On the Omega-Limit Sets of Planar Nonautonomous Differential Equations with Nonpositive Lyapunov Exponents

The well-known Poincare-Bendixson theorem tells us that the structure of the omega-limit sets of planar autonomous differential equations can be described by fixed points, limit cycles, or finite number of fixed points together with homoclinic and heteroclinic orbits connecting them. However, this is very different for planar nonautonomous differential equations. In this paper, we study the omega-limit sets of three classes of planar nonautonomous differential equations, and their corresponding dynamical behavior. (i) The first type is linear, the omega-limit set is an annulus, the Lyapunov exponents are zero, there is no periodic orbit except for one fixed point, and the orbits are transitive in the omega-limit sets, implying the existence of infinitely many transitive components depending on initial conditions. (ii) The second type has butterfly-like omega-limit sets, two zero Lyapunov exponents, and neither periodic orbits nor sensitive dependence on initial conditions. (iii) The last type has a unique omega-limit set (a global attractor) independent on initial conditions, and has two negative Lyapunov exponents, where the omega-limit set is an annulus or a subset (homeomorphic to a disk) of an annulus for different parameter regions.