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    Erratum and addendum to "A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies" (Volume 40, Number 4, 2020, 2285-2313)
doi: 10.3934/dcds.2021193
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"Large" strange attractors in the unfolding of a heteroclinic attractor

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

Received  March 2021 Revised  October 2021 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.

Citation: Alexandre Rodrigues. "Large" strange attractors in the unfolding of a heteroclinic attractor. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021193
References:
[1]

V. S. AfraimovichS.-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.  Google Scholar

[2]

M. Aguiar, Vector fields with heteroclinic networks, Ph.D. thesis, Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto, 2003. Google Scholar

[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.  Google Scholar

[4]

I. BaldomáS. 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.  Google Scholar

[5]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[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.  Google Scholar

[7]

H. BroerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hénon, A two dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.  doi: 10.1007/BF02392766.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

[29]

A. ShilnikovG. 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.  Google Scholar

[30]

A. ShilnikovL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 275-304.  doi: 10.1007/s002200100582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

show all references

References:
[1]

V. S. AfraimovichS.-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.  Google Scholar

[2]

M. Aguiar, Vector fields with heteroclinic networks, Ph.D. thesis, Departamento de Matemática Aplicada, Faculdade de Ciências da Universidade do Porto, 2003. Google Scholar

[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.  Google Scholar

[4]

I. BaldomáS. 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.  Google Scholar

[5]

M. Benedicks and L. Carleson, The dynamics of the Hénon map, Ann. of Math., 133 (1991), 73-169.  doi: 10.2307/2944326.  Google Scholar

[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.  Google Scholar

[7]

H. BroerC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hénon, A two dimensional mapping with a strange attractor, Comm. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.  doi: 10.1007/BF02392766.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys., 20 (1971), 167-192.  doi: 10.1007/BF01646553.  Google Scholar

[29]

A. ShilnikovG. 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.  Google Scholar

[30]

A. ShilnikovL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

Q. Wang and L.-S. Young, From invariant curves to strange attractors, Commun. Math. Phys., 225 (2002), 275-304.  doi: 10.1007/s002200100582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. Math., 147 (1998), 585-650.  doi: 10.2307/120960.  Google Scholar

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
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) $
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[ $
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
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 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
30])">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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