Lyapunov–Schmidt Method for Dynamical Systems

Vanderbauwhede, André

doi:10.1007/978-0-387-30440-3_314

Cited by 7 publications

(1 citation statement)

References 36 publications

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“…At first order of the path parameter a , the tangents

{boldX}_{1}^{b i} = {({}, \begin{matrix} array {boldU}_{1}^{b i} \\ array {normalλ}_{1}^{b i} \end{matrix})}_{i = 1, 2}

are solutions of the following linear system:

{normalL}_{t}^{c} ((), U_{1}^{b i}) = λ_{1} boldF

(⟨⟩, U_{1}^{b i}, U_{1}^{b i}) + {((), λ_{1}^{b i})}^{2} = 1 .

The singularity at the bifurcation point is treated classically with the help of the Lyapunov‐Schmidt reduction. () The 2 tangents are written as follows:

{boldU}_{1}^{b i} = {normalλ}_{1}^{b i} boldW + {normalη}_{1}^{b i} boldΦ,

with

{normalλ}_{1}^{b i}, {normalη}_{1}^{b i} \in double-struckR

being 2 scalars to be determined. To do so, solve the following linear system defined at the second order:

{normalL}_{t}^{c} ((), U_{2}^{b i}) = λ_{2} boldF - boldQ ((), U_{1}^{b i}, U_{1}^{b i})

(⟨⟩, U_{2}^{b i}, U_{1}^{b i})

…”

Section: Methodsmentioning

confidence: 99%

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Guevel

Allain

Girault

et al. 2017

Numerical Methods in Fluids

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Summary This paper deals with bifurcation analysis methods based on the asymptotic‐numerical method. It is used to investigate 3‐dimensional (3D) instabilities in a sudden expansion. To do so, high‐performance computing is implemented in ELMER, ie, an open‐source multiphysical software. In this work, velocity‐pressure mixed vectors are used with asymptotic‐numerical method–based methods, remarks are made for the branch‐switching method in the case of symmetry‐breaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and top‐bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, Ai=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post‐bifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.

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“…At first order of the path parameter a , the tangents

{boldX}_{1}^{b i} = {({}, \begin{matrix} array {boldU}_{1}^{b i} \\ array {normalλ}_{1}^{b i} \end{matrix})}_{i = 1, 2}

are solutions of the following linear system:

{normalL}_{t}^{c} ((), U_{1}^{b i}) = λ_{1} boldF

(⟨⟩, U_{1}^{b i}, U_{1}^{b i}) + {((), λ_{1}^{b i})}^{2} = 1 .

The singularity at the bifurcation point is treated classically with the help of the Lyapunov‐Schmidt reduction. () The 2 tangents are written as follows: