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Torus-Breakdown Near a Heteroclinic Attractor: A Case Study

Luísa Castro

Center for Health Technology and Services Research (CINTESIS), Faculdade de Medicina da Universidade do Porto, Portugal

School of Health Sciences, Polytechnic of Porto, 4200-319 Porto, Portugal

E-mail Address: luisacastro@med.up.pt

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and

Alexandre Rodrigues

Centro de Matemática da Universidade do Porto, 4169-007 Porto, Portugal

E-mail Address: alexandre.rodrigues@fc.up.pt

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https://doi.org/10.1142/S0218127421300299Cited by:6 (Source: Crossref)

Abstract

There are few explicit examples in the literature of vector fields exhibiting observable chaos that may be proved analytically. This paper reports numerical experiments performed for an explicit two-parameter family of $𝕊 𝕆 (2) \oplus ℤ_{2}$ -symmetric vector fields whose organizing center exhibits an attracting heteroclinic network linking two saddle-foci. Each vector field in the family is the restriction to $𝕊^{3}$ of a polynomial vector field in $ℝ^{4}$ . We investigate global bifurcations due to symmetry-breaking and we detect strange attractors via a mechanism called Torus-Breakdown. We explain how an attracting torus gets destroyed by following the changes in the unstable manifold of a saddle-focus.

Although a complete understanding of the corresponding bifurcation diagram and the mechanisms underlying the dynamical changes is out of reach, we uncover complex patterns for the symmetric family under analysis, using a combination of theoretical tools and computer simulations. This article suggests a route to obtain rotational horseshoes and strange attractors; additionally, we make an attempt to elucidate some of the bifurcations involved in an Arnold tongue.

Keywords:

