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April  2017, 10(2): 273-288. doi: 10.3934/dcdss.2017013

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

Sergey Gonchenko 1, and  Ivan Ovsyannikov 1,2,,

1. 

Lobachevsky State University of Nizhny Novgorod, 23 Gagarin av., Nizhny Novgorod, 603950, Russia
2. 

Universität Bremen, Bibliothekstrasse 1,28359 Bremen, Germany

* Corresponding author:Ivan Ovsyannikov.

Received  October 2015 Revised  November 2016 Published  January 2017

We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map $\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$ which, as known [14], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.

Keywords: Homoclinic tangency, rescaling, 3D Hénon map, Poincare map, Lorenz attractor.
Mathematics Subject Classification: Primary: 37C29, 37C70, 37G25.
Citation: Sergey Gonchenko, Ivan Ovsyannikov. Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 273-288. doi: 10.3934/dcdss.2017013
References:
[1]

V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212.   Google Scholar

[2]

V. S. Aframovich and L. P. Shilnikov, Strange attractors and quasiattractors, in Nonlinear Dynamics and Turbulence eds. G. I. Barenblatt, G. Iooss and D. D. Joseph, Boston, Pitmen, 1983. Google Scholar

[3]

A. Arneodo, P. Coullet and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys. , 79 (1981), 573-579, available at http://projecteuclid.org/euclid. cmp/1103909142. doi: 10.1007/BF01209312.  Google Scholar

[4]

E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincar Anal. Non Linéaire, 15 (1998), 539-579.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[5]

A. S. Gonchenko and S. V. Gonchenko, Lorenz-like attractors in nonholonomic models of a rattleback, Nonlinearity, 28 (2015), 3403-3417.  doi: 10.1088/0951-7715/28/9/3403.  Google Scholar

[6]

A. S. Gonchenko and S. V. Gonchenko, Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, Physica D: Nonlinear Phenomena, 337 (2016), 43-57, arXiv: 1510.02252. doi: 10.1016/j.physd.2016.07.006.  Google Scholar

[7]

A. S. Gonchenko, S. V. Gonchenko and A. O. Kazakov, Richness of chaotic dynamics in nonholonomic models of a Celtic stone, Regul. Chaotic Dyn., 18 (2013), 521-538.  doi: 10.1134/S1560354713050055.  Google Scholar

[8]

A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov and D. V. Turaev, Simple scenarios of onset of chaos in three-dimensional maps Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24 (2014), 1440005, 25pp. doi: 10.1142/S0218127414400057.  Google Scholar

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A. S. Gonchenko, S. V. Gonchenko, I. I. Ovsyannikov and D. V. Turaev, Examples of Lorenz-Like Attractors in Hénon-Like Maps, Math. Model. Nat. Phenom., 8 (2013), 48-70.  doi: 10.1051/mmnp/20138504.  Google Scholar

[10]

A. S. Gonchenko, S. V. Gonchenko and L. P. Shilnikov, Towards scenarios of chaos appearance in three-dimensional maps, Rus. J. Nonlin. Dyn., 8 (2012), 3-28.   Google Scholar

[11]

S. V. Gonchenko, V. S. Gonchenko and J. C. Tatjer, Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps, Regul. Chaotic Dyn., 12 (2007), 233-266.  doi: 10.1134/S156035470703001X.  Google Scholar

[12]

S. V. Gonchenko, J. D. Meiss and I. I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11 (2006), 191-212.  doi: 10.1070/RD2006v011n02ABEH000345.  Google Scholar

[13]

S. V. Gonchenko and I. I. Ovsyannikov, On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors, Mat. Model. of Nat. Phenom, 8 (2013), 71-83.  doi: 10.1051/mmnp/20138505.  Google Scholar

[14]

S. V. Gonchenko, I. I. Ovsyannikov, C. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[15]

S. V. Gonchenko, I. I. Ovsyannikov and J. C. Tatjer, Birth of discrete lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points, Regul. Chaotic Dyn., 19 (2014), 495-505.  doi: 10.1134/S1560354714040054.  Google Scholar

[16]

S. V. Gonchenko and L. P. Shilnikov, Moduli of systems with a structurally unstable homoclinic Poincare curve, (Russian), Izv. Ross. Akad. Nauk Ser. Mat. , 56 (1992), 1165-1197, English translation: Russian Acad. Sci. Izv. Math. , 41 (1993), 417-445. doi: 10.1070/IM1993v041n03ABEH002270.  Google Scholar

