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Existence of SRB measures for some Rössler systems

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Contributed by Theophile Caby ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

We prove that for the Rössler systemẋ = −(y + z),ẏ = x + ay,ż = b + z(x − c) with a = b = 0.2, the set of parameters c within 6 · 10 −5 of the classical value c = 5.7 for which the ow has a Hénon-like strange attractor supporting a SinaiRuelleBowen (SRB) measure has positive Lebesgue measure. The key step, proved with computer assistance, is that at an explicit parameter within 5.5 · 10 −5 of 5.7 the period-one saddle of the Poincaré map has a quadratic homoclinic tangency that unfolds generically; the conclusion then follows from the theorems of MoraViana, BenedicksYoung and BenedicksViana. The same method gives two further such tangencies near 5.7 and one at the band-merging parameter c ≈ 4.507. The proof reduces the generic-tangency conditions to nitely many inequalities, through explicit C 2 graph-transform bounds for the local invariant manifolds of a Poincaré map whose area contraction is of order 10 −11 10 −14 , and a scalar tangency function. The inequalities are checked with interval arithmetic in CAPD and rechecked with an independent implementation based on the Kv library; the analytic tangency criterion is formalised in Lean 4. Whether an SRB measure exists at c = 5.7 itself remains open.

Provenance statement

Claude Opus 5.5 max

References

  1. V. Araújo, M. J. Pacifico, E. R. Pujals, and M. Viana, Singular-hyperbolic attractors are chaotic , Trans. Amer. Math. Soc. 361 (2009), 2431--2485.
  2. Z. Arai and K. Mischaikow, Rigorous computations of homoclinic tangencies , SIAM J. Appl. Dyn. Syst. 5 (2006), 280--292.
  3. P. G. Barrientos and J. D. Rojas, Typical coexistence of infinitely many strange attractors , preprint, arXiv:2102.08316.arXiv
  4. M. Benedicks, Non uniformly hyperbolic dynamics: Hénon maps and related dynamical systems , survey, arXiv:math/0304450.arXiv
  5. M. Benedicks and L. Carleson, The dynamics of the Hénon map , Ann. of Math. 133 (1991), 73--169.
  6. M. Benedicks and M. Viana, Solution of the basin problem for Hénon-like attractors , Invent. Math. 143 (2001), 375--434.
  7. M. Benedicks and L.-S. Young, Sinai--Bowen--Ruelle measures for certain Hénon maps , Invent. Math. 112 (1993), 541--576.
  8. P. Berger, Strong regularity. Abundance of non-uniformly hyperbolic Hénon-like endomorphisms , Astérisque 410 (2019), 53--177.
  9. X. Cabré, E. Fontich, and R. de la Llave, The parameterization method for invariant manifolds I: manifolds associated to non-resonant subspaces , Indiana Univ. Math. J. 52 (2003), 283--328.
  10. M. J. Capiński and A. Wasieczko-Zając, Computer assisted proof of Shil'nikov homoclinics: with application to the Lorenz-84 model , SIAM J. Appl. Dyn. Syst. 16 (2017), 1453--1473.
Show all 39 referencesShow fewer references
  1. E. Catsigeras and H. Enrich, SRB-like measures for C0C^0 dynamics , Bull. Pol. Acad. Sci. Math. 59 (2011), 151--164.
  2. B. Coomes, H. Koçak, and K. J. Palmer, A computable criterion for the existence of homoclinic tangencies , J. Dynam. Differential Equations (2026), doi:10.1007/s10884-026-10533-y.DOI
  3. Y. Ding and L. Zheng, Existence of homoclinic orbit of Shilnikov type and the application in Rössler system , Math. Comput. Simulation 206 (2023), 770--779, doi:10.1016/j.matcom.2022.12.013.DOI
  4. Z. Galias, Is the classical Rössler attractor periodic? A validated numerical study , Chaos 36 (2026), no. 5, 053121, doi:10.1063/5.0301581.DOI
