Bifurcations of Poiseuille flow between parallel plates: Three-dimensional solutions with large spanwise wavelength

Afendikov, A. L.; Mielke, Alexander

doi:10.1007/bf00379917

Cited by 17 publications

(26 citation statements)

References 19 publications

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“…[Doe93,AfM95]) led to several previous studies of its homoclinic orbits, see [Hol86,Doe96,KaM96] and references therein. In this situation homoclinic solutions are physically especially relevant, since they are spatially localized (exponentially decaying) and thus have finite energy, see [DF*96, Cha98,MHK98].…”

Section: Introductionmentioning

confidence: 95%

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Afendikov

Mielke

1999

Journal of Differential Equations

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We study reversible, SO(2)-invariant vector fields in R 4 depending on a real parameter = which possess for ==0 a primary family of homoclinic orbits T : H 0 , : # S 1 . Under a transversality condition with respect to = the existence of homoclinic n-pulse solutions is demonstrated for a sequence of parameter values = (n)The existence of cascades of 2 l 3 m -pulse solutions follows by showing their transversality and then using induction. The method relies on the construction of an SO(2)-equivariant Poincare map which, after factorization, is a composition of two involutions: A logarithmic twist map and a smooth global map. Reversible periodic orbits of this map corresponds to reversible periodic or homoclinic solutions of the original problem. As an application we treat the steady complex Ginzburg Landau equation for which a primary homoclinic solution is known explicitly. Academic Press

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Section: Introductionmentioning

confidence: 95%

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Afendikov

Mielke

1999

Journal of Differential Equations

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“…Spatial Dynamical System Formulation. We obtain a spatial dynamical system formulation by substituting the ansatz (4) u (x, t) = v x · k − ct, x ε into the reaction diffusion equation (1). This leads to an equation in terms of the profile function v…”

Section: Reduction To Finite Dimensionsmentioning

confidence: 99%

“…Hence they are solutions which are fixed orthogonal to k and move in the k direction with speed c as time varies. However due to the periodic behaviour in the nonlinearity such travelling wave solutions will not in general exist for equation (1). Therefore we need to modify the type of solution we look for to accommodate this periodic behaviour.…”

Section: Introductionmentioning

confidence: 99%

“…Thus we start by substituting the generalised travelling wave ansatz (4) into the reaction diffusion equation to obtain a degenerate second order elliptic equation on a infinite cylinder with periodic boundary conditions in terms of the profile function v. After this we use the idea of Kirchgässner [14] to formulate this equation as a infinite dimensional dynamical system treating the unbounded direction as a time variable. This idea has been used in many papers to find and analyse solutions to partial differential equations on unbounded domains a few examples are [1,8,9,12,20,21] and most relevantly [16], and [17], which used it to study travelling wave solutions and pinning for a reaction diffusion equation on a infinite cylinder with periodic behaviour along the cylinder. We then apply the local centre manifold theorem to this spatial dynamical system, this requires some work as the linear part of our spatial dynamical system is not bisectorial.…”

Section: Introductionmentioning

confidence: 99%

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Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

Boden

Matthies

2014

J Dyn Diff Equat

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Abstract. The existence of travelling wave type solutions is studied for a scalar reaction diffusion equation in R 2 with a nonlinearity which depends periodically on the spatial variable. We treat the coefficient of the linear term as a parameter and we formulate the problem as an infinite spatial dynamical system. Using a centre manifold reduction we obtain a finite dimensional dynamical system on the centre manifold with fully degenerate linear part. By phase space analysis and Conley index methods we find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds.The analysis provides an approach to the homogenisation problem as the period of the periodic dependence in the nonlinearity tends to zero.

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“…Even if the Cauchy problem is not well-posed, this method of spatial dynamics has a long history, see [Kir82] and further work (see, e.g. [IM91,AM95,FS03] and the references therein).…”

Section: Examplesmentioning

confidence: 99%

Exponential Estimates in Averaging and Homogenisation

Matthies

2006

Analysis, Modeling and Simulation of Multiscale Problems

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Abstract. Many partial differential equations with rapid spatial or temporal scales have effective descriptions which can be derived by homogenisation or averaging. In this article we deal with examples, where quantitative estimates of the error is possible for higher order homogenisation and averaging. In particular, we provide theorems, which allow homogenisation and averaging beyond all orders by giving exponential estimates of appropriately averaged and homogenised descriptions.Methods include iterated averaging transformations, optimal truncation of asymptotic expansions and highly regular solutions (Gevrey regularity). Prototypical examples are reaction-diffusion equations with heterogeneous reaction terms or rapid external forcing, nonlinear Schrödinger equations describing dispersion management, and second-order linear elliptic equations.

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Bifurcations of Poiseuille flow between parallel plates: Three-dimensional solutions with large spanwise wavelength

Cited by 17 publications

References 19 publications

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Bifurcation of Homoclinic Orbits to a Saddle-Focus in Reversible Systems with SO(2)-Symmetry

Existence and Homogenisation of Travelling Waves Bifurcating from Resonances of Reaction–Diffusion Equations in Periodic Media

Exponential Estimates in Averaging and Homogenisation

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