Quasi convergence and stability for strongly order-preserving semiflows. (English) Zbl 0704.34054

Smith, Hal L.; Thieme, Horst R.

SIAM J. Math. Anal. 21, No. 3, 673-692 (1990).

Summary: Hirsch’s results concerning quasi convergence of almost all trajectories of strongly monotone semiflows are derived under weaker assumptions adopted from Matano. The proofs are based on a sequential limit set trichotomy, which follows from the nonordering principle and the limit set dichotomy. The assumption excluding totally ordered arcs of equilibria, which is required for the set of asymptotically stable points to be dense, is verified for dynamical systems that are analytic on the state space.

MSC:

37-XX	Dynamical systems and ergodic theory
34D20	Stability of solutions to ordinary differential equations
34G20	Nonlinear differential equations in abstract spaces
47H20	Semigroups of nonlinear operators

Keywords:

monotone dynamical system; limit set dichotomy; nonordering principle; global asymptotic stability; quasi convergence; strongly monotone semiflows; sequential limit set trichotomy