| The main object in this dissertation is the monotone recurrence relation.We divide the content into three parts.In the first part.we have the conclusion in Chapter 2 by using pseudo solutions as the main tool:The rotation set is closed.For each rational p/q in lowest terms in the rotation set,there exists a(p,q)-periodie Birklioff solution,and for each irrational ω in the rotation set,there exists a continuous curve,or a Denjoy minimal set,each element of which is a Birkhoff solution with rotation number ω.By constructing a supersolution and a subsolution which exchange rotation numbers,we obtain the next conclusion:If there is a solution with bounded action whose rotation interval is nontrivial,then the system has positive topological entropy.Moreover,we construct connecting orbits shadowing different rotation numbers.In the second part,by adding a parameter F of driving force in Chapter 3,we consider the system generated by the monotone recurrence relation.We call the range of F with pinned states the pinning set Sω,where ω denotes the mean spacing of neighboring particles or rotation number.Actually,Sω is a closed interval.We call its two endpoints the lower depinning force Fd-(ω)and the upper depinning force Fd+(ω),respectively.We get the relationship between depinning forces for the generalized monotone recurrence relation and invariant ordered circles(IOC):For the periodic case,the pinning set Sp/q degenerate to a single point F0 if and only if there exists an IOC in the set of(p,q)-periodic Birkhoff configurations such that all elements are equilibria of the system with F=Fo.For the irrational rotation number case,if there is an IOC with irrational rotation number ω such that each element of the IOC is an equilibrium of the system with F=F0,then the pinning set Sω={F0}.Furthermore,we show the depinning force is continuous at irrational numbers and it depends continuously on parameters in C0 topology due to a fundamental estimate of the depinning force.In Chapter 4,we obtain the existence of one-sided limits of the depinning force at rational points.We consider right limits Fd±{p/q+)of depinning forces at p/q.We obtain the result by defining IOCs of type p/q+:For Fd-(p/q+)≤F≤Fd+(p/q+),there exists an equilibrium of type p/q+connecting(p,q)-periodic equilibria.Furthermore,we verify that the shift map has positive topological entropy on the set of equilibria if F ∈(Fd-(p/q+),Fd+(p/q+)).In the third part,corresponding to Chapter 5 and Chapter 6,we study the monotone recurrence relation with the variational structure.There are two main results.Firstly,we show that one-sided limits of the depinning force at rational points can be a criterion of the existence of minimal foliations.Secondly,we apply the Morse theory of critical points to one kind of variational system and obtain Fd-(0/1+)and Fd+(0/1+)are critical values for traveling fronts and stationary fronts. |
