We present a mechanism for the emergence of strange attractors in a one-parameter family of differential equations defined on a 3-dimensional sphere. When the parameter is zero, its flow exhibits an attracting heteroclinic network (Bykov network) made by two 1-dimensional connections and one 2-dimensional separatrix between two saddles-foci with different Morse indices. After slightly increasing the parameter, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We will show that, for a set of parameters close enough to zero with positive Lebesgue measure, the dynamics exhibits strange attractors winding around the "ghost'' of a torus and supporting Sinai-Ruelle-Bowen (SRB) measures. We also prove the existence of a sequence of parameter values for which the family exhibits a superstable sink and describe the transition from a Bykov network to a strange attractor.
Citation: |
Figure 5. Adapted bifurcation diagram of [25] for the family (4): below the graph of $ t_2 $, the dynamics is governed by an attracting invariant 2-dimensional torus. Above the graph of $ t_1 $, one observes rotational horseshoes. In between, rotational horseshoes exist but they may be not observable in numerics; the dynamics is probably dominated either by a sink or bistability between a sink and an attracting torus
Figure 6. Illustration (based on plots obtained by using Maple) of $ h_a(x) \pmod{2\pi} $ for $ a = 0.1 $, $ \xi = 0 $ and $ \Phi_2(x, 0) = \sin x $ ($ \Rightarrow q = 2 $). (a) $ K_\omega \approx 0 $ – attracting torus. (b) $ K_\omega = 0.5 $ – attracting torus inside a resonance wedge with a saddle and a sink. (c): $ K_\omega = 2 $ – torus breakdown. (d): $ K_\omega = 5 $ – mixing properties ("big lobe'' of [30])
[1] |
V. S. Afraimovich, S.-B. Hsu and H. E. Lin, Chaotic behavior of three competing species of May–Leonard model under small periodic perturbations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 435-447.
doi: 10.1142/S021812740100216X.![]() ![]() ![]() |
[2] |
M. Aguiar, Vector fields with heteroclinic networks, Ph.D. thesis, Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto, 2003.
![]() |
[3] |
P. Ashwin and P. Chossat, Attractors for robust heteroclinic cycles with continua of connections, J. Nonlinear Sci., 8 (1998), 103-129.
doi: 10.1007/s003329900045.![]() ![]() ![]() |
[4] |
I. Baldomá, S. Ibáñez and T. Seara, Hopf-Zero singularities truly unfold chaos, Commun. Nonlinear Sci. Numer. Simul., 84 (2020), 105162.
doi: 10.1016/j.cnsns.2019.105162.![]() ![]() ![]() |
[5] |
M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169.
doi: 10.2307/2944326.![]() ![]() ![]() |
[6] |
M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle measures for certain Hénon maps, Invent. Math., 112 (1993), 541-576.
doi: 10.1007/BF01232446.![]() ![]() ![]() |
[7] |
H. Broer, C. Simó and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms, Nonlinearity, 11 (1998), 667-770.
doi: 10.1088/0951-7715/11/3/015.![]() ![]() ![]() |
[8] |
V. V. Bykov, Orbit Structure in a neighborhood of a separatrix cycle containing two saddle-foci, Translations of the American Mathematical Society - Series 2, 200 (2000), 87-97.
doi: 10.1090/trans2/200/08.![]() ![]() ![]() |
[9] |
M. L. Castro and A. A. P. Rodrigues, Torus-breakdown near a heteroclinic attractor: A case study, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 31 (2021), 2130029.
doi: 10.1142/S0218127421300299.![]() ![]() ![]() |
[10] |
B. Deng, The shilnikov problem, exponential expansion, strong $\lambda$–lemma, $C^1$ linearisation and homoclinic bifurcation, J. Diff. Eqs., 79 (1989), 189-231.
doi: 10.1016/0022-0396(89)90100-9.![]() ![]() ![]() |
[11] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.
doi: 10.1007/978-1-4612-1140-2.![]() ![]() ![]() |
[12] |
M. Hénon, A two dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.
doi: 10.1007/BF01608556.![]() ![]() ![]() |
[13] |
A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
doi: 10.1088/0951-7715/15/4/304.![]() ![]() ![]() |
[14] |
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook of Dynamical Systems, 3 (2010), 379-524.
doi: 10.1016/S1874-575X(10)00316-4.![]() ![]() |
[15] |
M. Jakobson, Absolutely continuous invariant measures for one parameter families of one-dimensional maps, Comm. Math. Phys., 81 (1981), 39-88.
doi: 10.1007/BF01941800.![]() ![]() ![]() |
[16] |
I. S. Labouriau and A. A. P. Rodrigues, Global generic dynamics close to symmetry, J. Diff. Eqs., 253 (2012), 2527-2557.
doi: 10.1016/j.jde.2012.06.009.![]() ![]() ![]() |
[17] |
I. S. Labouriau and A. A. P. Rodrigues, Dense heteroclinic tangencies near a Bykov cycle, J. Diff. Eqs., 259 (2015), 5875-5902.
