Attractors of dissipative homeomorphisms of the infinite surface homeomorphic to a punctured sphere. (English) Zbl 1520.37034
Global attractors of dynamical systems are fundamental in the study of many physical models, since they capture all the long-term dynamics of the system itself, such as periodic orbits, invariant closed curves and chaotic patterns.
Consider the surface \(M_s \approx \mathbb{S}^2 \setminus\{p_1, \dotsc, p_s\}\), which is the infinite surface homeomorphic to the \(s\)-punctured sphere. A homeomorphism of \(M_s\) is called dissipative if there exists a compact set \(\widetilde{M}_s \subset M_s\) which attracts uniformly all compact sets. For such a homeomorphism, one can show that the attractor \(\mathcal{A}\), the maximal invariant compact set, is always connected and is equal to the set of all bounded (forward and backward) orbits of the homeomorphism.
Here the authors focus on the class of Levinson homeomorphisms \(\mathcal{LH}(M_s)\) which are dissipative and whose attractor has empty interior. Expanding on the work [J. Math. Kyoto Univ. 49, No. 2, 339–346 (2009; Zbl 1185.37057)] of F. Nakajima on planar diffeomorphisms, the authors give conditions for a Levinson homeomorphism to have an attractor which is not arcwise connected, despite being connected. In particular, they remove the smoothness assumption, generalize the ambient space, and provide isotopy classes of homeomorphisms for which the existence of two fixed points, one of which is an inverse saddle, forces the attractor to be not arcwise connected.
Finally, they show that these results are sharp for the cases of the cylinder and the pair of paints (\(M_2\) and \(M_3\), respectively) and apply their results to some classical models in nonconservative dynamics, such as the ones considered in [R. Martins, J. Differ. Equations 212, No. 2, 351–365 (2005; Zbl 1074.34055)].
Consider the surface \(M_s \approx \mathbb{S}^2 \setminus\{p_1, \dotsc, p_s\}\), which is the infinite surface homeomorphic to the \(s\)-punctured sphere. A homeomorphism of \(M_s\) is called dissipative if there exists a compact set \(\widetilde{M}_s \subset M_s\) which attracts uniformly all compact sets. For such a homeomorphism, one can show that the attractor \(\mathcal{A}\), the maximal invariant compact set, is always connected and is equal to the set of all bounded (forward and backward) orbits of the homeomorphism.
Here the authors focus on the class of Levinson homeomorphisms \(\mathcal{LH}(M_s)\) which are dissipative and whose attractor has empty interior. Expanding on the work [J. Math. Kyoto Univ. 49, No. 2, 339–346 (2009; Zbl 1185.37057)] of F. Nakajima on planar diffeomorphisms, the authors give conditions for a Levinson homeomorphism to have an attractor which is not arcwise connected, despite being connected. In particular, they remove the smoothness assumption, generalize the ambient space, and provide isotopy classes of homeomorphisms for which the existence of two fixed points, one of which is an inverse saddle, forces the attractor to be not arcwise connected.
Finally, they show that these results are sharp for the cases of the cylinder and the pair of paints (\(M_2\) and \(M_3\), respectively) and apply their results to some classical models in nonconservative dynamics, such as the ones considered in [R. Martins, J. Differ. Equations 212, No. 2, 351–365 (2005; Zbl 1074.34055)].
Reviewer: Mauro Artigiani (Bogotá)
MSC:
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
| 37C75 | Stability theory for smooth dynamical systems |
References:
| [1] | Barge, H. and Sanjurjo, J. M., Dissipative flows, global attractors and shape theory, Topol. Appl.258 (2019) 392-401. · Zbl 1414.37013 |
| [2] | Bartuccelli, M. V., Deane, J. H. and Gentile, G., Globally and locally attractive solutions for quasi-periodically forced systems, J. Math. Anal. Appl.328(1) (2007) 699-714. · Zbl 1116.34038 |
| [3] | Bartuccelli, M. V., Gentile, G. and Georgiou, K. V., On the dynamics of a vertically driven damped planar pendulum, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.457(2016) (2001) 3007-3022. · Zbl 1001.70023 |
| [4] | Byszewski, J., Graff, G. and Ward, T., Dold sequences, periodic points, and dynamics, Bull. Lond. Math. Soc.53(5) (2021) 1263-1298. · Zbl 1504.11038 |
| [5] | Dold, A., Fixed point indices of iterated maps, Invent. Math.74(3) (1983) 419-435. · Zbl 0583.55001 |
| [6] | Farb, B. and Margalit, D., A Primer on Mapping Class Groups, , Vol. 49 (Princeton University Press, Princeton, NJ, 2012). |
| [7] | Hale, J., Dissipation and compact attractors, J. Dynam. Differential Equations18(3) (2006) 485-523. · Zbl 1119.37046 |
| [8] | Levi, M., Nonchaotic behavior in the Josephson junction, Phys. Rev. A (3)37(3) (1988) 927-931. |
| [9] | Levinson, N., Transformation theory of non-linear differential equations of the second order, Ann. of Math.45(2) (1944) 723-737. · Zbl 0061.18910 |
| [10] | Martins, R., The effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction, J. Differential Equations212(2) (2005) 351-365. · Zbl 1074.34055 |
| [11] | Martins, R., One-dimensional attractor for a dissipative system with a cylindrical phase space, Discrete Contin. Dyn. Syst.14(3) (2006) 533-547. · Zbl 1119.34043 |
| [12] | Nakajima, F., Connected and not arcwise connected invariant sets for some 2-dimensional dynamical systems, J. Math. Kyoto Univ.49(2) (2009) 339-346. · Zbl 1185.37057 |
| [13] | Ortega, R., Periodic Differential Equations in the Plane: A Topological Perspective, , Vol. 29 (Walter de Gruyter GmbH & Co KG, Berlin2019). · Zbl 1427.34052 |
| [14] | Pliss, V. A., Nonlocal Problems of the Theory of Oscillations (Academic Press, New York, London, 1966). · Zbl 0151.12104 |
| [15] | Richeson, D. and Wiseman, J., Bounded homeomorphisms of the open annulus, New York J. Math.9 (2003) 55-68. · Zbl 1020.37021 |
| [16] | Shen, Z., Zhou, S. and Shen, W., One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation, J. Differential Equations248(6) (2010) 1432-1457. · Zbl 1190.60047 |
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