Skip to main content Accessibility help
×
Home
Hostname: page-component-568f69f84b-cgcw8 Total loading time: 0.236 Render date: 2021-09-17T14:50:16.454Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Global attractors of analytic plane flows

Published online by Cambridge University Press:  01 June 2009

VÍCTOR JIMÉNEZ LÓPEZ
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email: vjimenez@um.es)
DANIEL PERALTA-SALAS
Affiliation:
Departamento de Matemáticas, Universidad Carlos III, 28911 Leganés, Spain (email: dperalta@math.uc3m.es)

Abstract

In this paper the global attractors of analytic and polynomial plane flows are characterized up to homeomorphisms. Following on from previous results for continuous and differentiable dynamical systems, our theorem completes the characterization of the global attractors of plane flows.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andronov, A. A., Leontovich, E. A., Gordon, I. I. and Maĭer, A. G.. Qualitative Theory of Second-order Dynamic Systems. Halsted Press, New York, Toronto, 1973.Google Scholar
[2]Bhatia, N. P. and Szegö, G. P.. Stability Theory of Dynamical Systems. Springer, Berlin, 1970.CrossRefGoogle Scholar
[3]Borsuk, K.. Theory of Shape. Polish Scientific Publishers, Warsaw, 1970.Google Scholar
[4]Garay, B. M.. Strong cellularity and global asymptotic stability. Fund. Math. 138 (1991), 147154.CrossRefGoogle Scholar
[5]Giraldo, A., Morón, M. A., Ruiz del Portal, F. R. and Sanjurjo, J. M. R.. Shape of global attractors in topological spaces. Nonlinear Anal. 60 (2005), 837847.CrossRefGoogle Scholar
[6]Giraldo, A. and Sanjurjo, J. M. R.. On the global structure of invariant regions of flows with asymptotically stable attractors. Math. Z. 232 (1999), 739746.CrossRefGoogle Scholar
[7]Günther, B.. Construction of differentiable flows with prescribed attractor. Topology Appl. 62 (1995), 8791.CrossRefGoogle Scholar
[8]Günther, B. and Segal, J.. Every attractor of a flow on a manifold has the shape of a finite polyhedron. Proc. Amer. Math. Soc. 119 (1993), 321329.CrossRefGoogle Scholar
[9]Gutiérrez, C.. Smoothing continuous flows on two-manifolds and recurrences. Ergod. Th. & Dynam. Sys. 6 (1986), 1744.CrossRefGoogle Scholar
[10]López, V. Jiménez and Llibre, J.. A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane. Adv. Math. 216 (2007), 677710.CrossRefGoogle Scholar
[11]Krantz, S. G. and Parks, H. R.. A Primer of Real Analytic Functions, 2nd edn. Birkhäuser, Boston, 2002.CrossRefGoogle Scholar
[12]Kuratowski, K.. Topology, II. Academic Press, New York, 1968.Google Scholar
[13]Robinson, J. C.. Global attractors: topology and finite-dimensional dynamics. J. Dynam. Differential Equations 11 (1999), 557581.CrossRefGoogle Scholar
[14]Schecter, S. and Singer, M. F.. A class of vectorfields on S 2 that are topologically equivalent to polynomial vectorfields. J. Differential Equations 57 (1985), 406435.CrossRefGoogle Scholar
[15]Sullivan, D.. Combinatorial Invariants of Analytic Spaces (Proceedings of Liverpool Singularities Symposium, I (1969/70)). Springer, Berlin, 1971, pp. 165168.Google Scholar
3
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Global attractors of analytic plane flows
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Global attractors of analytic plane flows
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Global attractors of analytic plane flows
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *