Prevalent Behavior Of Monotone Dynamical Systems And The Birkhoff Center Of Competitive Systems

This thesis mainly focuses on the systems in the measure-theoretic sense,and analyzes the prevalent behavior in monotone dynamical systems and the Birkhoff center of competitive systems.Prevalence is a useful perspective to analyze the system in the measure-theoretic sense,and is parallel to another classical notion in the topological sense called genericity.Firstly,we investigate deeply the discrete strong monotone dynamic system and obtain that prevalent points with pre-compact orbit converge to a linearly stable cycle.This result analyzes the behavior of discrete monotone dynamical systems in the measure-theoretic sense,which is parallel to generic convergence theorem for the discrete monotone dynamical systems.It shows genericity and prevalence are highly unified in monotone systems.Then,we utilize the result to the parabolic equations whose nonlinear term is periodic,and obtain that the solutions of the equations converge to the periodic solutions.Secondly,we investigate the global dynamics from the measure-theoretic perspective for smooth flows with invariant cones of rank k.For such systems,it is shown that prevalent(or equivalently,almost all)orbits will be pseudo-ordered or convergent to equilibria.Here,the pseudo-ordered orbit is an orbit which contains at least two different ordered points.This reduces to Hirsch’s prevalent convergence Theorem if the rank k=1 and implies an almost-sure Poincare-Bendixson Theorem for the case k=2.These results are then applied to obtain an almost-sure Poincare-Bendixson Theorem for high-dimensional differential eauqtions.In the study,we utilize the k-exponential separation and k-Lyapunov exponent to obtain the Probe lemma.The lemma effectively analyzes the structure of every point’s probe neighborhood,and obtain that if the observed point has no pseudo ordered orbit and does not converge to equilibria,then the all points in its probe neighborhood have pseudo ordered orbit or converge to equilibria.In dynamic systems,invariant measure is a crucial perspective to analyze the system in the measure-theoretic sense.And Birkhoff center contains the core information of systems including the support of any invariant measure.In the end of this thesis,we are going to analyze the order-structure of the Birkhoff center of competitive systems.We obtain that the connect part of Birkhoff center is either strongly ordered or unordered.In this way,we obtain the dichotomy of the support of any invariant measure:the support of any ergodic measure is either strongly ordered or unordered.Then,we utilize it to the competitive systems in R3 and obtain that the topological entropy of such systems is zero.