The Research On Chaoticity Of Several Systems

Chaos is an interdisciplinary academic branch which is developed for nearly fourdecades. Its essence is to study the uncertainty in nonlinear system, especially theregularities inherent in the uncertainty. These properties can be used to control thesystems. With the development of basic science and applied science, the research onchaos theory and applied has become one of the main topics of nonlinear science.On the basis of the discussion about the development of chaos, this dissertationfirst reviews the common definitions of chaos. Then the dissertation focus on thechaotic properties of continuous self-maps in five systems, especially the characteristicsof Li-Yorke chaos, Devaney chaos, spatio-temporal chaos,（F₁,F₂）-chaos, distributionalchaos, dense chaotic, sensitivity, and Li-Yorke sensitivity. The following six aspects ofresults are obtained:1. On topological spaces, ω-chaos and four Devaney chaos （DevC, EDevC,MDevC, WMDevC） are preserved under topological conjugation. So as in generalmetric spaces. While, on general metric spaces, topological conjugate do not keepLi-Yorke chaos. It needs conditions ‘compact space+topological conjugate’ or‘uniformly conjugation’. Additionally, it is proved that, on general metric spaces, theAuslander-Yorke chaos, sensitivity, distributional chaos, distributional chaos in asequence, dense chaos, and dense δ-chaos are all maintained under uniformlyconjugation.2. Considering of periodic points of continuous self-maps on linear orderedtopological system, it is pointed that there exist periodic points if horseshoes existed,and odd periodic points imply the existence of horseshoes. According to the study of theunstable manifolds, we prove that the interval with endpoints of two adjacent fixedpoints is contained in the unilateral unstable manifold of one of the endpoints. If thecontinuous self-map has finitely many periodic points, then the unstable manifold of afixed point p is divided into two zones, namely, the left and right unilateral unstablemanifolds of p. Through the research of dense orbits, we point that dense orbits implytopological transitivity. And if the set of even iterate points is dense, then the set ofk （mod s）iterate points is dense too.3. On a class of coupled map lattice related to the Belousov-Zhabotinsky oscillating reaction, with the metric defined in the fourth chapter, a sufficient conditionfor the system is（F₁, F₂）-Chaos （Li-Yorke chaos, or distributional chaos） is obtained.If the induced mapping is limited on the diagonal of the space, a similar sufficientcondition will be obtained for the system is dense chaos, dense δ-chaos,spatio-temporal chaos, sensitivity, or Li-Yorke sensitivity. However, these sufficientconditions do not always established if the metric changes. Chaotic original maps cannot guarantee the chaoticity of the system. The dissertation also points out that thetopological entropy of the system is not less than topological entropy of the originalmapping. If the topological entropy of original mapping is greater than0, then thesystem is ω-chaotic.4. According to study distributional chaotic of a weighted shifts operator. Weproved that the weighted shifts operator is distributional ε-chaos and uniformlydistributional chaos （where ε is any value greater than0and less than diameter of thespace）. And these chaotic properties are preserved under product operation. Then, it isproved that the principal measure of this weighted shifts operator is1.5. In non-autonomous system, it is proved that the chaotic off₁,∞is a necessaryand sufficient condition for the chaotic offn,∞. If the map sequencef₁,∞is P-chaos,then the product mapping sequencef[m]1,∞（m is a positive integer） is P-chaos too.There P-chaos denote Li-Yorke chaos, distributional chaos, sensitivity, Li-Yorkesensitivity, or Li-Yorke sensitivity. And iff₁,∞converges uniformly, then the converseof the above conclusion is true.6. The Li-Yorke chaotic （distributional chaotic, or distributional chaotic in asequence） on the whole space of Cournot mapping is equivalent to the one limited onMPE-set. And an example is illustrated to show that this conclusion does not hold forsensitivity or Li-Yorke sensitivity.Finally, the dissertation summarized the work, prospected the work which isrequired in-depth study. Thus, the dissertation laid some foundation for the futureresearch.