# American Institute of Mathematical Sciences

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:
 [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.
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
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
(a) An example of reversible map with a couple of symmetric homoclinic tangencies (homoclinic figure-8). (b) A neighbourhood of the contour $O\cup\Gamma_1\cup\Gamma_2$
(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)
Domains of definitions and associated coordinates for the first return map ${T_{2m1k}} = T_2T_0^mT_1T_0^k$
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) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$
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) ${T_{2m1k}} = T_2 T_0^m T_1 T_0^k$
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"
Elements of the bifurcation diagram for the map $H$: painted regions correspond to the existence of symmetric elliptic and saddle fixed points of $H$
Two examples of creation of secondary homoclinic tangencies to the point $O$ together with their Smale horseshoes
 [1] Victoria Rayskin. Homoclinic tangencies in $R^n$. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 465-480. doi: 10.3934/dcds.2005.12.465 [2] Thorsten Riess. Numerical study of secondary heteroclinic bifurcations near non-reversible homoclinic snaking. Conference Publications, 2011, 2011 (Special) : 1244-1253. doi: 10.3934/proc.2011.2011.1244 [3] Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 [4] Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765 [5] 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 [6] Daniel Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1039-1055. doi: 10.3934/dcdsb.2009.11.1039 [7] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [8] Antonio Pumariño, Joan Carles Tatjer. Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 971-1005. doi: 10.3934/dcdsb.2007.8.971 [9] Qingdao Huang, Hong Qian. The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1457-1464. doi: 10.3934/dcdss.2011.4.1457 [10] E. Canalias, Josep J. Masdemont. Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 261-279. doi: 10.3934/dcds.2006.14.261 [11] Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415 [12] Yuncheng You. Dynamics of three-component reversible Gray-Scott model. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1671-1688. doi: 10.3934/dcdsb.2010.14.1671 [13] J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure & Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521 [14] Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57 [15] Genghong Lin, Zhan Zhou. Homoclinic solutions of discrete $\phi$-Laplacian equations with mixed nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1723-1747. doi: 10.3934/cpaa.2018082 [16] Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883 [17] Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485 [18] Antonio Pumariño, José Ángel Rodríguez, Joan Carles Tatjer, Enrique Vigil. Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 523-541. doi: 10.3934/dcdsb.2014.19.523 [19] Shin Kiriki, Yusuke Nishizawa, Teruhiko Soma. Heterodimensional tangencies on cycles leading to strange attractors. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 285-300. doi: 10.3934/dcds.2010.27.285 [20] Alexis De Vos, Yvan Van Rentergem. Young subgroups for reversible computers. Advances in Mathematics of Communications, 2008, 2 (2) : 183-200. doi: 10.3934/amc.2008.2.183

2017 Impact Factor: 1.179

## Metrics

• PDF downloads (37)
• HTML views (61)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]