Systems of differential equations that are competitive or cooperative. IV: Structural stability in three-dimensional systems. (English) Zbl 0734.34042

Hirsch, Morris W.

SIAM J. Math. Anal. 21, No. 5, 1225-1234 (1990).

Summary: [For parts I-III see ibid. 13, 167-179 (1982; Zbl 0494.34017); ibid. 16, 423-439 (1985; Zbl 0658.34023); Nonlinearity 1, No.1, 51-71 (1988; Zbl 0658.34024); for part V see J. Differ. Equations 80, No.1, 94-106 (1989; Zbl 0712.34045).]
It is shown that among three-dimensional systems that are competitive or cooperative, those satisfying the generic Kupka-Smale conditions also satisfy the Morse-Smale conditions and are therefore structurally stable. This identifies a new and easily recognizable class of systems approximable by structurally stable systems.

MSC:

34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
37C10	Dynamics induced by flows and semiflows

Keywords:

structural stability; dynamical system; three-dimensional systems; competitive; cooperative; generic Kupka-Smale conditions; Morse-Smale conditions

Citations:

Zbl 0494.34017; Zbl 0658.34023; Zbl 0658.34024; Zbl 0712.34045