Topological bifurcations of minimal invariant sets for set-valued dynamical systems
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- by Jeroen S. W. Lamb, Martin Rasmussen and Christian S. Rodrigues
- Proc. Amer. Math. Soc. 143 (2015), 3927-3937
- DOI: https://doi.org/10.1090/S0002-9939-2015-12544-0
- Published electronically: April 2, 2015
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Abstract:
We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are naturally satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.References
- Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330, DOI 10.1007/978-3-642-69512-4
- Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
- Vítor Araújo, Attractors and time averages for random maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire 17 (2000), no. 3, 307–369 (English, with English and French summaries). MR 1771137, DOI 10.1016/S0294-1449(00)00112-8
- L. Arnold, Random Dynamical Systems, Springer, Berlin, Heidelberg, New York, 1998.
- Zvi Artstein, Invariant measures of set-valued maps, J. Math. Anal. Appl. 252 (2000), no. 2, 696–709. MR 1800186, DOI 10.1006/jmaa.2000.7095
- Peter Ashwin, Minimal attractors and bifurcations of random dynamical systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1987, 2615–2634. MR 1807833, DOI 10.1098/rspa.1999.0419
- Peter H. Baxendale, A stochastic Hopf bifurcation, Probab. Theory Related Fields 99 (1994), no. 4, 581–616. MR 1288071, DOI 10.1007/BF01206233
- Carlos J. Braga Barros and Josiney A. Souza, Attractors and chain recurrence for semigroup actions, J. Dynam. Differential Equations 22 (2010), no. 4, 723–740. MR 2734477, DOI 10.1007/s10884-010-9164-3
- Ryan T. Botts, Ale Jan Homburg, and Todd R. Young, The Hopf bifurcation with bounded noise, Discrete Contin. Dyn. Syst. 32 (2012), no. 8, 2997–3007. MR 2903997, DOI 10.3934/dcds.2012.32.2997
- Fritz Colonius, Tobias Gayer, and Wolfgang Kliemann, Near invariance for Markov diffusion systems, SIAM J. Appl. Dyn. Syst. 7 (2008), no. 1, 79–107. MR 2399558, DOI 10.1137/040618539
- Fritz Colonius, Ale Jan Homburg, and Wolfgang Kliemann, Near invariance and local transience for random diffeomorphisms, J. Difference Equ. Appl. 16 (2010), no. 2-3, 127–141. MR 2640434, DOI 10.1080/10236190802653646
- Fritz Colonius and Wolfgang Kliemann, The dynamics of control, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 2000. With an appendix by Lars Grüne. MR 1752730, DOI 10.1007/978-1-4612-1350-5
- Fritz Colonius and Wolfgang Kliemann, Limits of input-to-state stability, Systems Control Lett. 49 (2003), no. 2, 111–120. MR 2011661, DOI 10.1016/S0167-6911(02)00315-8
- Fritz Colonius, Albert Marquardt, Edwin Kreuzer, and Wolfgang Sichermann, A numerical study of capsizing: comparing control set analysis and Melnikov’s method, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 18 (2008), no. 5, 1503–1514. MR 2427137, DOI 10.1142/S0218127408021129
- Fritz Colonius and Tobias Wichtrey, Control systems with almost periodic excitations, SIAM J. Control Optim. 48 (2009), no. 2, 1055–1079. MR 2486105, DOI 10.1137/070704733
- Klaus Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1992. MR 1189795, DOI 10.1515/9783110874228
- T. Gayer, Control sets and their boundaries under parameter variation, J. Differential Equations 201 (2004), no. 1, 177–200. MR 2057543, DOI 10.1016/j.jde.2004.02.006
- Tobias Gayer, Controllability and invariance properties of time-periodic systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), no. 4, 1361–1375. MR 2152078, DOI 10.1142/S021812740501265X
- L. Grüne and P. E. Kloeden, Discretization, inflation and perturbation of attractors, Ergodic theory, analysis, and efficient simulation of dynamical systems, Springer, Berlin, 2001, pp. 399–416. MR 1850315
- L. Grüne, Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization, Springer Lecture Notes in Mathematics, vol. 1783, Springer, Berlin, Heidelberg, 2002.
