Title:
Almost Periodically Forced Circle Flows
Almost Periodically Forced Circle Flows
| dc.contributor.author | Huang, Wen | |
| dc.contributor.author | Yi, Yingfei | |
| dc.contributor.corporatename | Georgia Institute of Technology. School of Mathematics | |
| dc.date.accessioned | 2009-08-04T16:46:33Z | |
| dc.date.available | 2009-08-04T16:46:33Z | |
| dc.date.issued | 2007 | |
| dc.description | 2000 Mathematics Subject Classification. Primary 37B05; Secondary 34C28, 54H20. | en |
| dc.description.abstract | We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle. Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multifrequency-driven dynamical irregularities and complexities in non-monotone dynamical systems. | en |
| dc.description.sponsorship | The first author is partially supported by NSFC grant 10531010, 973 project 2006CB805903, and FANEDD grant 200520. The second author is partially supported by NSFC grant 10428101 and NSF grants DMS0204119, DMS0708331. | en |
| dc.identifier.uri | http://hdl.handle.net/1853/29407 | |
| dc.language.iso | en_US | en |
| dc.publisher | Georgia Institute of Technology | en |
| dc.relation.ispartofseries | CDSNS2007-413 | en |
| dc.subject | Almost automorphic dynamics | en |
| dc.subject | Almost periodically forced circle flows | en |
| dc.subject | Forced nonlinear oscillators | en |
| dc.subject | Minimal sets | en |
| dc.subject | Topological dynamics | en |
| dc.title | Almost Periodically Forced Circle Flows | en |
| dc.type | Text | |
| dc.type.genre | Pre-print | |
| dspace.entity.type | Publication | |
| local.contributor.corporatename | College of Sciences | |
| local.contributor.corporatename | School of Mathematics | |
| relation.isOrgUnitOfPublication | 85042be6-2d68-4e07-b384-e1f908fae48a | |
| relation.isOrgUnitOfPublication | 84e5d930-8c17-4e24-96cc-63f5ab63da69 |