Symmctry Breaking for 3 Coupled Systems

概要

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Studying a system of 3Hybridly coupled oscillators.it has been found that the periodic solution may become unstable by a symmetry breaking bifurcation. At the bifurcation setting, the fixed point of the Poincare mapping corresponding to the symmetric solution has two eigenvalues equal to 1.The bifurcation can be described by D_0+3D_1⟷D_2+3D_1(1)where D_n is a direct type fixed point whose unstable subsapce is n dimmensional. The 3fixed points D_1,corresponding to dissymetric solutions, merge with the D_0 fixed point at the bifurcation value. In order to understand this penomenon, we investigate the symmetry breaking bifurcations occuring for a system of three coupled quadratic maps: [numerical formula] Figure 1shows the differents symmetry the system possesses according to the setting of the parameter a and E. On this figure D3 is the symmetry of a triangle in the space, C3 is the symmetry of a triangle in the plane,'Switching' is the symmetry of the triangle with respect to one of its bissectrice. [figure]
1996-03-11