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How to Analytically Prove the Existence of Strange Attractors Using Measure Theory

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Nonlinear Analysis and Boundary Value Problems (NABVP 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 292))

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Abstract

We describe the attractors for a two-parameter family of two-dimensional piecewise affine maps using measure theory. These piecewise affine maps arise when studying the unfolding of homoclinic tangencies for certain class of three dimensional diffeomorphisms. We also prove the existence, for each natural number n,  of an open set of parameters in which the respective transformation exhibits at least \(2^n\) two-dimensional strange attractors.

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References

  1. Benedicks, M., Carleson, L.: The dynamics of the Hénon map. Ann. Math. 133, 73–169 (1991)

    Article  MathSciNet  Google Scholar 

  2. Buzzi, J.: Absolutely continuous invariant probability measures for arbitrary expanding piecewise \(\mathbb{R}\) analytic mappings of the plane. Ergod. Theory Dyn. Syst. 20, 697–708 (2000)

    Article  MathSciNet  Google Scholar 

  3. Colli, E.: Infinitely many coexisting strange attractors. Annales de l’Institut Henri Poincaré - Analyse non-linéaire 15, 539–579 (1998)

    Article  MathSciNet  Google Scholar 

  4. Lorenz, E.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  5. Mora, L., Viana, M.: Abundance of strange attractors. Acta Math. 171(1), 1–71 (1993)

    Article  MathSciNet  Google Scholar 

  6. Newhouse, S.: Diffeomorphisms with infinitely many sinks. Topology 13, 9–18 (1974)

    Article  MathSciNet  Google Scholar 

  7. Palis, J., Takens, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  8. Pumariño, A., Rodríguez, J.A., Tatjer, J.C., Vigil, E.: Expanding Baker maps as models for the dynamics emerging from 3-D homoclinic bifurcations. Discret. Contin. Dyn. Syst. - Ser. B 19(2), 523–541 (2014)

    Article  MathSciNet  Google Scholar 

  9. Pumariño, A., Rodríguez, J.A., Tatjer, J.C., Vigil, E.: Chaotic dynamics for 2-d tent maps. Nonlinearity 28, 407–434 (2015)

    Article  MathSciNet  Google Scholar 

  10. Pumariño, A., Rodríguez, J.A., Vigil, E.: Expanding baker maps: coexistence of strange attractors. Discret. Contin. Dyn. Syst. - Ser. A 37, 523–550 (2017)

    Article  MathSciNet  Google Scholar 

  11. Pumariño, A., Rodríguez, J.A., Vigil, E.: Renormalization of two-dimensional piecewise linear maps: abundance of 2-D strange attractors. Discret. Contin. Dyn. Syst. - Ser. A 38, 941–966 (2018)

    Article  MathSciNet  Google Scholar 

  12. Pumariño, A., Rodríguez, J.A., Vigil, E.: Persistent two-dimensional strange attractors for a two-parameter family of expanding Baker Maps. Discret. Contin. Dyn. Syst. - Ser. B 24(2), 657–670 (2019)

    Article  MathSciNet  Google Scholar 

  13. Pumariño, A., Tatjer, J.C.: Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms. Nonlinearity 19, 2833–2852 (2006)

    Article  MathSciNet  Google Scholar 

  14. Pumariño, A., Tatjer, J.C.: Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphism. Discret. Contin. Dyn. Syst. - Ser. B 8(4), 971–1005 (2007)

    Article  MathSciNet  Google Scholar 

  15. Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys. 20(3), 167–192 (1971)

    Article  MathSciNet  Google Scholar 

  16. Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223–248 (2000)

    Article  MathSciNet  Google Scholar 

  17. Tatjer, J.C.: Three-dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergod. Theory Dyn. Syst. 21, 249–302 (2001)

    Article  MathSciNet  Google Scholar 

  18. Tsujii, M.: Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143, 349–373 (2001)

    Article  MathSciNet  Google Scholar 

  19. Tucker, W.: The Lorenz attractor exists. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 328(12), 1197–1202 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work has been supported by project MINECO-15-MTM2014-56953-P.

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Authors and Affiliations

  1. Departmento de Matemáticas, Universidad de Oviedo, c Federico García Lorca 18, 33007, Oviedo, Spain

    Antonio Pumariño, José A. Rodríguez & Enrique Vigil

Authors
  1. Antonio Pumariño
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  2. José A. Rodríguez
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  3. Enrique Vigil
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Corresponding author

Correspondence to Enrique Vigil .

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Editors and Affiliations

  1. Departamento de Matemática Aplicada II, Universidade de Vigo, Ourense, Spain

    Iván Area

  2. Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Santiago de Compostela, A Coruña, Spain

    Alberto Cabada

  3. Departamento de Matemáticas, Universidade de Vigo, Ourense, Spain

    José Ángel Cid

  4. E.T.S. Ingenieros Industriales, Universidad Nacional de Educación a Distancia (UNED), Madrid, Spain

    Daniel Franco

  5. Departamento de Matemática Aplicada II, Universidade de Vigo, Vigo, Pontevedra, Spain

    Eduardo Liz

  6. Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Santiago de Compostela, A Coruña, Spain

    Rodrigo López Pouso

  7. Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, Santiago de Compostela, A Coruña, Spain

    Rosana Rodríguez-López

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Pumariño, A., Rodríguez, J.A., Vigil, E. (2019). How to Analytically Prove the Existence of Strange Attractors Using Measure Theory. In: Area, I., et al. Nonlinear Analysis and Boundary Value Problems . NABVP 2018. Springer Proceedings in Mathematics & Statistics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-26987-6_3

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