TY - JOUR

T1 - On strong transitions between structures of differing symmetry accompanying weakly supercritical convection

AU - Malomed, B. A.

AU - Tribel'skii, M. I.

PY - 1988

Y1 - 1988

N2 - A complete classification of the phase space of the dynamical system which describes the motion of a liquid when there is weakly supercritical convection is carried out within the framework of a six-mode Galerkin approximation. It is shown that all the phase trajectories are attracted to the corresponding stationary states. The domains of attraction to each of these states are found. The minimum value of a perturbation, which converts a weakly stable solution of one symmetry into a stable solution of another symmetry when the parameters of the problem are close to their bifurcation values, is estimated. It is known that, in certain cases when there is weakly supercritical convection, two types of stationary spatially periodic flows may arise which, in typical situations, are cellular hexagonal structures and structures in the form of two dimensional axles. When this is so, both types of structures are found to be stable with respect to small perturbations in a certain domain of the values of the parameters of the problem so that strong transitions between them become possible under the action of perturbations of finite amplitude /1-3/. In such cases the question as to the minimum amplitude of a perturbation which converts a structure of one symmetry into a structure of another symmetry, and questions relating to this regarding a complete classification of the possible asymptotic states of the dynamical system under consideration and the domains of attraction of different initial conditions to these asymptotic states are of interest. This paper is concerned with investigating these questions.

AB - A complete classification of the phase space of the dynamical system which describes the motion of a liquid when there is weakly supercritical convection is carried out within the framework of a six-mode Galerkin approximation. It is shown that all the phase trajectories are attracted to the corresponding stationary states. The domains of attraction to each of these states are found. The minimum value of a perturbation, which converts a weakly stable solution of one symmetry into a stable solution of another symmetry when the parameters of the problem are close to their bifurcation values, is estimated. It is known that, in certain cases when there is weakly supercritical convection, two types of stationary spatially periodic flows may arise which, in typical situations, are cellular hexagonal structures and structures in the form of two dimensional axles. When this is so, both types of structures are found to be stable with respect to small perturbations in a certain domain of the values of the parameters of the problem so that strong transitions between them become possible under the action of perturbations of finite amplitude /1-3/. In such cases the question as to the minimum amplitude of a perturbation which converts a structure of one symmetry into a structure of another symmetry, and questions relating to this regarding a complete classification of the possible asymptotic states of the dynamical system under consideration and the domains of attraction of different initial conditions to these asymptotic states are of interest. This paper is concerned with investigating these questions.

UR - http://www.scopus.com/inward/record.url?scp=45549115101&partnerID=8YFLogxK

U2 - 10.1016/0021-8928(88)90060-3

DO - 10.1016/0021-8928(88)90060-3

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AN - SCOPUS:45549115101

SN - 0021-8928

VL - 52

SP - 59

EP - 62

JO - Journal of Applied Mathematics and Mechanics

JF - Journal of Applied Mathematics and Mechanics

IS - 1

ER -