Travelling-waves of the Kuramoto-Sivashinsky, equation: period-multiplying bifurcations

Kent, Phillip; Elgin, J.N.

doi:10.1088/0951-7715/5/4/004

Cited by 38 publications

(37 citation statements)

References 18 publications

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“…Thereby, the doubled, tripled, and quadrupled orbits form, in fact, one family, in which the shape of the orbit continuously changes, acquiring and losing loops in the phase space. A similar effect for solutions of the KS equation was described in [42].…”

Section: Bifurcations Of Periodic Solutionssupporting

confidence: 57%

“…Since for D > √ 2 the contour is absent, those branches detach from it in the course of increase of D and interconnect. As a result, for moderate and high values of D one observers the picture typical of the KS equation [42] (Figure 7(a)). In terms of geometric characteristics of the orbit, the discussed families of secondary solutions merge into a single curve which has two turning points.…”

Section: Bifurcations Of Periodic Solutionsmentioning

confidence: 77%

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Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

Zaks¹,

Podolny²,

Nepomnyashchy³

et al. 2005

SIAM J. Appl. Math.

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We investigate bifurcations of stationary periodic solutions of a convective CahnHilliard equation, ut+Duux+(u−u 3 +uxx)xx = 0, describing phase separation in driven systems, and study the stability of the main family of these solutions. For the driving parameter D < D 0 = √ 2/3, the periodic stationary solutions are unstable. For D > D 0 , the periodic stationary solutions are stable if their wavelength belongs to a certain stability interval. It is therefore shown that in a driven phase-separating system that undergoes spinodal decomposition the coarsening can be stopped by the driving force, and formation of stable periodic structures is possible. The modes that destroy the stability at the boundaries of the stability interval are also found.

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Section: Bifurcations Of Periodic Solutionssupporting

confidence: 57%

Section: Bifurcations Of Periodic Solutionsmentioning

confidence: 77%

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

Zaks¹,

Podolny²,

Nepomnyashchy³

et al. 2005

SIAM J. Appl. Math.

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“…The main reason we present Theorem 1.3 is that it shows, together with Theorem 1.2, that for a countable set of parameter values there exist infinitely many heteroclinic connections between equilibrium points in both directions, called Bykov cycles. This provides a Shilnikov-like structure [9], [10].…”

Section: Wilczakmentioning

confidence: 99%

“…The heteroclinic solutions presented in Fig. 2 correspond to admissible sequences (4, 5, 2), (4, 5, 2, 3, 4, 1), and (4,11,10,9,4), respectively. The analytic form (3) of the heteroclinic solution found by Kuramoto and Tsuzuki [12] is the simplest one of the family of heteroclinic solutions resulting from Theorem 4.3 and it corresponds to the shortest admissible sequence i 0 = 4.…”

Section: Wilczakmentioning

confidence: 99%

The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof

Wilczak

2006

Found Comput Math

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In this paper we present a new topological tool which allows us to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system y + y + 0.5y 2 = c 2 [16]. We prove that there exists a countable set of parameter values c for which a pair of the Shilnikov homoclinic orbits to the equilibrium points (±c √ 2, 0, 0) appear. This result was conjectured by Michelson [16]. We also show that there exists a countable set of parameter values for which there exists a heteroclinic orbit connecting the equilibrium (−c √ 2, 0, 0) possessing a one-dimensional unstable manifold with the equilibrium (c √ 2, 0, 0) possessing a one-dimensional stable manifold. The method used in the proof can be applied to other reversible systems.To verify the assumptions of the main topological theorem for the Michelson system, we use rigorous computations based on interval arithmetic.

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“…In the limit of infinite substrate diffusivity (A → 0), C 3 → ∞ and so there is also a Hopf bifurcation at c = 2 (µ = ±i R (2)). This simultaneous bifurcation had been referred to as a Gavrilov-Guckenheimer point [4,42,43], and is known to arise in the isothermal Benney equation [4] as well as the Kuramoto-Sivashinsky equation [44], which is an approximation of the isothermal falling film near the critical Reynolds number [4]. Including the substrate diffusivity has the effect of separating the transcritical and Hopf bifurcations.…”

Section: B Bifurcations From the Trivial Statementioning

confidence: 99%

Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

2016

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Consider the dynamics of a thin film flowing down a heated substrate. The substrate heating generates a temperature distribution on the free surface which in turn induces surface-tension gradients and corresponding thermocapillary stresses that affect the free surface and therefore the fluid flow. We study here the effect of finite substrate thermal diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and heat transport equations indicates that if the substrate diffusivity is sufficiently small, the film becomes unstable at a finite wavelength and at a Reynolds number smaller than that predicted in the long-wavelength limit. This property is captured in a reducedorder system of equations derived using a weighted-residual integral-boundary-layer method. This reduced-order model is also used to compute the bifurcation diagrams of solution branches connecting the trivial flat film to traveling waves including solitary pulses. The effect of finite diffusivity is to separate a simultaneous Hopf/transcritical bifurcation into its individual component bifurcations. The appropriate Hopf bifurcation then connects only to the solution branch of negative-hump pulses, with wave speed less than the linear wave speed while the branch of positive-single-hump pulses merges with the branch of positive-two-hump pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions, whose period cannot be increased indefinitely. Numerical simulation of the reduced-order system shows the development of a train of coherent structures each of which resembles a stationary positive-hump pulse, and, in the regime of finite-wavelength instability, wavelength selection and saturation to periodic traveling waves.

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Travelling-waves of the Kuramoto-Sivashinsky, equation: period-multiplying bifurcations

Cited by 38 publications

References 18 publications

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

Periodic Stationary Patterns Governed by a Convective Cahn--Hilliard Equation

The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof

Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity

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