[17]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits, Dokl. Akad. Nauk, 330 (1993), 144-147.   Google Scholar

[18]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits, Chaos, 6 (1996), 15-31.  doi: 10.1063/1.166154.  Google Scholar

[19]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions, Nonlinearity, 21 (2008), 923-972.  doi: 10.1088/0951-7715/21/5/003.  Google Scholar

[20]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regul. Chaotic Dyn., 14 (2009), 137-147.  doi: 10.1134/S1560354709010092.  Google Scholar

[21]

S. V. Gonchenko, O. V. Sten'kin and D. V. Turaev, Complexity of homoclinic bifurcations and $Ω$-moduli, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 969-989.  doi: 10.1142/S0218127496000539.  Google Scholar

[22]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds. Lecture Notes in Math. Springer-Verlag, Berlin, 583 (1977), ⅱ+149 pp.  Google Scholar

[23]

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

[24]

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

[25]

S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. , 50 (1979), 101-151. Available from: http://www.numdam.org/item?id=PMIHES_1979__50__101_0.  Google Scholar

[26]

S. E. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. , 57 (1983), 5-71. Available from: http://www.numdam.org/item?id=PMIHES_1983__57__5_0.  Google Scholar

[27]

I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve, Mat. Sb. (N.S.), 130 (1986), 552-570.   Google Scholar

[28]

I. M. Ovsyannikov and L. P. Shilnikov, Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos, (Russian), Mat. Sb. , 182 (1991), 1043-1073; English translation in Math. USSR-Sb. 73 (1992), 415-443.  Google Scholar

[29]

J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math., 140 (1994), 207-250.  doi: 10.2307/2118546.  Google Scholar

[30]

A. L. Shilnikov, On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model. Homoclinic chaos, Phys. D, 62 (1993), 338-346.  doi: 10.1016/0167-2789(93)90292-9.  Google Scholar

[31]

A. L. Shilnikov, L. P. Shilnikov and D. V. Turaev, Normal forms and Lorenz attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3 (1993), 1123-1139.  doi: 10.1142/S0218127493000933.  Google Scholar

[32]

L. P. Shilnikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Mat. Sb. (N.S.), 77 (1968), 461-472.   Google Scholar

[33]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part Ⅰ World Scientific, 1998. doi: 10.1142/9789812798596.  Google Scholar

[34]

T. Shimizu and N. Morioka, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A, 76 (1980), 201-204.  doi: 10.1016/0375-9601(80)90466-1.  Google Scholar

[35]

J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory Dynam. Systems, 21 (2001), 249-302.  doi: 10.1017/S0143385701001146.  Google Scholar

[36]

G. Tigan and D. Turaev, Analytical search for homoclinic bifurcations in Morioka-Shimizu model, Phys. D, 240 (2011), 985-989.  doi: 10.1016/j.physd.2011.02.013.  Google Scholar

[37]

D. V. Turaev, On dimension of nonlocal bifurcational problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 919-948.  doi: 10.1142/S0218127496000515.  Google Scholar

[38]

D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, (Russian), Mat. Sb. , 189 (1998), 137-160; English translation: Sb. Math. , 189 (1998), 29-314. doi: 10.1070/SM1998v189n02ABEH000300.  Google Scholar

[39]

D. V. Turaev and L. P. Shilnikov, Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors, (Russian) Dokl. Akad. Nauk. , 418 (2008), 23-27; English translation: Dokl. Math. , 77 (2008), 17-21. doi: 10.1134/S1064562408010055.  Google Scholar

show all references

References:
[1]

V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, On attracting structurally unstable limit sets of Lorenz attractor type, Trudy Moskov. Mat. Obshch., 44 (1982), 150-212.   Google Scholar

[2]

V. S. Aframovich and L. P. Shilnikov, Strange attractors and quasiattractors, in Nonlinear Dynamics and Turbulence eds. G. I. Barenblatt, G. Iooss and D. D. Joseph, Boston, Pitmen, 1983. Google Scholar

[3]

A. Arneodo, P. Coullet and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys. , 79 (1981), 573-579, available at http://projecteuclid.org/euclid. cmp/1103909142. doi: 10.1007/BF01209312.  Google Scholar

[4]

E. Colli, Infinitely many coexisting strange attractors, Ann. Inst. H. Poincar Anal. Non Linéaire, 15 (1998), 539-579.  doi: 10.1016/S0294-1449(98)80001-2.  Google Scholar

[5]

