September  2018, 38(9): 4483-4507. doi: 10.3934/dcds.2018196

Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

1. 

Departament de Matemàtiques and Lab of Geometry and Dynamical Systems, Universitat Politècnica de Catalunya, Av. Doctor Marañón, 44-50, Barcelona, 08028, Spain

2. 

Departament de Matemàtiques, Universitat de Barcelona. Gran Via de les Corts Catalanes, 585, Barcelona, 08007, Spain

3. 

Lobachevsky University of Nizhny Novgorod. Gagarina av. 23, Nizhny Novgorod, 603950, Russia

4. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Diagonal, 647, Barcelona, 08028, Spain

* Corresponding author: A. Delshams

Received  September 2017 Revised  April 2018 Published  June 2018

Fund Project: This work has been supported by the Russian Scientific Foundation grant: sections 1-4, 6 and 7 were carried out under the project 14-41-00044, and section 5 under the project 14-12-00811. AD, MG and JTL have been also partially supported by the MICIIN/FEDER grant MTM2015-65715-P and by the Catalan grant 2017SGR1049 (AD, JTL). MG has been partially supported by Juan de la Cierva-Formación Fellowship FJCI-2014-21229, the grant MTM2016-80117-P (MINECO/FEDER, UE) and the Knut and Alice Wallenberg Foundation grant 2013-0315. SG also thanks RFBR (grant 16-01-00364) and the Russian Ministry of Science and Education, project 1.3287.2017

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

Citation: Amadeu Delshams, Marina Gonchenko, Sergey V. Gonchenko, J. Tomás Lázaro. Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196
References:
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P. Berger, Generic family with robustly infinitely many sinks, Inv. Math., 205 (2016), 121-172.  doi: 10.1007/s00222-015-0632-6.  Google Scholar

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A. DelshamsS. V. GonchenkoV. S. GonchenkoJ. T. Lázaro and O. V. Sten'kin, Abundance of attracting, repelling and elliptic orbits in 2-dimensional reversible maps, Nonlinearity, 26 (2013), 1-33.  doi: 10.1088/0951-7715/26/1/1.  Google Scholar

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A. DelshamsM. S. Gonchenko and S. V. Gonchenko, On dynamics and bifurcations of area-preserving maps with homoclinic tangencies, Nonlinearity, 28 (2015), 3027-3071.  doi: 10.1088/0951-7715/28/9/3027.  Google Scholar

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P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20 (2000), 393-438.  doi: 10.1017/S0143385700000195.  Google Scholar

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P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. & Dynam. Sys., 20 (2002), 393-438.  doi: 10.1017/S0143385700000195.  Google Scholar

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S. V. Gonchenko, On stable periodic motions in systems close to a system with a nontransversal homoclinic curve, Russian Math. Notes, 33 (1983), 745-755.   Google Scholar

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S. V. Gonchenko and L. P. Shilnikov, Invariants of Ω-conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory, Ukrainian Math. J., 42 (1990), 134-140.  doi: 10.1007/BF01071004.  Google Scholar

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S. V. GonchenkoL. P. Shilnikov and D. V. Turaev, On models with non-rough Poincare homoclinic curves, Physica D, 62 (1993), 1-14.  doi: 10.1016/0167-2789(93)90268-6.  Google Scholar

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S. V. GonchenkoD. V. Turaev and L. P. Shilnikov, On Newhouse domains of 2-dimensional diffeomorphisms with a structurally unstable heteroclinic cycle, Proc. Steklov Inst. Math., 216 (1997), 70-118.   Google Scholar

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S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, Homoclinic tangencies of an arbitrary order in Newhouse domains, Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., 67 (1999), 69-128 [English translation in J. Math. Sci. 105 (2001), 1738-1778].  Google Scholar

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S. V. Gonchenko and L. P. Shilnikov, On 2-dimensional area-preserving mappings with homoclinic tangencies, Doklady Mathematics, 63 (2001), 395-399.   Google Scholar

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S. V. Gonchenko and V. S. Gonchenko, On bifurcations of birth of closed invariant curves in the case of 2-dimensional diffeomorphisms with homoclinic tangencies, Proc. Steklov Inst., 244 (2004), 80-105.   Google Scholar

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

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S. V. Gonchenko and M. S. Gonchenko, On cascades of elliptic periodic points in 2-dimensional symplectic maps with homoclinic tangencies, J. Regular and Chaotic Dynamics, 14 (2009), 116-136.  doi: 10.1134/S1560354709010080.  Google Scholar

[20]

S. V. GonchenkoV. S. Gonchenko and L. P. Shilnikov, On homoclinic origin of Henon-like maps, Regular and Chaotic Dynamics, 15 (2010), 462-481.  doi: 10.1134/S1560354710040052.  Google Scholar

[21]

