Surface wave mode interactions: effects of symmetry and degeneracy

Simonelli, Fábio; Gollub, Jerry P.

doi:10.1017/s0022112089000443

Cited by 139 publications

(154 citation statements)

References 24 publications

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“…It should be emphasized that because of the approximate D4 symmetry the applicability of our equations will extend beyond the present set of problems. We mention just two examples, the dynamics of Faraday waves in a nearly square container [35] and the long-term dynamics of the solar sunspot cycle arising from the interaction of dipole and quadrupole magnetic fields [36].…”

Section: Fig 16 (Continued)mentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

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The effect of distant end walls on the bifurcation to traveling waves is considered. Previous approaches have treated the problem by assuming that.it is a weak perturbation of the translation invariant problem. When the problem is formulated instead in a finite box of length L and the limit L --+ 00 is taken, one obtains amplitude equations that differ from the usual Ginzburg-Landau description by the presence of an additional nonlinear term. This formulation leads to a description in terms of the amplitudes of the primary box modes, which are odd and even parity standing waves. For large L, the equations that result take the form of a Hopf bifurcation with approximate D4 symmetry. These equations are able to describe, qualitatively, not only traveling and "blinking" states, but also asymmetrical blinking states and "repeated transients," all of which have been observed in binary fluid convection experiments. PACS number(s): 47.20. Bp, 47.20.Ky, 03.40.Kf

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Section: Fig 16 (Continued)mentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

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“…This nonlinear coupling leads to triad interactions and energy transfer between trapped modes. Interesting phenomena involving triad interactions have been studied in the context of acoustic-gravity waves (Kadri & Stiassnie 2013;Kadri & Akylas 2016), edge waves (Li 2007), Bragg scattering by bottom ripples and interfacial waves (Alam 2012), while contributions to the analysis of mode-mode interactions can be found in Mei & Zhou (1991) and Zardi & Seminara (1995) for pulsating bubbles and Ciliberto & Gollub (1985), Simonelli & Gollub (1989), Kambe & Umeki (1990), Umeki (1991) for Faraday waves. The coupled evolution equations of both modal amplitudes are obtained and now involve coupling terms.…”

Section: Introductionmentioning

confidence: 99%

Weakly nonlinear theory for oscillating wave surge converters in a channel

2017

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We present a weakly nonlinear theory on the natural modes' resonance of an array of oscillating wave surge converters (OWSCs) in a channel. We first derive the evolution equation of the Stuart-Landau type for the gate oscillations in uniform and modulated incident waves and then evaluate the nonlinear effects on the energy conversion performance of the array. We show that the gates are unstable to side-band perturbations so that a Benjamin-Feir instability similar to the case of Stokes' waves is possible. The non-autonomous dynamical system presents period doubling bifurcations and strange attractors. We also analyse the competition of two natural modes excited by one incident wave. For weak damping and power take-off coefficient, the dynamical effects on the generated power of the OWSCs are investigated. We show that the occurrence of subharmonic resonance significantly increases energy production.

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“…31,109,110 They also include applications to biology in contexts as varied as the markings on big cats, giraffes, and zebras 96 to patterns of geometric visual hallucinations, 13,40 and central place theory in the formation of cities. 80,95 However, we shall not describe these applications here.…”

Section: Continuous Symmetries and Patternsmentioning

confidence: 99%

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts Rene Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network. Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems-systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have ...

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Surface wave mode interactions: effects of symmetry and degeneracy

Cited by 139 publications

References 24 publications

Oscillatory bifurcation with broken translation symmetry

Oscillatory bifurcation with broken translation symmetry

Weakly nonlinear theory for oscillating wave surge converters in a channel

Recent advances in symmetric and network dynamics

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