zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite-time attractivity and bifurcation for nonautonomous differential equations. (English) Zbl 1213.34070
The aim of the paper is to introduce nonautonomous and finite-time versions of central concepts from the theory of dynamical systems such as attractivity and bifurcation. The discussion includes an appropriate spectral theory for linear systems as well as finite-time analogues of the well-known transcritical and pitchfork bifurcations. The introduced notions are illustrated by several examples.

34D09Dichotomy, trichotomy
37B55Nonautonomous dynamical systems
37G35Attractors and their bifurcations
34C23Bifurcation (ODE)
Full Text: DOI
[1] Abraham R. H., Marsden J. E. and Ratiu T., Manifolds, Tensor Analysis, and Applications, Springer, New York, (1988) · Zbl 0875.58002
[2] Aulbach B., Gewöhnliche Differenzialgleichungen, Spektrum Akademischer Verlag, Heidelberg, (in German), (2004)
[3] Aulbach B. and Siegmund S., A spectral theory for nonautonomous difference equations, Proceedings of the Fifth Conference on Difference Equations and Applications, Temuco/Chile 2000, Gordon & Breach Publishers, (2000) · Zbl 1062.39014
[4] --, The dichotomy spectrum for noninvertible systems of linear difference equations, Journal of Difference Equations and Applications 7(6), 895--913, (2001) · Zbl 1001.39003 · doi:10.1080/10236190108808310
[5] Berger A., Doan T. S. and Siegmund S., Nonautonomous finite-time dynamics, Discrete and Continuous Dynamical Systems B, 9(3--4), 463--492, (2008) · Zbl 1148.37010 · doi:10.3934/dcdsb.2008.9.463
[6] --, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246, 1098--1118, (2009) · Zbl 1169.34040 · doi:10.1016/j.jde.2008.06.036
[7] Chicone C., Ordinary Differential Equations with Applications, Texts in Applied Mathematics, vol. 34, Springer, New York, (1999) · Zbl 0937.34001
[8] Coddington E. A. and Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York Toronto London, (1955) · Zbl 0064.33002
[9] Colonius F., Kloeden P. E. and Siegmund S., (eds.), Foundations of Nonautonomous Dynamical Systems, Special Issue of Stochastics and Dynamics, vol. 4, (2004) · Zbl 1075.37501
[10] Coppel W. A., Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, (1965) · Zbl 0154.09301
[11] Fabbri R. and Johnson R. A., On a saddle-node bifurcation in a problem of quasiperiodic harmonic forcing, EQUADIFF 2003. Proceedings of the International Conference on Differential Equations, Hasselt, Belgium (Dumortier F., Broer H., Mawhin J., Vanderbauwhede A. and Lunel V., eds.), pp. 839--847, (2005) · Zbl 1107.37040
[12] Glendinning P., Non-smooth pitchfork bifurcations, Discrete and Continuous Dynamical Systems B, 4(2), 457--464, (2004) · Zbl 1056.37069 · doi:10.3934/dcdsb.2004.4.457
[13] Guckenheimer J. and Holmes P., Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, Springer, New York, (1983) · Zbl 0515.34001
[14] Haller G., Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10(1), 99--108, (2000) · Zbl 0979.37012 · doi:10.1063/1.166479
[15] Johnson R. A., Kloeden P. E. and Pavani R., Two-step transition in nonautonomous bifurcations: An explanation, Stochastics and Dynamics, 2(1), 67--92, (2002) · Zbl 1009.34037 · doi:10.1142/S0219493702000297
[16] Johnson R. A. and Mantellini F., A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles, Discrete and Continuous Dynamical Systems, 9(1), 209--224, (2003) · Zbl 1044.37039
[17] Kloeden P. E., Pitchfork and transcritical bifurcations in systems with homogeneous nonlinearities and an almost periodic time coefficient, Communications on Pure and Applied Analysis, 3(2), 161--173, (2004) · Zbl 1228.34058 · doi:10.3934/cpaa.2004.3.161
[18] Kloeden P. E. and Siegmund S., Bifurcations and continuous transitions of attractors in autonomous and nonautonomous systems, International J. Bifurcation and Chaos, 15(3), 743--762, (2005) · Zbl 1079.34041 · doi:10.1142/S0218127405012454
[19] Langa J. A., Robinson J. C. and Suárez A., Stability, instability and bifurcation phenomena in non-autonomous differential equations, Nonlinearity, 15(3), 887--903, (2002) · Zbl 1004.37032 · doi:10.1088/0951-7715/15/3/322
[20] --, Bifurcations in non-autonomous scalar equations, J. Differential Equations, 221(1), 1--35, (2006) · Zbl 1096.34026 · doi:10.1016/j.jde.2005.06.023
[21] Lekien F., Shadden S. C. and Marsden J. E., Lagrangian coherent structures in n-dimensional systems, J. Mathematical Physics 48(6), 065404, 19 pp, (2007) · Zbl 1144.81374 · doi:10.1063/1.2740025
[22] Lyapunov A. M., The General Problem of the Stability of Motion, Mathematical Society of Kharkov, Kharkov, (in Russian), (1892) · Zbl 0041.32204
[23] --, Sur les figures d’equilibre peu differentes des ellipsodies d’une masse liquide homogène donnee d’un mouvement de rotation, Academy of Science St. Petersburg, St. Petersburg, (in French), (1906)
[24] Núñez C. and Obaya R., A non-autonomous bifurcation theory for deterministic scalar differential equations, Discrete and Continuous Dynamical Systems B, 9(3--4), 701--730, (2008) · Zbl 1151.37021 · doi:10.3934/dcdsb.2008.9.701
[25] Poincaré H., Les méthodes nouvelles de la mécanique céleste, Gauthier-Villars, Paris, (3 volumes, in French), (1892-1899)
[26] Pötzsche C., Nonautonomous continuation and bifurcation of bounded solutions I: Difference equations, Manuscript.
[27] --, Robustness of hyperbolic solutions under parametric perturbations, to appear in: Journal of Difference Equations and Applications.
[28] Rasmussen M., Towards a bifurcation theory for nonautonomous difference equations, Journal of Difference Equations and Applications, 12(3--4), 297--312, (2006) · Zbl 1136.37314 · doi:10.1080/10236190500489400
[29] --, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Springer Lecture Notes in Mathematics, vol. 1907, Springer, Berlin, Heidelberg, New York, (2007) · Zbl 1131.37001
[30] --, Nonautonomous bifurcation patterns for one-dimensional differential equations, J. Differential Equations, 234(1), 267--288, (2007) · Zbl 1125.34032 · doi:10.1016/j.jde.2006.11.002
[31] Sacker R. J. and Sell G. R., Existence of dichotomies and invariant splittings for linear differential systems I, J. Differential Equations 15, 429--458, (1974) · Zbl 0294.58008 · doi:10.1016/0022-0396(74)90067-9
[32] --, A spectral theory for linear differential systems, J. Differential Equations, 27, 320--358, (1978) · Zbl 0372.34027 · doi:10.1016/0022-0396(78)90057-8
[33] Shadden S. C., Lekien F. and Marsden J. E., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, Nonlinear Phenomena, 212(3--4), 271--304, (2005) · Zbl 1161.76487 · doi:10.1016/j.physd.2005.10.007
[34] Siegmund S., Dichotomy spectrum for nonautonomous differential equations, J. Dynamics and Differential Equations, 14(1), 243--258, (2002) · Zbl 0998.34045 · doi:10.1023/A:1012919512399
[35] Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathematics, vol. 2, Springer, New York, (1990) · Zbl 0701.58001