1. On the first Liapunov coefficient formula of 3D Lotka-Volterra equations with applications to multiplicity of limit cycles
- Author
-
Jifa Jiang, Wenxi Wu, Shuo Huang, and Fengli Liang
- Subjects
Hopf bifurcation ,Series (mathematics) ,Applied Mathematics ,010102 general mathematics ,Lotka–Volterra equations ,Multiplicity (mathematics) ,Symbolic computation ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,symbols.namesake ,Limit cycle ,symbols ,Applied mathematics ,Limit (mathematics) ,0101 mathematics ,Analysis ,Mathematics - Abstract
This paper provides the first Liapunov coefficient formula of 3D Lotka-Volterra equations. This formula gives applications to stability of positive equilibrium and to detecting sub/super criticality of Hopf bifurcation. For 3D competitive Lotka-Volterra equations, combining this formula with the Poincare-Bendixson theorem, we obtain criteria on multiplicity of limit cycles among Zeeman's classes 27-31, and present a series of examples to admit at least two limit cycles, which are rigorously proved by the first Liapunov coefficient formula, rather than by symbolic computation using Maple. A new Hopf bifurcation that all 2 × 2 principal minors of the community matrix are positive is found, and numerical simulation reveals its global limit cycle bifurcations are plenty.
- Published
- 2021