A. S. Gonchenko and S. V. Gonchenko, Lorenz-like attractors in nonholonomic models of a rattleback, Nonlinearity, 28 (2015), 3403-3417.  doi: 10.1088/0951-7715/28/9/3403.  Google Scholar

[6]

A. S. Gonchenko and S. V. Gonchenko, Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, Physica D: Nonlinear Phenomena, 337 (2016), 43-57, arXiv: 1510.02252. doi: 10.1016/j.physd.2016.07.006.  Google Scholar

[7]

A. S. Gonchenko, S. V. Gonchenko and A. O. Kazakov, Richness of chaotic dynamics in nonholonomic models of a Celtic stone, Regul. Chaotic Dyn., 18 (2013), 521-538.  doi: 10.1134/S1560354713050055.  Google Scholar

[8]

A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov and D. V. Turaev, Simple scenarios of onset of chaos in three-dimensional maps Internat. J. Bifur. Chaos Appl. Sci. Engrg. 24 (2014), 1440005, 25pp. doi: 10.1142/S0218127414400057.  Google Scholar

[9]

A. S. Gonchenko, S. V. Gonchenko, I. I. Ovsyannikov and D. V. Turaev, Examples of Lorenz-Like Attractors in Hénon-Like Maps, Math. Model. Nat. Phenom., 8 (2013), 48-70.  doi: 10.1051/mmnp/20138504.  Google Scholar

[10]

A. S. Gonchenko, S. V. Gonchenko and L. P. Shilnikov, Towards scenarios of chaos appearance in three-dimensional maps, Rus. J. Nonlin. Dyn., 8 (2012), 3-28.   Google Scholar

[11]

S. V. Gonchenko, V. S. Gonchenko and J. C. Tatjer, Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps, Regul. Chaotic Dyn., 12 (2007), 233-266.  doi: 10.1134/S156035470703001X.  Google Scholar

[12]

S. V. Gonchenko, J. D. Meiss and I. I. Ovsyannikov, Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation, Regul. Chaotic Dyn., 11 (2006), 191-212.  doi: 10.1070/RD2006v011n02ABEH000345.  Google Scholar

[13]

S. V. Gonchenko and I. I. Ovsyannikov, On global bifurcations of three-dimensional diffeomorphisms leading to Lorenz-like attractors, Mat. Model. of Nat. Phenom, 8 (2013), 71-83.  doi: 10.1051/mmnp/20138505.  Google Scholar

[14]

S. V. Gonchenko, I. I. Ovsyannikov, C. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 3493-3508.  doi: 10.1142/S0218127405014180.  Google Scholar

[15]

S. V. Gonchenko, I. I. Ovsyannikov and J. C. Tatjer, Birth of discrete lorenz attractors at the bifurcations of 3D maps with homoclinic tangencies to saddle points, Regul. Chaotic Dyn., 19 (2014), 495-505.  doi: 10.1134/S1560354714040054.  Google Scholar

[16]

S. V. Gonchenko and L. P. Shilnikov, Moduli of systems with a structurally unstable homoclinic Poincare curve, (Russian), Izv. Ross. Akad. Nauk Ser. Mat. , 56 (1992), 1165-1197, English translation: Russian Acad. Sci. Izv. Math. , 41 (1993), 417-445. doi: 10.1070/IM1993v041n03ABEH002270.  Google Scholar

[17]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits, Dokl. Akad. Nauk, 330 (1993), 144-147.   Google Scholar

[18]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits, Chaos, 6 (1996), 15-31.  doi: 10.1063/1.166154.  Google Scholar

[19]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On dynamical properties of multidimensional diffeomorphisms from Newhouse regions, Nonlinearity, 21 (2008), 923-972.  doi: 10.1088/0951-7715/21/5/003.  Google Scholar

[20]

S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev, On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors, Regul. Chaotic Dyn., 14 (2009), 137-147.  doi: 10.1134/S1560354709010092.  Google Scholar

[21]

S. V. Gonchenko, O. V. Sten'kin and D. V. Turaev, Complexity of homoclinic bifurcations and $Ω$-moduli, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 969-989.  doi: 10.1142/S0218127496000539.  Google Scholar

[22]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds. Lecture Notes in Math. Springer-Verlag, Berlin, 583 (1977), ⅱ+149 pp.  Google Scholar

[23]

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

[24]

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

[25]

S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. , 50 (1979), 101-151. Available from: http://www.numdam.org/item?id=PMIHES_1979__50__101_0.  Google Scholar

[26]

S. E. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. , 57 (1983), 5-71. Available from: http://www.numdam.org/item?id=PMIHES_1983__57__5_0.  Google Scholar

