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Existence and nonexistence of homoclinic trajectories of the Liénard system
Pattern formation in reactiondiffusion systems with $D_2$symmetric kinetics
1.  Department of Mathematics, University of North Carolina at Wilmington, Wilmington, NC 28403, United States 
2.  MCNC  North Carolina Supercomputing Center, 3021 Cornwallis Road, Research Triangle Park, NC 27709, United States 
Because the stationary patterns of such reactiondiffusion systems are the symmetric cycles of its steadystate system, we investigate the bifurcations of manifolds of symmetric cycles near equilibria in general $D_2$symmetric reversible systems. This is done through an analysis of the bifurcation regimes at strong resonances using 1dimensional universal unfoldings of $D_2$symmetric reversible normal forms. We prove there are two disjoint manifolds at "odd" resonance and four disjoint manifolds at "even" resonance. The number of these disjoint manifolds, in turn, determines the number of different types of stationary patterns.
Applications of our analysis to the study of pattern formation in reactiondiffusion systems are illustrated with a predatorprey model arising from mathematical ecology. Numerical results are obtained as a verification of our analysis.
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