S. V. GonchenkoJ. S. W. LambI. Rios and D. V. Turaev, Attractors and repellers near generic elliptic points of reversible maps, Doclady Mathematics, 89 (2014), 65-67.   Google Scholar

[22]

S. V. Gonchenko and D. V. Turaev, On three types of dynamics, and the notion of attractor, Tr. Mat. Inst. Steklova, 297 (2017), 133-157.  doi: 10.1134/S0371968517020078.  Google Scholar

[23]

J. S. W. Lamb and O. V. Stenkin, Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits, Nonlinearity, 17 (2004), 1217-1244.  doi: 10.1088/0951-7715/17/4/005.  Google Scholar

[24]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[25]

D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.  Google Scholar

[26]

S. E. Newhouse, Non density of Axiom A(a) on $S^2$, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 191-202.   Google Scholar

[27]

S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[28]

S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 101-151.   Google Scholar

[29]

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

[30]

N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergod. Th. Dyn.Sys., 15 (1995), 735-757.  doi: 10.1017/S0143385700008634.  Google Scholar

[31]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar

[32]

L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Commun.Math.Phys., 106 (1986), 635-657.  doi: 10.1007/BF01463400.  Google Scholar

[33]

D. V. Turaev, On the genericity of the Newhouse phenomenon, in EQUADIFF 2003, World Sci. Publ., Hackensack, 2005. Google Scholar

show all references

References:
[1]

P. Berger, Generic family with robustly infinitely many sinks, Inv. Math., 205 (2016), 121-172.  doi: 10.1007/s00222-015-0632-6.  Google Scholar

[2]

A. DelshamsS. V. GonchenkoV. S. GonchenkoJ. T. Lázaro and O. V. Sten'kin, Abundance of attracting, repelling and elliptic orbits in 2-dimensional reversible maps, Nonlinearity, 26 (2013), 1-33.  doi: 10.1088/0951-7715/26/1/1.  Google Scholar

[3]

A. DelshamsM. S. Gonchenko and S. V. Gonchenko, On dynamics and bifurcations of area-preserving maps with homoclinic tangencies, Nonlinearity, 28 (2015), 3027-3071.  doi: 10.1088/0951-7715/28/9/3027.  Google Scholar

[4]

P. Duarte, Abundance of elliptic isles at conservative bifurcations, Dyn. Stab. Syst., 14 (1999), 339-356.  doi: 10.1080/026811199281930.  Google Scholar

[5]

P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20 (2000), 393-438.  doi: 10.1017/S0143385700000195.  Google Scholar

[6]

P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. & Dynam. Sys., 20 (2002), 393-438.  doi: 10.1017/S0143385700000195.  Google Scholar

[7]

N. K. Gavrilov and L. P. Shilnikov, On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve (Part 1), Math. USSR Sb., 17 (1972), 467-485; (Part 2), Math. USSR Sb, 90 (1973), 139-156.  Google Scholar

[8]

S. V. Gonchenko, On stable periodic motions in systems close to a system with a nontransversal homoclinic curve, Russian Math. Notes, 33 (1983), 745-755.   Google Scholar

[9]

S. V. Gonchenko and L. P. Shilnikov, Invariants of Ω-conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory, Ukrainian Math. J., 42 (1990), 134-140.  doi: 10.1007/BF01071004.  Google Scholar

[10]

S. V. GonchenkoL. P. Shilnikov and D. V. Turaev, On models with non-rough Poincare homoclinic curves, Physica D, 62 (1993), 1-14.  doi: 10.1016/0167-2789(93)90268-6.  Google Scholar

[11]

S. V. GonchenkoD. V. Turaev and L. P. Shilnikov, On the existence of Newhouse regions near systems with non-rough Poincaré homoclinic curve (multidimensional case), Russian Acad. Sci. Dokl. Math., 47 (1993), 268-273.  doi: 10.1016/0167-2789(93)90268-6.  Google Scholar

[12]

S. V. GonchenkoO. V. Stenkin and D. V. Turaev, Complexity of homoclinic bifurcations and Ω-moduli, Int. Journal of Bifurcation and Chaos, 6 (1996), 969-989.  doi: 10.1142/S0218127496000539.  Google Scholar

[13]

S. V. GonchenkoD. V. Turaev and L. P. Shilnikov, On Newhouse domains of 2-dimensional diffeomorphisms with a structurally unstable heteroclinic cycle, Proc. Steklov Inst. Math., 216 (1997), 70-118.   Google Scholar

[14]

S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, Homoclinic tangencies of an arbitrary order in Newhouse domains, Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., 67 (1999), 69-128 [English translation in J. Math. Sci. 105 (2001), 1738-1778].  Google Scholar

[15]

S. V. Gonchenko and L. P. Shilnikov, On 2-dimensional area-preserving mappings with homoclinic tangencies, Doklady Mathematics, 63 (2001), 395-399.   Google Scholar

[16]

