Nonlinear Dynamics and Chaos in Parametrically Excited Surface Waves

Umeki, Makoto; Kambe, Tsutomu

doi:10.1143/jpsj.58.140

Cited by 61 publications

(32 citation statements)

References 15 publications

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“…To obtain such an interaction we select appropriately the driving frequency and amplitude. Theoretical studies of such mode interactions include those based on amplitude equations for nearly inviscid flows but without the inclusión of streaming flow [86], [87], [88] and generic studies based on the 0(2) symmetry of the system [62]; for a comparison and critique of these approaches, see [62] and the comment by Miles [51]. In this section we retain the exact 0(2) symmetry of the system and focus on the role of the streaming flow generated by the mode interaction.…”

Section: Mode-mode Interaction In Circular Containersmentioning

confidence: 99%

“…In [62] it is shown that if the amplitude equations are computed to a sufflciently high order (m + n -1 = 10 in the Ciliberto-Gollub experiment), the 0(2) symmetry of the original system is restored. [45], [87], [88]. However, these equations fail to reproduce essential features of the experimental bifurcation diagram.…”

Section: Comparison With Previous Theoretical Approachesmentioning

confidence: 99%

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Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Higuera

Vega

Knobloch³

2002

J. Nonlinear Sci.

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Summary.We derive a set of asymptotically exact coupled amplitude-streaming flow (CASF) equations governing the evolution of weakly nonlinear nearly inviscid multimode Faraday waves and the associated streaming flow in flnite geometries. The streaming flow is found to play a particularly important role near mode interactions. Such interactions come about either through a suitable choice of parameters or through breaking of degeneracy among modes related by symmetry. An example of the flrst case is provided by the interaction of two nonaxisymmetric modes in a circular container with different azimuthal wavenumbers. The second case arises when the shape of the container is changed from square to slightly rectangular, or from circular to slightly noncircular but with a plañe of symmetry. The generation of streaming flow in each of these cases is discussed in detail and the properties of the resulting CASF equations are described. A preliminary analysis suggests that these equations can resolve discrepancies between existing theory and experimental results in the flrst two of the above cases.

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Section: Mode-mode Interaction In Circular Containersmentioning

confidence: 99%

Section: Comparison With Previous Theoretical Approachesmentioning

confidence: 99%

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Higuera

Vega

Knobloch³

2002

J. Nonlinear Sci.

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“…The functions Ψ + (ζ) and Ψ − (ζ), solutions to equation (3), can be expressed through the Jost solutions and the elements of the scattering matrix, defined as…”

Section: The Riemann-hilbert Problemmentioning

confidence: 99%

“…This equation describes nonlinear Faraday resonance in a vertically oscillating water trough [1,2,3,4,5,6,7,8,9,10,11,12]; an easy-plane ferromagnet with a combination of a stationary and a high-frequency magnetic field in the easy plane [13]; and the effect of phase-sensitive amplifiers on solitons propagating in optical fibres [14,15,16]. The equation (1)- (2) has two stationary soliton solutions, one of which is unstable for all h and γ and hence usually disregarded [13].…”

Section: Introductionmentioning

confidence: 99%

Soliton–radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation

Shchesnovich

Barashenkov

2002

Physica D: Nonlinear Phenomena

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We use the Riemann-Hilbert problem to study the interaction of the soliton with radiation in the parametrically driven, damped nonlinear Schrödinger equation. The analysis is reduced to the study of a finite-dimensional dynamical system for the amplitude and phase of the soliton and the complex amplitude of the long-wavelength radiation. In contrast to previously utilised Inverse Scattering-based perturbation techniques, our approach is valid for arbitrarily large driving strengths and damping coefficients. We show that, contrary to suggestions made in literature, the complexity observed in the soliton's dynamics cannot be accounted for just by its coupling to the long-wavelength radiation.

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“…[6,7], and for the Faraday experiment Refs. [21][22][23][24][25][26]); but in the present context we operate on the more abstract level of symmetry groups.…”

Section: Comparison With Other Systemsmentioning

confidence: 99%

Mode competition in a system of two parametrically driven pendulums; the role of symmetry

Banning

Weele

Kettenis

1997

Physica A: Statistical Mechanics and its Applications

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This paper is the final part in a series of four on the dynamics of two coupled, parametrically driven pendulums. In the previous three parts (Banning and van der Weele, Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case, Physica A 220 (1995) 485-533; Banning et al., Mode competition in a system of two parametrically driven pendulums; the dissipative case, Physica A 245 (1997) l 1-48; Banning et al., Mode competition in a system of two parametrically driven pendulums with nonlinear coupling, Physica A 245 (1997) 49-98) we have given a detailed survey of the different oscillations in the system, with particular emphasis on mode interaction. In the present paper we use group theory to highlight the role of symmetry. It is shown how certain symmetries can obstruct period doubling and Hopf bifurcations; the associated routes to chaos cannot proceed until these symmetries have been broken. The symmetry approach also reveals the general mechanism of mode interaction and enables a useful comparison with other systems.

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Nonlinear Dynamics and Chaos in Parametrically Excited Surface Waves

Cited by 61 publications

References 15 publications

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Coupled Amplitude-Streaming Flow Equations for Nearly Inviscid Faraday Waves in Small Aspect Ratio Containers

Soliton–radiation coupling in the parametrically driven, damped nonlinear Schrödinger equation

Mode competition in a system of two parametrically driven pendulums; the role of symmetry

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