[27]

I. M. Ovsyannikov and L. P. Shil'nikov, On systems with a saddle-focus homoclinic curve, Mat. Sb. (N.S.), 130 (1986), 552-570.   Google Scholar

[28]

I. M. Ovsyannikov and L. P. Shilnikov, Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos, (Russian), Mat. Sb. , 182 (1991), 1043-1073; English translation in Math. USSR-Sb. 73 (1992), 415-443.  Google Scholar

[29]

J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math., 140 (1994), 207-250.  doi: 10.2307/2118546.  Google Scholar

[30]

A. L. Shilnikov, On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model. Homoclinic chaos, Phys. D, 62 (1993), 338-346.  doi: 10.1016/0167-2789(93)90292-9.  Google Scholar

[31]

A. L. Shilnikov, L. P. Shilnikov and D. V. Turaev, Normal forms and Lorenz attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 3 (1993), 1123-1139.  doi: 10.1142/S0218127493000933.  Google Scholar

[32]

L. P. Shilnikov, On the generation of a periodic motion from trajectories doubly asymptotic to an equilibrium state of saddle type, Mat. Sb. (N.S.), 77 (1968), 461-472.   Google Scholar

[33]

L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part Ⅰ World Scientific, 1998. doi: 10.1142/9789812798596.  Google Scholar

[34]

T. Shimizu and N. Morioka, On the bifurcation of a symmetric limit cycle to an asymmetric one in a simple model, Phys. Lett. A, 76 (1980), 201-204.  doi: 10.1016/0375-9601(80)90466-1.  Google Scholar

[35]

J. C. Tatjer, Three-dimensional dissipative diffeomorphisms with homoclinic tangencies, Ergodic Theory Dynam. Systems, 21 (2001), 249-302.  doi: 10.1017/S0143385701001146.  Google Scholar

[36]

G. Tigan and D. Turaev, Analytical search for homoclinic bifurcations in Morioka-Shimizu model, Phys. D, 240 (2011), 985-989.  doi: 10.1016/j.physd.2011.02.013.  Google Scholar

[37]

D. V. Turaev, On dimension of nonlocal bifurcational problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6 (1996), 919-948.  doi: 10.1142/S0218127496000515.  Google Scholar

[38]

D. V. Turaev and L. P. Shilnikov, An example of a wild strange attractor, (Russian), Mat. Sb. , 189 (1998), 137-160; English translation: Sb. Math. , 189 (1998), 29-314. doi: 10.1070/SM1998v189n02ABEH000300.  Google Scholar

[39]

D. V. Turaev and L. P. Shilnikov, Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors, (Russian) Dokl. Akad. Nauk. , 418 (2008), 23-27; English translation: Dokl. Math. , 77 (2008), 17-21. doi: 10.1134/S1064562408010055.  Google Scholar

Figure 1.  Schematic examples of pseudohyperbolic attractors. (a) Shilnikov-Turaev wild spiral attractor for a four-dimensional flow. (b) A discrete super-Lorenz attractor that contains a saddle fixed point $O$, pieces of its unstable (one-dimensional) manifolds are drown. Fixed points $O_1$ and $O_2$ (with $\dim W^u(O_i) = 2$ do not belong to the attractor, they are posed in "holes".
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Figure 2.  Plots of attractors of map (1) observed numerically in [14] for $M_1=0,B=0.7$ and $M_2=0.85$ (left) or $M_2=0.815$ (right). In the left panel, the projection on the $(x,y)$-plane is also displayed. In the right panel, a "figure-eight" saddle closed invariant curve inside the lacuna is shown.
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Figure 3.  The main steps of the creation of a discrete (c) Lorenz attractor; (d) figure-eight attractor.
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Figure 4.  Two examples of 3D diffeomorphisms with nontransversal heteroclinic cycles containing two fixed points $O_1$ and $O_2$ of type $(2,1)$: (a) $O_1$ is a saddle-focus, this case was studied in [20]; (b) $O_1$ and $O_2$ are both saddle-foci, this case was studied in [13].
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Figure 5.  Examples of 3D maps with quadratic homoclinic tangencies to a saddle fixed point $O$ of conservative type (whose bifurcations lead to the birth of discrete Lorenz attractors): (a) $O$ is a conservative saddle-focus [12]; (b) $O$ is a general saddle and the tangency is not simple [15]; (c) $O$ is a resonant saddle ($\lambda_1 = -\lambda_2$), studied in the present paper.
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