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On the Burnside problem in Diff(M)
Putting a boundary to the space of Liénard equations
1. | Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S., Université de Bourgogne, B.P. 47 870, 21078 Dijon Cedex, France |
In fact it is not even known if there exists a finite bound $L(n)$ independent of $a,$ for the number of limit cycles. In this paper, I want to investigate this question of finiteness. I show that there exists a finite bound $L(K,n)$ if one restricts the parameter in a compact $K$ and that there is a natural way to put a boundary to the space of Liénard equations. This boundary is made of slow-fast equations of Liénard type, obtained as singular limits of the Liénard equations for large values of the parameter. Then the existence of a global bound $L(n)$ can be related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.
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