- A. J. Homburg and T. Young, Hard bifurcations in dynamical systems with bounded random perturbations, Regul. Chaotic Dyn. 11 (2006), no. 2, 247–258. MR 2245080, DOI 10.1070/RD2006v011n02ABEH000348
- Ale Jan Homburg and Todd R. Young, Bifurcations of random differential equations with bounded noise on surfaces, Topol. Methods Nonlinear Anal. 35 (2010), no. 1, 77–97. MR 2677432
- R. A. Johnson, P. E. Kloeden, and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation, Stoch. Dyn. 2 (2002), no. 1, 67–92. MR 1897353, DOI 10.1142/S0219493702000297
- P. E. Kloeden, General control systems, Mathematical control theory (Proc. Conf., Australian Nat. Univ., Canberra, 1977) Lecture Notes in Math., vol. 680, Springer, Berlin, 1978, pp. 119–137. MR 515715
- Peter E. Kloeden and Pedro Marín-Rubio, Negatively invariant sets and entire trajectories of set-valued dynamical systems, Set-Valued Var. Anal. 19 (2011), no. 1, 43–57. MR 2770896, DOI 10.1007/s11228-009-0123-2
- Desheng Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim. 46 (2007), no. 1, 35–60. MR 2299619, DOI 10.1137/060662101
- Richard McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J. 41 (1992), no. 4, 1165–1209. MR 1206344, DOI 10.1512/iumj.1992.41.41058
- R. P. McGehee and T. Wiandt, Conley decomposition for closed relations, J. Difference Equ. Appl. 12 (2006), no. 1, 1–47. MR 2197583, DOI 10.1080/00207210500171620
- Emilio Roxin, On generalized dynamical systems defined by contingent equations, J. Differential Equations 1 (1965), 188–205. MR 201756, DOI 10.1016/0022-0396(65)90019-7
- Emilio Roxin, Control Theory and Its Applications, Stability and Control: Theory, Methods and Applications, vol. 4, Gordon and Breach Science Publishers, Amsterdam, 1997.
- Hicham Zmarrou and Ale Jan Homburg, Bifurcations of stationary measures of random diffeomorphisms, Ergodic Theory Dynam. Systems 27 (2007), no. 5, 1651–1692. MR 2358982, DOI 10.1017/S0143385707000077
- Hicham Zmarrou and Ale Jan Homburg, Dynamics and bifurcations of random circle diffeomorphisms, Discrete Contin. Dyn. Syst. Ser. B 10 (2008), no. 2-3, 719–731. MR 2425065, DOI 10.3934/dcdsb.2008.10.719
Bibliographic Information
- Jeroen S. W. Lamb
- Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- MR Author ID: 319947
- Martin Rasmussen
- Affiliation: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- MR Author ID: 751819
- Christian S. Rodrigues
- Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany
- Received by editor(s): November 26, 2013
- Received by editor(s) in revised form: May 12, 2014
- Published electronically: April 2, 2015
- Additional Notes: The first author was supported by an FAPESP-Brazil Visiting Professorship (2009-18338-2)
The first and second author gratefully acknowledge partial support by EU IRSES project DynEurBraz and the warm hospitality of IMECC UNICAMP during the development of this paper
The second author was supported by an EPSRC Career Acceleration Fellowship and a Marie Curie Intra-European Fellowship of the European Community
The third author has received ERC funding under EU’s Seventh Framework Programme FP7 (grant agreement number: 267087) - Communicated by: Yingfei Yi
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3927-3937
- MSC (2010): Primary 37G35, 37H20, 37C70, 49K21; Secondary 37B25, 34A60
- DOI: https://doi.org/10.1090/S0002-9939-2015-12544-0
- MathSciNet review: 3359583