# American Institute of Mathematical Sciences

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

 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.

Citation: Răzvan M. Tudoran. Dynamical systems with a prescribed globally bp-attracting set and applications to conservative dynamics. Discrete and 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. [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. [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. [4] B. Günther, Construction of differentiable flows with prescribed attractor, Topology Appl., 62 (1995), 87-91.  doi: 10.1016/0166-8641(94)00047-7. [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, 1988. [6] P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38. SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222. [7] J. M. Lee, Introduction to Smooth Manifolds, Second edition, Graduate Texts in Mathematics, 218. Springer, New York, 2013. [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. [9] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011. [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. [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. [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.

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##### 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. [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. [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. [4] B. Günther, Construction of differentiable flows with prescribed attractor, Topology Appl., 62 (1995), 87-91.  doi: 10.1016/0166-8641(94)00047-7. [5] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, 1988. [6] P. Hartman, Ordinary Differential Equations, Classics in Applied Mathematics, 38. SIAM, Philadelphia, PA, 2002. doi: 10.1137/1.9780898719222. [7] J. M. Lee, Introduction to Smooth Manifolds, Second edition, Graduate Texts in Mathematics, 218. Springer, New York, 2013. [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. [9] J. C. Robinson, Dimensions, Embeddings, and Attractors, Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011. [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. [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. [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.
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