S. V. Gonchenko and V. S. Gonchenko, On bifurcations of birth of closed invariant curves in the case of 2-dimensional diffeomorphisms with homoclinic tangencies, Proc. Steklov Inst., 244 (2004), 80-105.   Google Scholar

[17]

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

[18]

S. V. GonchenkoL. P. Shilnikov and D. 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

[19]

S. V. Gonchenko and M. S. Gonchenko, On cascades of elliptic periodic points in 2-dimensional symplectic maps with homoclinic tangencies, J. Regular and Chaotic Dynamics, 14 (2009), 116-136.  doi: 10.1134/S1560354709010080.  Google Scholar

[20]

S. V. GonchenkoV. S. Gonchenko and L. P. Shilnikov, On homoclinic origin of Henon-like maps, Regular and Chaotic Dynamics, 15 (2010), 462-481.  doi: 10.1134/S1560354710040052.  Google Scholar

[21]

S. V. GonchenkoJ. S. W. LambI. Rios and D. V. Turaev, Attractors and repellers near generic elliptic points of reversible maps, Doclady Mathematics, 89 (2014), 65-67.   Google Scholar

[22]

S. V. Gonchenko and D. V. Turaev, On three types of dynamics, and the notion of attractor, Tr. Mat. Inst. Steklova, 297 (2017), 133-157.  doi: 10.1134/S0371968517020078.  Google Scholar

[23]

J. S. W. Lamb and O. V. Stenkin, Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits, Nonlinearity, 17 (2004), 1217-1244.  doi: 10.1088/0951-7715/17/4/005.  Google Scholar

[24]

E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.  Google Scholar

[25]

D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.  Google Scholar

[26]

S. E. Newhouse, Non density of Axiom A(a) on $S^2$, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 191-202.   Google Scholar

[27]

S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar

[28]

S. E. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Etudes Sci., 50 (1979), 101-151.   Google Scholar

[29]

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

[30]

N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergod. Th. Dyn.Sys., 15 (1995), 735-757.  doi: 10.1017/S0143385700008634.  Google Scholar

[31]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar

[32]

L. Tedeschini-Lalli and J. A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Commun.Math.Phys., 106 (1986), 635-657.  doi: 10.1007/BF01463400.  Google Scholar

[33]

D. V. Turaev, On the genericity of the Newhouse phenomenon, in EQUADIFF 2003, World Sci. Publ., Hackensack, 2005. Google Scholar

Figure 1.  Two different examples of planar reversible maps with symmetric nontransversal (quadratic tangency) heteroclinic cycles: (a) with a nontransversal symmetric heteroclinic orbit to a symmetric couple of saddle points, and (b) with a symmetric couple of nontransversal heteroclinic orbits to symmetric saddle points
Figure 2.  Three examples of planar reversible maps with symmetric nontransversal homoclinic tangencies: (a) a symmetric quadratic homoclinic tangency; (b) a symmetric cubic homoclinic tangency; (c) a symmetric couple of nontransversal homoclinic (figure-8) orbits to the same symmetric saddle point
Figure 3.  (a) An example of reversible map with a couple of symmetric homoclinic tangencies (homoclinic figure-8). (b) A neighbourhood of the contour $O\cup\Gamma_1\cup\Gamma_2$
Figure 4.  (a) A reversible diffeomorphism with a symmetric transversal homoclinic orbit; (b) creation of a symmetric couple of nontransversal homoclinic orbits $\Gamma_1$ and $\Gamma_2$ (a "fish" configuration)
Figure 8.  Domains of definitions and associated coordinates for the first return map ${T_{2m1k}} = T_2T_0^mT_1T_0^k$
Figure 5.  A geometric structure of the homoclinic points $M_1^+$, $M_{1}^{-}$, $M_2^+$ and $M_2^-$ and their neighbourhoods in the figure-8 homoclinic configuration. Schematic actions of the first return maps: (a) $T_{1k} = T_1 T_0^k$, (b) $T_{2k} = T_2 T_0^k$ and (c) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$
Figure 6.  A geometric structure of the homoclinic points $M_1^+$, $M_{1}^{-}$, $M_2^+$ and $M_2^-$ and their neighbourhoods in the "fish" homoclinic configuration. Several schematic actions of the first return maps are represented: (a) $T_{1k} = T_1 T_0^k$, (b) $T_{2k} = T_2 T_0^k$ and (c) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$
Figure 7.  Domains of definition and range of the successor map from $\Pi_i^{+}$ into $\Pi_j^-$, $i, j = 1, 2$, under iterations of $T_{0}$ in the cases of (a) homoclinic figure-8; (b) homoclinic "fish"
Figure 9.  Elements of the bifurcation diagram for the map $H$: painted regions correspond to the existence of symmetric elliptic and saddle fixed points of $H$
Figure 10.  Two examples of creation of secondary homoclinic tangencies to the point $O$ together with their Smale horseshoes
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