| Citation: |
Xianbo Sun. EXACT BOUND ON THE NUMBER OF LIMIT CYCLES ARISING FROM A PERIODIC ANNULUS BOUNDED BY A SYMMETRIC HETEROCLINIC LOOP[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 378-390. doi: 10.11948/20190294
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EXACT BOUND ON THE NUMBER OF LIMIT CYCLES ARISING FROM A PERIODIC ANNULUS BOUNDED BY A SYMMETRIC HETEROCLINIC LOOP
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Abstract
In this paper, the bound on the number of limit cycles by Poincaré bifurcation in a small perturbation of some seventh-degree Hamiltonian system is concerned. The lower and upper bounds on the number of limit cycles have been obtained in two previous works, however, the sharp bound is still unknown. We will employ some new techniques to determine which is the exact bound between $ 3 $ and $ 4 $. The asymptotic expansions are used to determine the four vertexes of a tetrahedron, and the sharp bound can be reached when the parameters belong to this tetrahedron.-
Keywords:
- Limit cycle /
- Abelian integral /
- heteroclinic loop /
- sharp bound
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References
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Xianbo Sun. EXACT BOUND ON THE NUMBER OF LIMIT CYCLES ARISING FROM A PERIODIC ANNULUS BOUNDED BY A SYMMETRIC HETEROCLINIC LOOP[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 378-390. doi: 10.11948/20190294
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Figure 1. Periodic annului bounded by heteroclinic loops (red curves) for system
$ (1.7)_{\varepsilon = 0} $ . (a)$ \zeta = 1, \ \alpha<\beta<0 $ ; (b)$ \zeta = 1, \ \alpha = \beta>1 $ ; (c)$ \zeta = 1, \ \alpha = 1, \ \beta>1 $ ; (d)$ \zeta = -1, \ \alpha<0, \beta = 1 $ ; (e)$ \zeta = 1, \ \alpha = \beta = 1 $ ; (f)$ \zeta = -1, \ \alpha = 0, \beta = 1 $ ; (g)$ \zeta = 1, \ \alpha<0, \beta = 0 $ ; (h)$ \zeta = 1, \ \alpha = \beta = 0 $ . For cases$ a $ and$ b $ , the heteroclinic loop connects two hyperbolic saddles and the inner boundary is an elementary center. For cases$ c $ ,$ d $ and$ e $ , the heteroclinic loop connects two nilpotent saddles (cusps) and the inner boundary is an elementary center. For case$ f $ , the heteroclinic loop connects two nilpotent cusps and the inner boundary is a nilpotent center. For cases$ g $ and$ h $ , the heteroclinic loop connects two hyperbolic saddles and the inner boundary is a nilpotent center -
Figure 2. The level set of
$ {\cal H}(x, y) $