American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations
  • CPAA Home
  • This Issue
  • Next Article
    Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations
June  2006, 5(2): 383-393. doi: 10.3934/cpaa.2006.5.383

Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry

Toshiyuki Ogawa 1,

1. 

Graduate School of Engineering Science, Osaka University, 560-8531 Toyonaka, Japan

Received  February 2005 Revised  June 2005 Published  March 2006

Rayleigh-Bénard convection in a small rectangular domain is studied by the standard bifurcation analysis. The dynamics on the center manifold is calculated up to 3rd order. By this normal form, it is possible to determine the local bifurcation structures in principle. One can easily imagine that mixed mode solutions such as hexagonal, patchwork-quilt patterns are unstable from the knowledge of amplitude equation:Ginzburg-Landau equation. However they are not necessarily similar to each other in a small rectangular domain. Several non-trivial stable mixed mode patterns are introduced.
Keywords: bifurcation analysis, normal form., Rayleigh-Bénard convection.
Mathematics Subject Classification: 35B32, 37G05 ,76E0.
Citation: Toshiyuki Ogawa. Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry. Communications on Pure and Applied Analysis, 2006, 5 (2) : 383-393. doi: 10.3934/cpaa.2006.5.383
[1]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure and Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[2]

Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591
[3]

B. A. Wagner, Andrea L. Bertozzi, L. E. Howle. Positive feedback control of Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619
[4]

Jhean E. Pérez-López, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis. Journal of Computational Dynamics, 2020, 7 (1) : 159-181. doi: 10.3934/jcd.2020006
[5]

Tingyuan Deng. Three-dimensional sphere $S^3$-attractors in Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 577-591. doi: 10.3934/dcdsb.2010.13.577
[6]

Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53
[7]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2621-2634. doi: 10.3934/dcdsb.2021151
[8]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[9]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[10]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[11]

Marco Cabral, Ricardo Rosa, Roger Temam. Existence and dimension of the attractor for the Bénard problem on channel-like domains. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 89-116. doi: 10.3934/dcds.2004.10.89
[12]

O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449
[13]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[14]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[15]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[16]

Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022085
[17]

Pooja Girotra, Jyoti Ahuja, Dinesh Verma. Analysis of Rayleigh Taylor instability in nanofluids with rotation. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 495-512. doi: 10.3934/naco.2021018
[18]

Yong Duan, Jian-Guo Liu. Convergence analysis of the vortex blob method for the $b$-equation. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1995-2011. doi: 10.3934/dcds.2014.34.1995
[19]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281
[20]

Stefan Siegmund. Normal form of Duffing-van der Pol oscillator under nonautonomous parametric perturbations. Conference Publications, 2001, 2001 (Special) : 357-361. doi: 10.3934/proc.2001.2001.357

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (112)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy