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

Amadeu Delshams; Marina Gonchenko; Sergey V. Gonchenko; J. Tomás Lázaro

doi:10.3934/dcds.2018196

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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

Amadeu Delshams ^1,, , Marina Gonchenko ^2, , Sergey V. Gonchenko ^3, and J. Tomás Lázaro ^4,

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.

Keywords: Newhouse phenomenon, homoclinic and heteroclinic tangencies, reversible mixed dynamics.

Mathematics Subject Classification: 37-XX, 37G20, 37G40, 34C37.

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 and Continuous Dynamical Systems, 2018, 38 (9) : 4483-4507. doi: 10.3934/dcds.2018196

References:

[1]	P. Berger, Generic family with robustly infinitely many sinks, Inv. Math., 205 (2016), 121-172. doi: 10.1007/s00222-015-0632-6.
[2]	A. Delshams, S. V. Gonchenko, V. S. Gonchenko, J. 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.
[3]	A. Delshams, M. 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.
[4]	P. Duarte, Abundance of elliptic isles at conservative bifurcations, Dyn. Stab. Syst., 14 (1999), 339-356. doi: 10.1080/026811199281930.
[5]	P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20 (2000), 393-438. doi: 10.1017/S0143385700000195.
[6]	P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. & Dynam. Sys., 20 (2002), 393-438. doi: 10.1017/S0143385700000195.
[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.
[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.
[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.
[10]	S. V. Gonchenko, L. 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.
[11]	S. V. Gonchenko, D. 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.
[12]	S. V. Gonchenko, O. 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.
[13]	S. V. Gonchenko, D. 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.
[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].
[15]	S. V. Gonchenko and L. P. Shilnikov, On 2-dimensional area-preserving mappings with homoclinic tangencies, Doklady Mathematics, 63 (2001), 395-399.
[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.
[17]	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, Regular and Chaotic Dynamics, 12 (2007), 233-266. doi: 10.1134/S156035470703001X.
[18]	S. V. Gonchenko, L. 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.
[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.
[20]	S. V. Gonchenko, V. 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.
[21]	S. V. Gonchenko, J. S. W. Lamb, I. Rios and D. V. Turaev, Attractors and repellers near generic elliptic points of reversible maps, Doclady Mathematics, 89 (2014), 65-67.
[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.
[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.
[24]	E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.
[25]	D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.
[26]	S. E. Newhouse, Non density of Axiom A(a) on $S^2$, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 191-202.
[27]	S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.
[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.
[29]	J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many sinks, Ann. Math., 140 (1994), 207-250. doi: 10.2307/2118546.
[30]	N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergod. Th. Dyn.Sys., 15 (1995), 735-757. doi: 10.1017/S0143385700008634.
[31]	M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.
[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.
[33]	D. V. Turaev, On the genericity of the Newhouse phenomenon, in EQUADIFF 2003, World Sci. Publ., Hackensack, 2005.

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.
[2]	A. Delshams, S. V. Gonchenko, V. S. Gonchenko, J. 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.
[3]	A. Delshams, M. 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.
[4]	P. Duarte, Abundance of elliptic isles at conservative bifurcations, Dyn. Stab. Syst., 14 (1999), 339-356. doi: 10.1080/026811199281930.
[5]	P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. Dyn. Sys., 20 (2000), 393-438. doi: 10.1017/S0143385700000195.
[6]	P. Duarte, Persistent homoclinic tangencies for conservative maps near the identity, Ergod. Th. & Dynam. Sys., 20 (2002), 393-438. doi: 10.1017/S0143385700000195.
[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.
[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.
[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.
[10]	S. V. Gonchenko, L. 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.
[11]	S. V. Gonchenko, D. 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.
[12]	S. V. Gonchenko, O. 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.
[13]	S. V. Gonchenko, D. 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.
[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].
[15]	S. V. Gonchenko and L. P. Shilnikov, On 2-dimensional area-preserving mappings with homoclinic tangencies, Doklady Mathematics, 63 (2001), 395-399.
[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.
[17]	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, Regular and Chaotic Dynamics, 12 (2007), 233-266. doi: 10.1134/S156035470703001X.
[18]	S. V. Gonchenko, L. 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.
[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.
[20]	S. V. Gonchenko, V. 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.
[21]	S. V. Gonchenko, J. S. W. Lamb, I. Rios and D. V. Turaev, Attractors and repellers near generic elliptic points of reversible maps, Doclady Mathematics, 89 (2014), 65-67.
[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.
[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.
[24]	E. Lombardi, Oscillatory Integrals and Phenomena Beyond all Algebraic Orders, Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104102.
[25]	D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.
[26]	S. E. Newhouse, Non density of Axiom A(a) on $S^2$, Proc. Amer. Math. Soc. Symp. Pure Math., 14 (1970), 191-202.
[27]	S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18. doi: 10.1016/0040-9383(74)90034-2.
[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.
[29]	J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many sinks, Ann. Math., 140 (1994), 207-250. doi: 10.2307/2118546.
[30]	N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergod. Th. Dyn.Sys., 15 (1995), 735-757. doi: 10.1017/S0143385700008634.
[31]	M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.
[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.
[33]	D. V. Turaev, On the genericity of the Newhouse phenomenon, in EQUADIFF 2003, World Sci. Publ., Hackensack, 2005.

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

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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

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Figure 3. (a) An example of reversible map with a couple of symmetric homoclinic tangencies (homoclinic figure-8). (b) A neighbourhood of the contour $O\cup\Gamma_1\cup\Gamma_2$

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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)

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Figure 8. Domains of definitions and associated coordinates for the first return map ${T_{2m1k}} = T_2T_0^mT_1T_0^k$

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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) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$

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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) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$

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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"

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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 $H$

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Figure 10. Two examples of creation of secondary homoclinic tangencies to the point $O$ together with their Smale horseshoes

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

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