MODE INTERACTIONS WITH SPHERICAL SYMMETRY

1994 ◽  
Vol 04 (04) ◽  
pp. 885-904 ◽  
Author(s):  
S.B.S.D. CASTRO
Keyword(s):  
Steady State ◽  
Spherical Harmonics ◽  
Spherical Symmetry ◽  
Mode Interaction ◽  
Natural Representation ◽  
Mode Interactions ◽  
Study Mode ◽  
Bifurcation Problems ◽  
Steady State Bifurcation ◽  
Secondary Bifurcations

We study mode interaction steady-state bifurcation problems with spherical symmetry. Using the representation of O(3) in terms of spherical harmonics, we study the interactions of the modes of dimension 1, 3 and 5. When studying mode interactions involving the 3- and the 5-dimensional modes, we come across a very natural representation, which turns out to be that of SO(3) instead of O(3). Given that the study of problems with this latter symmetry has already been done, we then study problems with SO(3) symmetry. For all these problems, we stress the existence of secondary bifurcations giving rise to the existence of limit cycles and the occurrence of heteroclinic connections between equilibria.

Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions

1997 ◽  
Vol 07 (07) ◽  
pp. 1691-1698 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Hopf Bifurcation ◽  
Geometric Structure ◽  
Trivial Solution ◽  
Mixed Mode ◽  
Mode Interaction ◽  
Hopf Bifurcations ◽  
Bifurcation Points ◽  
Mode Interactions ◽  
The Stability ◽  
Secondary Bifurcations

In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.

Numerical Methods for Steady State/Hopf Mode Interactions

1997 ◽  
Vol 07 (03) ◽  
pp. 585-605 ◽  
Author(s):  
F. Amdjadi ◽  
P. J. Aston
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Symmetry Breaking ◽  
Mode Interaction ◽  
Extended System ◽  
Test Function ◽  
Alternative Approach ◽  
Mode Interactions ◽  
Sivashinsky Equation ◽  
Two Parameter

Numerical methods for dealing with steady state/Hopf mode interactions using extended systems are considered. In particular, it is shown that such a mode interaction corresponds to a symmetry breaking bifurcation of a Hopf extended system as well as a Hopf bifurcation of a symmetry breaking extended system. Non-degeneracy conditions associated with these bifurcations are derived and interpreted in the context of the mode interaction. The alternative approach of using a single test function instead of a full extended system is considered in detail in one of the cases. Numerical results for a two-parameter version of the Kuramoto–Sivashinsky equation are presented to illustrate the theory.

Rayleigh–Bénard convection and pattern formation in magnetohydrodynamics

1998 ◽  
Vol 60 (3) ◽  
pp. 529-539 ◽  
Author(s):  
RENU BAJAJ ◽  
S. K. MALIK
Keyword(s):  
Magnetic Field ◽  
Steady State ◽  
Thermal Instability ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
Steady State Bifurcation ◽  
The Stability ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
The Magnetic Field

A nonlinear thermal instability in a layer of electrically conducting fluid in the presence of a magnetic field is discussed. Steady-state bifurcation results in the formation of patterns: rolls, squares and hexagons. The stability of various patterns is also investigated. It is found that in the absence of a magnetic field only rolls are stable, but when the magnetic field strength exceeds a certain finite value, squares and hexagons also become stable.

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