Imperfect homoclinic bifurcations

Glendinning, Paul; Abshagen, J.; Mullin, T.

doi:10.1103/physreve.64.036208

Cited by 29 publications

(24 citation statements)

References 17 publications

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“…This leads to spontaneous merging of two limit cycles to a single limit cycle in the appropriate phase space, as the bifurcation parameter is raised above a critical value. The gluing bifurcation is observed in liquid crystals [3], in fluids [4,5], and in electronics circuits [6]. Recently, Pal et.…”

Section: Introductionmentioning

confidence: 99%

Homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

Maity

Kumar

Pal

2013

EPL

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We present the effects of small Coriolis force on homoclinic gluing and ungluing bifurcations in low-Prandtl-number(0.025 ≤ P r) rotating Rayleigh-Bénard system with stress-free top and bottom boundaries. We have performed direct numerical simulations for a a wide range of Taylor number (5 ≤ T a ≤ 50) and reduce Rayleigh number r (≤ 1.25). We observe homoclinic ungluing bifurcation, marked by the spontaneous breaking of a larger limit cycle into two possible set of limit cycles in the phase space, for lower values of T a. Two unglued limit cycles merge together for higher values of T a, as Ra is raised sufficiently. The range of T a for which both gluing and ungluing bifurcation can be seen depends on the Prandtl number P r. The variation of the bifurcation points with T a is also investigated. We also present a low-dimensional model which qualitatively captures the dynamics of the system near the homoclinic bifurcation points. The model is used to study the variation of the homoclinic bifurcation points with Prandtl number for different values of T a.

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Section: Introductionmentioning

confidence: 99%

Homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

Maity

Kumar

Pal

2013

EPL

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“…In fact, we find that the finite value of ε 'unfolds' the saddle-node/global bifurcation present at ε = 0 (cf. [16]), with the result that the global bifurcation splits into two successive codimension-one global bifurcations, one involving the small amplitude asymmetric oscillations and the other the large amplitude R 1 R 2 -symmetric oscillations, with the saddle-node bifurcation now occurring on one or other of these oscillation branches. The behavior near each of these global bifurcations is determined by the eigenvalue ratio δ = |λ s /λ u |, where λ s ≈ −0.00051 is the least stable eigenvalue of M + 2 and λ u ≈ 0.4889 is its unstable eigenvalue, both computed at µ = µ c for ε = 0.01.…”

Section: Case Ii: Relaxation Oscillations and Canardsmentioning

confidence: 99%

Dynamics of nearly inviscid Faraday waves in almost circular containers

Higuera

Knobloch

Vega

2005

Physica D: Nonlinear Phenomena

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Parametrically driven surface gravity-capillary waves in an elliptically distorted circular cylinder are studied. In the nearly inviscid regime, the waves couple to a streaming flow driven in oscillatory viscous boundary layers. In a cylindrical container, the streaming flow couples to the spatial phase of the waves, but in a distorted cylinder, it couples to their amplitudes as well. This coupling may destabilize pure standing oscillations, and lead to complex time-dependent dynamics at onset. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting 'canards'.

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“…It occurs when two limit cycles simultaneously become homoclinic orbits of the same saddle point. This phenomenon has been recently observed in a variety of systems including liquid crystals [11], fluid dynamical systems [12], biological systems [13], optical systems [14], and electrical circuits [15], and is a topic of current research. The pattern dynamics in the vicinity of a homoclinic bifurcation has, however, not been investigated in a fluid dynamical system.…”

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confidence: 99%

Pattern dynamics near inverse homoclinic bifurcation in fluids

et al. 2013

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We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns diverges, and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well. [9]. The selection of a pattern is a consequence of at least one broken symmetry of the system. Unbroken symmetries often introduce multiple patterns, which may lead to a transition from local to global pattern dynamics. The gluing [10] of two limit cycles on two sides of a saddle point in the phase space of a given system is an example of a local to nonlocal bifurcation. It occurs when two limit cycles simultaneously become homoclinic orbits of the same saddle point. This phenomenon has been recently observed in a variety of systems including liquid crystals [11], fluid dynamical systems [12], biological systems [13], optical systems [14], and electrical circuits [15], and is a topic of current research. The pattern dynamics in the vicinity of a homoclinic bifurcation has, however, not been investigated in a fluid dynamical system.A Rayleigh-Bénard system [16,17], where a thin layer of a fluid is heated uniformly from below and cooled uniformly from above, is a classical example of an extended dissipative system which shows a plethora of patternforming instabilities [2], chaos [18], and turbulence [19]. Low-Prandtl-number [20] and very low-Prandtl-number convection [21,22] show three-dimensional oscillatory behavior close to the instability onset. In addition, the Rayleigh-Bénard system possesses symmetries under translation and rotation in the horizontal plane that can introduce multiple sets of patterns. The possibility of a homoclinic bifurcation and the pattern dynamics in its vicinity are unexplored in three dimensional (3D) Rayleigh-Bénard convection.We report, in this article, for the first time the possibility of an inverse homoclinic bifurcation in direct numerical simulations (DNS) of three dimensional (3D) Rayleigh-Bénard convection (RBC) in low-Prandtlnumber fluids, and the results of our investigations of fluid patterns close to the bifurcation. We observe spontaneous breaking of a periodic competition of two mutually perpendicular sets of cross rolls to one set of oscillating cross rolls, as the Rayleigh number Ra is raised above a critical value Ra h . The time period of the oscillating patterns diverges, and shows scaling behavior in the close vicinity of the transition point. The exponents of scaling are asymmetric on the two sides of the transition po...

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Imperfect homoclinic bifurcations

Cited by 29 publications

References 17 publications

Homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

Homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation

Dynamics of nearly inviscid Faraday waves in almost circular containers

Pattern dynamics near inverse homoclinic bifurcation in fluids

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