Journal
JOURNAL OF DIFFERENTIAL EQUATIONS
Volume 254, Issue 8, Pages 3501-3529Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jde.2013.01.027
Keywords
Bifurcation; Oscillator network; Pattern generation; Hierarchical system
Categories
Funding
- NSFC [10971057, 11271115]
- Ministry of Education of China
- Hunan Provincial Natural Science Foundation [10JJ1001]
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In this paper, we consider a hierarchically organized network composed of three interacting systems each of which consists of three coupled oscillators. This hierarchical network is equivariant under the symmetry group D-3 X Z(3). Using the lattice of isotropy subgroups, we study the reduced equations restricted to invariant fixed-point subspaces and prove that it is possible for the oscillator network to have 4 distinct equilibria or 45 distinct periodic solutions with maximal isotropy subgroups. These are classified and their bifurcation directions are determined in terms of the quadratic coefficients and relevant quantities. A center manifold reduction from the hierarchical network to the normal form equations is then performed in order to investigate the codimension two bifurcations. Using this reduction we find a great variety of equilibria, periodic and quasi-periodic oscillation patterns of maximal and submaximal symmetry which can be classified in a two-level pattern hierarchy. (C) 2013 Elsevier Inc. All rights reserved.f
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