American Institute of Mathematical Sciences

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May  2022, 42(5): 2355-2379. doi: 10.3934/dcds.2021193

"Large" strange attractors in the unfolding of a heteroclinic attractor

Alexandre Rodrigues

Centro de Matemática da Univ. do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Received  March 2021 Revised  October 2021 Published  May 2022 Early access  December 2021

Fund Project: *AR was partially supported by CMUP (UID/MAT/00144/2020), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. AR also acknowledges financial support from Program INVESTIGADOR FCT (IF/00107/2015)

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.

Keywords: Heteroclinic bifurcations, Heteroclinic network, Strange attractors, Rank-one dynamics, Superstable sinks.
Mathematics Subject Classification: Primary: 34C28; 34C37; 37D05; Secondary: 37D45; 37G35.
Citation: Alexandre Rodrigues. "Large" strange attractors in the unfolding of a heteroclinic attractor. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2355-2379. doi: 10.3934/dcds.2021193
References:
[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.

show all references

References:
[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.

Figure 1.  Example of a Misiurewicz-type map $ h: {\mathbb{S}}^1 \rightarrow {\mathbb{R}} $. For $ \delta>0 $, the set $ C_{\delta} $ is a neighbourhood of the set of critical points $ C $. These maps are among the simplest examples with non-uniform expansion
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Figure 2.  Illustration of the transition map $ \Psi_{2 \rightarrow 1} $ from $ {\text{Out}}(O_2) $ to $ {\text{In}} (O_1) $. The graph of $ \Phi_2(x, 0) $ may be seen as the first hit of $ W^u(O_2) $ to $ {\text{In}} (O_1) $
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Figure 3.  Graph of $ k(x) = -K_\omega \ln (x) $ and illustration of the sequences $ (\lambda_n)_n $ and $ (\lambda_{(n, a)})_n $ for a fixed $ a \in [0, 2\pi[ $
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Figure 4.  Graph of the map $ h_a $ for $ K_\omega \gg K_\omega^0 $ with $ q = 2 $ (number of critical points). Indicated is a superstable periodic orbit of period 2 as it contains a critical point
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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
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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])
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