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Bi-center Problem in a Class of Z2-equivariant Quintic Vector Fields
  
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KeyWord:Nilpotent singular point; Center–focus problem; Bi-center; Lyapunov constant
Author NameAffiliation
Hongwei Li School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276005, China 
Feng Li School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276005, China 
Pei Yu Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada 
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Abstract:
      In this paper, we study the center problem for Z2-equivariant quintic vector fields. First of all, for convenience in analysis, the system is simplified by using some transformations. When the system has two nilpotent points at (0,±1) with multiplicity three, the first seven Lyapunov constants at the singular points are calculated by applying the inverse integrating factor method. Then, fifteen center conditions are obtained for the two nilpotent singular points of the system to be centers, and the sufficiency of the first seven center conditions are proved. Finally, the first five Lyapunov constants are calculated at the two nilpotent points (0,±1) with multiplicity five by using the method of normal forms, and the center problem of this system is partially solved.