The Theory Of C~1-smooth Nonautonomous Monotone Dynamical Systems

This thesis systematically and completely investigates the dynamics of C1-smooth nonautonomous monotone dynamical systems.In differential equation models,the evolution of systems is often affected by periodic or near-periodic forces,which are characterized by nonautonomous systems.When the system is driven by an external periodic force,the Poincare mapping of the equation generates a discrete-time dynamical system;While,if the system is driven by a near-periodic(for instance,almost periodic)external frequency,the equation naturally generates a skew-product(semi)flow.In the first part,we focus on the dynamics of C1-smooth strongly monotone discrete-time dynamical systems.We prove the dynamics alternative,which concludes that any precompact orbit is either asymptotic to a linearly stable cycle;or manifestly unstable.The generic convergence to cycles is obtained as a by-product of the C1dynamics alternative.It follows that the C1-dynamics alternative turns out to be a perfect analogy of the celebrated Hirsch’s limit-set dichotomy for continuous-time semiflows.We further prove the sharpened C1-dynamics alternative,which concludes that there is a positive integer m such that,any precompact orbit is either asymptotic to a linearly stable cycle whose minimal period is bounded by m;or manifestly unstable.As a by-product of the sharpened C1-dynamics alternative,we obtain the sharpened C1-generic convergence for strongly monotone discrete dynamical systems,which concludes that generic orbit converges to cycles whose minimal periods are bounded by m.Furthermore,we study the C1-perturbed systems of C1-smooth strongly monotone discrete-time dynamical systems.We prove C1-robustness of the sharpened C1dynamics alternative,that is,for any C1-perturbed system,any orbit initiated nearby the attractor A is either asymptotic to a linearly stable cycle whose minimal period is bounded by m;or manifestly unstable.As a by-product,we prove that for any C1perturbed system,generic orbit initiated nearby the attractor A converges to cycles whose minimal periods are bounded by m.In the second part,we focus on the almost-periodically forced monotone skewproduct semiflows.Compared with discrete-time systems,the generic convergence theorem usually fails for monotone skew-product semiflows.This is a essential difference between the monotone skew-product semiflow and monotone discrete systems.Under the C1 smoothness assumption,we prove that any linearly stable minimal set in monotone skew-product semiflows must be almost automorphic.This extends the celebrated result of Shen and Yi for the almost automorphy of the classical C1,α-smooth systems to C1-smooth systems.Based on this,one can reduce the regularity of the differential equations in the application and obtain the almost automorphic phenomena in a wider range.