References

Afraimovich, V. S. & Shilnikov, L. P. [1983] “ Strange attractors and quasiattractors,” Nonlinear Dynamics and Turbulence, eds. Barenblatt, G. I., Iooss, G. & Joseph, D. D. (Pitman, Boston), pp. 1–51. Google Scholar
Afraimovich, V. S., Hsu, S.-B. & Lin, H. E. [2001] “ Chaotic behavior of three competing species of May–Leonard model under small periodic perturbations,” Int. J. Bifurcation and Chaos 11, 435–447. Link, Web of Science, Google Scholar
Afraimovich, V. S. & Hsu, S. B. [2003] Lectures on Chaotic Dynamical Systems, AMS/IP Studies in Advanced Mathematics, Vol. 28 (American Mathematical Society/International Press). Crossref, Google Scholar
Afraimovich, V. S. & Shilnikov, L. P. [1991] “On invariant two-dimensional tori, their breakdown and stochasticity,” Methods of the Qualitative Theory of Differential Equations (Gor’kov. Gos. University), pp. 3–26; Translated in: Amer. Math. Soc. Transl. 149, 201–212. Google Scholar
Aguiar, M. [2003] “Vector fields with heteroclinic networks,” PhD thesis, Applied Mathematics Department, Sciences Faculty of University of Porto. Google Scholar
Aguiar, M. A. D., Castro, S. B. S. D. & Labouriau, I. S. [2005] “ Dynamics near a heteroclinic network,” Nonlinearity 18, 391–414. Crossref, Web of Science, Google Scholar
Aguiar, M. A. D., Castro, S. B. S. D. & Labouriau, I. S. [2006] “ Simple vector fields with complex behaviour,” Int. J. Bifurcation and Chaos 16, 369–381. Link, Web of Science, Google Scholar
Anishchenko, V., Safonova, M. & Chua, L. [1993] “ Confirmation of the Afraimovich–Shilnikov torus-breakdown theorem via a torus circuit,” IEEE Trans. Circuits Syst.-I: Fund. Th. Appl. 40, 792–800. Crossref, Web of Science, Google Scholar
Arnold, V. [1961] “ Small denominators. I. Mapping the circle onto itself,” Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 25, 21–86. Google Scholar
Arnold, V. I., Afraimovich, V. S., Ilyashenko, U. S. & Shilnikov, L. P. [2013] Bifurcation Theory and Catastrophe Theory (Springer Science & Business Media). Google Scholar
Aronson, D., Chory, M., Hall, G. & McGehee, R. [1982] “ Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study,” Commun. Math. Phys. 83, 303–354. Crossref, Web of Science, Google Scholar
Baesens, C., Guckenheimer, J., Kim, S. & Mackay, R. [1991] “ Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos,” Physica D 49, 387–475. Crossref, Web of Science, Google Scholar
Bakri, T. & Verhulst, F. [2014] “ Bifurcations of quasi-periodic dynamics: Torus breakdown,” Z. Angew. Math. Phys. 65, 1053–1076. Crossref, Web of Science, Google Scholar
Barrio, R., Carvalho, M., Castro, L. & Rodrigues, A. [2020] “ Experimentally accessible orbits near a Bykov cycle,” Int. J. Bifurcation and Chaos 30, 2030030-1–24. Link, Web of Science, Google Scholar
Bykov, V. V. [2000] “ Orbit structure in a neighbourhood of a separatrix cycle containing two saddle-foci,” Amer. Math. Soc. Transl. 200, 87–97. Crossref, Google Scholar
Capiński, M., Fleurantin, E. & James, J. [2020] “ Computer assisted proofs of two-dimensional attracting invariant tori for ODEs,” Discr. Contin. Dyn. Syst. 12, 6681–6707. Crossref, Web of Science, Google Scholar
Denjoy, A. [1932] “ Sur les courbes définies par les équations différentielles a la surface du tore,” J. Math. Pures Appl. 11, 333–376. Google Scholar
Fleurantin, E. & James, J. M. [2020] “ Resonant tori, transport barriers, and chaos in a vector field with a Neimark–Sacker bifurcation,” Commun. Nonlin. Sci. Numer. Simulat. 85, 105226. Crossref, Web of Science, Google Scholar
Gaspard, P. [1993] “ Local birth of homoclinic chaos,” Physica D 62, 94–122. Crossref, Web of Science, Google Scholar
Golubitsky, M. & Stewart, I. [2003] The Symmetry Perspective, Vol. 200 (Birkhauser). Google Scholar
Gonchenko, S. V., Shilnikov, L. P. & Turaev, D. V. [1997] “ Quasiattractors and homoclinic tangencies,” Comput. Math. Appl. 34, 195–227. Crossref, Web of Science, Google Scholar
Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, Vol. 42 (Springer-Verlag, NY). Crossref, Google Scholar
Herman, M. [1977] Mesure de Lebesgue et Nombre de Rotation, Lecture Notes in Math., Vol. 597 (Springer), pp. 271–293. Crossref, Google Scholar
Hirsch, M. W., Pugh, C. & Shub, M. [1970] “ Invariant manifolds,” Bull. Amer. Math. Soc. 76, 1015–1019. Crossref, Web of Science, Google Scholar
Homburg, A. J. & Sandstede, B. [2010] “ Homoclinic and heteroclinic bifurcations in vector fields,” Handbook of Dynamical Systems, Vol. 3 (North Holland, Amsterdam), pp. 379–524. Crossref, Google Scholar
Kaneko, K. [1983] “ Doubling of torus,” Progr. Theoret. Phys. 69, 1806–1810. Crossref, Web of Science, Google Scholar
Knobloch, J., Lamb, J. S. W. & Webster, K. N. [2014] “ Using Lin’s method to solve Bykov’s problems,” J. Diff. Eqs. 257, 2984–3047. Crossref, Web of Science, Google Scholar
Labouriau, I. S. & Rodrigues, A. A. P. [2012] “ Global generic dynamics close to symmetry,” J. Diff. Eqs. 253, 2527–2557. Crossref, Web of Science, Google Scholar
Labouriau, I. S. & Rodrigues, A. A. P. [2015] “ Dense heteroclinic tangencies near a Bykov cycle,” J. Diff. Eqs. 259, 5875–5902. Crossref, Web of Science, Google Scholar
Mañé, M. [1987] Ergodic Theory and Differentiable Dynamics, Vol. 8 (Springer-Verlag, Berlin). Crossref, Google Scholar
Mora, L. & Viana, M. [1993] “ Abundance of strange attractors,” Acta Math. 171, 1–71. Crossref, Web of Science, Google Scholar
Newhouse, S. E. [1979] “ The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms,” Publ. Math. Inst. Hautes Études Sci. 50, 101–151. Crossref, Google Scholar
Passeggi, A., Potrie, R. & Sambarino, M. [2018] “ Rotation intervals and entropy on attracting annular continua,” Geom. Topol. 22, 2145–2186. Crossref, Web of Science, Google Scholar
Peckham, B. [1990] “ The necessity of the Hopf bifurcation for periodically forced oscillators,” Nonlinearity 3, 261–280. Crossref, Web of Science, Google Scholar
Rodrigues, A. A. P., Labouriau, I. S. & Aguiar, M. A. D. [2011] “ Chaotic double cycling,” Dyn. Syst. Int. J. 26, 199–233. Crossref, Web of Science, Google Scholar
Rodrigues, A. A. P. [2013] “ Repelling dynamics near a Bykov cycle,” J. Dyn. Diff. Eqs. 25, 605–625. Crossref, Web of Science, Google Scholar
Rodrigues, A. A. P. & Labouriau, I. S. [2014] “ Spiralling dynamics near heteroclinic networks,” Physica D 268, 34–49. Crossref, Web of Science, Google Scholar
Rodrigues, A. A. P. [2020] “ Unfolding a Bykov attractor: From an attracting torus to strange attractors,” J. Dyn. Diff. Eqs., https://doi.org/10.1007/s10884-020-09858-z. Crossref, Web of Science, Google Scholar
Ruelle, D. [1981] “ Differentiable dynamical systems and the problem of turbulence,” Bull. Amer. Math. Soc. 5, 29–42. Crossref, Web of Science, Google Scholar
Schwarz, G. [1980] “ Lifting smooth homotopies of orbit spaces,” Publ. Math. l’IHÉS 51, 37–135. Crossref, Google Scholar
Turaev, D. V. & Shilniikov, L. P. [1986] “ Bifurcation of torus-chaos quasi-attractors,” Mathematical Mechanisms of Turbulence (Russian) Akad. Nauk Ukrain., SSR (Inst. Mat., Kiev), pp. 113–121. Google Scholar
Wolf, A., Swift, J. B. H., Swinney, L. & Vastano, J. A. [1985] “ Determining Lyapunov exponents from a time series,” Physica D 16, 285–317. Crossref, Web of Science, Google Scholar