  5. A. Gierzkiewicz and P. Zgliczyński, Periodic orbits in Rössler system , preprint, arXiv:2101.10406 (2021).arXiv
  6. S. V. Gonchenko, A. S. Gonchenko, A. O. Kazakov, and A. D. Kozlov, Elements of contemporary theory of dynamical chaos: a tutorial. Part I. Pseudohyperbolic attractors , Int. J. Bifurcation Chaos 28 (2018), 1830036.
  7. S. V. Gonchenko, L. P. Shilnikov, and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I , Nonlinearity 21 (2008), 923--972.
  8. E. Igra, One dimensional dynamics and the Rössler attractor , preprint, arXiv:2312.13840 (2023).arXiv
  9. F. Immler, A verified ODE solver and the Lorenz attractor , J. Automat. Reason. 61 (2018), 73--111.
  10. T. Kapela, M. Mrozek, D. Wilczak, and P. Zgliczyński, CAPD::DynSys: a flexible C++ toolbox for rigorous numerical analysis of dynamical systems , Commun. Nonlinear Sci. Numer. Simul. 101 (2021), 105578.
  11. M. Kashiwagi, kv --- a C++ library for verified numerical computation , http://verifiedby.me/kv/, version 0.4.62.link
  12. R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken , Computing 4 (1969), 187--201.
  13. R. J. Lohner, Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems , in: Computational Ordinary Differential Equations (J. R. Cash and I. Gladwell, eds.), Clarendon Press, Oxford, 1992, 425--435.
  14. The mathlib Community, The Lean mathematical library , in: Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020), 367--381.
  15. L. Mora and M. Viana, Abundance of strange attractors , Acta Math. 171 (1993), 1--71.
  16. L. de Moura and S. Ullrich, The Lean 4 theorem prover and programming language , in: Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci. 12699 , Springer, 2021, 625--635.
  17. A. Neumaier, Interval Methods for Systems of Equations , Encyclopedia of Mathematics and its Applications 37, Cambridge University Press, 1990.
  18. J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations , Cambridge Studies in Advanced Mathematics 35, Cambridge University Press, 1993.
  19. N. Romero, Persistence of homoclinic tangencies in higher dimensions , Ergodic Theory Dynam. Systems 15 (1995), 735--757.
  20. O. E. Rössler, An equation for continuous chaos , Phys. Lett. A 57 (1976), 397--398.
  21. S. Sternberg, On the structure of local homeomorphisms of euclidean nn-space, II , Amer. J. Math. 80 (1958), 623--631.
  22. W. Tucker, A rigorous ODE solver and Smale's 14th problem , Found. Comput. Math. 2 (2002), 53--117.
  23. M. Viana, Dynamics: a probabilistic and geometric perspective , in: Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), Doc. Math. Extra Vol. ICM 1998, 557--578.
  24. D. Wilczak, S. Serrano, and R. Barrio, A mechanism for growth of topological entropy and global changes of the shape of chaotic attractors , Chaos 35 (2025), no. 10, doi:10.1063/5.0284636.DOI
  25. D. Wilczak and P. Zgliczyński, Computer assisted proof of the existence of homoclinic tangency for the Hénon map and for the forced damped pendulum , SIAM J. Appl. Dyn. Syst. 8 (2009), 1632--1663.
  26. D. Wilczak and P. Zgliczyński, Period doubling in the Rössler system---a computer assisted proof , Found. Comput. Math. 9 (2009), 611--649.
  27. D. Wilczak and P. Zgliczyński, CrC^r-Lohner algorithm , Schedae Informaticae 20 (2011), 9--46.
  28. L.-S. Young, What are SRB measures, and which dynamical systems have them? , J. Stat. Phys. 108 (2002), 733--754.
  29. P. Zgliczyński, Computer assisted proof of chaos in the Rössler equations and in the Hénon map , Nonlinearity 10 (1997), 243--252.

Version history

  1. v1Submitted by Theophile CabyInitial depositCurrentOct 30, 2026