doi: 10.1016/j.jde.2015.07.017.![]() ![]() ![]() |
[18] |
I. S. Labouriau and A. A. P. Rodrigues, Global bifurcations close to symmetry, J. Math. Anal. Appl., 444 (2016), 648-671.
doi: 10.1016/j.jmaa.2016.06.032.![]() ![]() ![]() |
[19] |
A. Mohapatra and W. Ott, Homoclinic loops, heteroclinic cycles, and rank one dynamics, SIAM J. Appl. Dyn. Syst., 14 (2015), 107-131.
doi: 10.1137/140995659.![]() ![]() ![]() |
[20] |
L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
doi: 10.1007/BF02392766.![]() ![]() ![]() |
[21] |
W. Ott, Strange attractors in periodically-kicked degenerate Hopf bifurcations, Comm. Math. Phys., 281 (2008), 775-791.
doi: 10.1007/s00220-008-0499-0.![]() ![]() ![]() |
[22] |
W. Ott and M. Stenlund, From limit cycles to strange attractors, Comm. Math. Phys., 296 (2010), 215-249.
doi: 10.1007/s00220-010-0994-y.![]() ![]() ![]() |
[23] |
W. Ott and Q. Wang, Periodic attractors versus nonuniform expansion in singular limits of families of rank one maps, Discrete Contin. Dyn. Syst., 26 (2010), 1035-1054.
doi: 10.3934/dcds.2010.26.1035.![]() ![]() ![]() |
[24] |
A. A. P. Rodrigues, Repelling dynamics near a Bykov cycle, J. Dynam. Differential Equations, 25 (2013), 605-625.
doi: 10.1007/s10884-013-9289-2.![]() ![]() ![]() |
[25] |
A. A. P. Rodrigues, Unfolding a Bykov attractor: From an attracting torus to strange attractors, J. Dynam. Differential Equations, 2020.
doi: 10.1007/s10884-020-09858-z.![]() ![]() |
[26] |
A. A. P. Rodrigues, Abundance of strange attractors near an attracting periodically perturbed network, SIAM J. Appl. Dyn. Syst., 20 (2021), 541-570.
doi: 10.1137/20M1335510.![]() ![]() ![]() |
[27] |
A. A. P. Rodrigues, Dissecting a resonance wedge on heteroclinic bifurcations, J. Stat. Phys., 184 (2021), Paper No. 25, 32 pp.
doi: 10.1007/s10955-021-02811-4.![]() ![]() ![]() |
[28] |
D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.
doi: 10.1007/BF01646553.![]() ![]() ![]() |
[29] |
A. Shilnikov, G. Nicolis and C. Nicolis, Bifurcation and predictability analysis of a low-order atmospheric circulation model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1701-1711.
doi: 10.1142/S0218127495001253.![]() ![]() ![]() |
[30] |
A. Shilnikov, L. Shilnikov and D. Turaev, On some mathematical topics in classical synchronization: A tutorial, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2143-2160.
doi: 10.1142/S0218127404010539.![]() ![]() ![]() |
[31] |
Q. Wang and W. Ott, Dissipative homoclinic loops of two-dimensional maps and strange attractors with one direction of instability, Comm. Pure Appl. Math., 64 (2011), 1439-1496.
doi: 10.1002/CPA.20379.![]() ![]() ![]() |
[32] |
Q. Wang and L.-S. Young, Strange attractors with one direction of instability, Commun. Math. Phys., 218 (2001), 1-97.
doi: 10.1007/s002200100379.![]() ![]() ![]() |
[33] |
Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 275-304.
doi: 10.1007/s002200100582.![]() ![]() ![]() |
[34] |
Q. Wang and L.-S. Young, Strange attractors in periodically-kicked limit cycles and Hopf bifurcations, Commun. Math. Phys., 240 (2003), 509-529.
doi: 10.1007/s00220-003-0902-9.![]() ![]() ![]() |
[35] |
Q. Wang and L.-S. Young, Nonuniformly expanding 1D maps, Commun. Math. Phys., 264 (2006), 255-282.
doi: 10.1007/s00220-005-1485-4.![]() ![]() ![]() |
[36] |
Q. Wang and L.-S. Young, Toward a theory of rank one attractors, Ann. of Math., 167 (2008), 349-480.
doi: 10.4007/annals.2008.167.349.![]() ![]() ![]() |
[37] |
Q. Wang and L.-S. Young, Dynamical profile of a class of rank-one attractors, Ergodic Theory Dynam. Systems, 33 (2013), 1221-1264.
doi: 10.1017/S014338571200020X.![]() ![]() ![]() |
[38] |
L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.
doi: 10.2307/120960.![]() ![]() ![]() |
Example of a Misiurewicz-type map
Illustration of the transition map
Graph of
Graph of the map
Adapted bifurcation diagram of [25] for the family (4): below the graph of
Illustration (based on plots obtained by using Maple) of