the waves plus one unit.…”

Section: Discussionmentioning

confidence: 99%

Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation

2013

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A methodology to compute azimuthal waves, appearing in thermal convection of a pure fluid contained in a rotating spherical shell, and to study their stability is presented. It is based on continuation, Newton-Krylov, and Arnoldi methods. An application to the study of a double-Hopf bifurcation of the basic state is shown for Ekman and Prandtl numbers E = 10 −4 and σ = 0.1, respectively, radius ratios η ∈ [0.32,0.35], Rayleigh numbers R ∈ [1.8 × 10 5 ,6 × 10 5 ], and nonslip and perfectly conducting boundary conditions. The knowledge of the bifurcation diagrams, including the unstable solutions, allows one to understand the coexistence of stable thermal Rossby waves of different azimuthal wave numbers at some parameter regions, and the origin of some new intermittent solutions found, as trajectories close to heteroclinic chains. Moreover, the structure of the eigenfunctions at the secondary bifurcations explains the existence of the amplitude and shape modulated waves.

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Section: Discussionmentioning

confidence: 99%