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May  2020, 40(5): 3013-3030. doi: 10.3934/dcds.2020159

Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics

Răzvan M. Tudoran

West University of Timişoara, Faculty of Mathematics and Computer Science, Department of Mathematics, Blvd. Vasile Pȃrvan, No. 4, 300223–Timişoara, Romȃnia

Received  November 2019 Revised  December 2019 Published  March 2020

Given an arbitrary fixed closed subset $ \mathcal{C}\subset\mathbb{R}^n $, we provide an explicit method to construct a dynamical system which admits the regular part of $ \mathcal{C} $ as globally bp-attracting set, i.e. a closed and invariant set which attracts every bounded positive orbit of the dynamical system. As application, we provide an explicit method of leafwise asymptotic bp-stabilization of the regular part of an a-priori given invariant set of a conservative system. The theoretical results are illustrated for the completely integrable case of the Rössler dynamical system.

Keywords: Invariant manifolds, asymptotic bp-stabilization, conservative dynamics.
Mathematics Subject Classification: Primary: 34C45; Secondary: 37J15; 70H33.
Citation: Răzvan M. Tudoran. Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 3013-3030. doi: 10.3934/dcds.2020159
References:
[1]

G.-I. Bischi, C. Mira and L. Gardini, Unbounded sets of attraction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1437-1469.  doi: 10.1142/S0218127400000980.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, 1988.  Google Scholar

[6]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38. SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[7]

J. M. Lee, Introduction to Smooth Manifolds, Second edition, Graduate Texts in Mathematics, 218. Springer, New York, 2013.  Google Scholar

[8]

T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics, London Math. Soc. Lecture Note Ser., Geometric mechanics and symmetry, Cambridge Univ. Press, Cambridge, 306 (2005), 23-156.  doi: 10.1017/CBO9780511526367.003.  Google Scholar

[9] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.   Google Scholar
[10]

R. M. Tudoran and A. G\^irban, On the completely integrable case of the Rössler system, J. Math. Phys., 53 (2012), 052701, 10 pp. doi: 10.1063/1.4708621.  Google Scholar

[11]

R. M. Tudoran, Affine distributions on Riemannian manifolds with applications to dissipative dynamics, J. Geom. Phys., 92 (2015), 55-68.  doi: 10.1016/j.geomphys.2015.01.017.  Google Scholar

[12]

R. M. Tudoran, Asymptotic bp-stabilization of a given closed invariant set of a smooth dynamical system, J. Differential Equations, 267 (2019), 3768-3777.  doi: 10.1016/j.jde.2019.04.013.  Google Scholar

show all references

References:
[1]

G.-I. Bischi, C. Mira and L. Gardini, Unbounded sets of attraction, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 1437-1469.  doi: 10.1142/S0218127400000980.  Google Scholar

[2]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[3]

B. Günther and J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc., 119 (1993), 321-329.  doi: 10.1090/S0002-9939-1993-1170545-4.  Google Scholar

[4]

B. Günther, Construction of differentiable flows with prescribed attractor, Topology Appl., 62 (1995), 87-91.  doi: 10.1016/0166-8641(94)00047-7.  Google Scholar

[5]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, 1988.  Google Scholar

[6]

P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38. SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222.  Google Scholar

[7]

J. M. Lee, Introduction to Smooth Manifolds, Second edition, Graduate Texts in Mathematics, 218. Springer, New York, 2013.  Google Scholar

[8]

T. S. Ratiu, R. M. Tudoran, L. Sbano, E. Sousa Dias and G. Terra, Chapter II: A crash course in geometric mechanics, London Math. Soc. Lecture Note Ser., Geometric mechanics and symmetry, Cambridge Univ. Press, Cambridge, 306 (2005), 23-156.  doi: 10.1017/CBO9780511526367.003.  Google Scholar

[9] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.   Google Scholar
[10]

R. M. Tudoran and A. G\^irban, On the completely integrable case of the Rössler system, J. Math. Phys., 53 (2012), 052701, 10 pp. doi: 10.1063/1.4708621.  Google Scholar

[11]

R. M. Tudoran, Affine distributions on Riemannian manifolds with applications to dissipative dynamics, J. Geom. Phys., 92 (2015), 55-68.  doi: 10.1016/j.geomphys.2015.01.017.  Google Scholar

[12]

R. M. Tudoran, Asymptotic bp-stabilization of a given closed invariant set of a smooth dynamical system, J. Differential Equations, 267 (2019), 3768-3777.  doi: 10.1016/j.jde.2019.04.013.  Google Scholar

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