一类2n次Kolmogorov系统的极限环

The Limit Cycle of a Class of 2n-degree Kolmogorov's System

关键词:  Kolmogorov系统  极限环  闭轨

Key words:  Kolmogorov's system  limit cycle  closed orbit

摘要:

研究一类2n次Kolmogorov系统dx/dt=x(a₀-a₁x+a₃x³-a₄x⁴+a₅xy^2n-1),dy/dt=y(b₁x²ⁿ-b₂),极限环的存在性问题.主要讨论a₅>0和a₅<0两种情形.当a₅>0时,系统在第一象限内不存在极限环;当a₅<0时,得到了平衡点的稳定性态,系统无闭轨的充分条件以及在第一象限内存在稳定极限环的条件.

Abstract:

The problem about the existence of a limit cycle of a class of 2n-degree Kolmogorov's system dx/dt=x (a₀-a₁x+a₃x³-a₄x⁴+a₅xy^2n-1), dy/dt=y (b₁x²ⁿ-b₂), is studied, and the system is discussed mainly when a₅>0 and a₅<0, respectively.A system has not limit cycles in the first quadrant when a₅>0.The stability of equilibrium, the sufficient conditions of nonexistence of closed orbits and the conditions of existence of a stable limit cycles of the system in the first quadrant are obtained when a₅<0.