# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020343

## Reversible perturbations of conservative Hénon-like maps

 1 Universitat Politècnica de Catalunya, Barcelona, Spain 2 Mathematical Center "Mathematics of Future Technologies", Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia 3 Laboratory of Dynamical Systems and Applications, National Research University Higher School of Economics, Nizhny Novgorod, Russia

Received  May 2020 Revised  August 2020 Published  October 2020

For area-preserving Hénon-like maps and their compositions, we consider smooth perturbations that keep the reversibility of the initial maps but destroy their conservativity. For constructing such perturbations, we use two methods, a new method based on reversible properties of maps written in the so-called cross-form, and the classical Quispel-Roberts method based on a variation of involutions of the initial map. We study symmetry breaking bifurcations of symmetric periodic orbits in reversible families containing quadratic conservative orientable and nonorientable Hénon maps as well as a product of two Hénon maps whose Jacobians are mutually inverse.

Citation: Marina Gonchenko, Sergey Gonchenko, Klim Safonov. Reversible perturbations of conservative Hénon-like maps. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020343
##### References:
 [1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2$^nd$ edition, Springer-Verlag, NY, 1996. doi: 10.1007/978-1-4612-1037-5.  Google Scholar [2] V. S. Biragov, Bifurcations in a two-parameter family of conservative mappings that are close to the Hénon mapping, Selecta Math. Soviet, 9 (1990), 273-282.   Google Scholar [3] R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113.  doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [4] A. Delshams, S. V. Gonchenko, V. S. Gonchenko, J. T. Lazaro and O. Sten'kin, Abundance of attracting, repelling and elliptic orbits in two-dimensional reversible maps, Nonlinearity, 26 (2013), 1-33.  doi: 10.1088/0951-7715/26/1/1.  Google Scholar [5] A. Delshams, M. Gonchenko, S. V. Gonchenko and and J. T. Lazaro, Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies, Discrete Contin. Dyn. 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Gonchenko, Reversible mixed dynamics: A concept and examples, Discontinuity, Nonlinearity, and Complexity, 5 (2016), 345-354.  doi: 10.5890/DNC.2016.12.003.  Google Scholar [11] M. S. Gonchenko, A. O. Kazakov, E. A. Samylina and A. I. Shyhmamedov, On the 1: 3 resonance under reversible perturbations of conservative cubic Hénon maps, preprint, 2020. Google Scholar [12] S. V. Gonchenko and D. V. Turaev, On three types of dynamics and the notion of attractor, Proc. Steklov Inst. Math., 297 (2017), 116-137.  doi: 10.1134/S0371968517020078.  Google Scholar [13] S. V. Gonchenko, A. S. Gonchenko and A. O. Kazakov, Richness of chaotic dynamics in nonholonomic models of a Celtic stone, Regu. Chaotic Dyn., 18 (2013), 521-538.  doi: 10.1134/S1560354713050055.  Google Scholar [14] S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, On the existence of Newhouse domains in a neighborhood of systems with a structurally unstable Poincare homoclinic curve (the higher-dimensional case), Dokl. Math., 47 (1993), 268-273.   Google Scholar [15] S. V. Gonchenko, D. V. Turaev and L. P. Shil'nikov, On Newhouse domains of two-dimensional diffeomorphisms that are close to a diffeomorphism with a structurally unstable heteroclinic contour, Proc. Steklov Inst. Math., 216 (1997), 70-118.   Google Scholar [16] S. V. Gonchenko, J. S. V. Lèmb, I. Rios and D. Turaev, Attractors and repellers in the neighborhood of elliptic points of reversible systems, Dokl. Math., 89 (2014), 65-67.   Google Scholar [17] S. V. Gonchenko, M. S. Gonchenko and I. O. Sinitsky, On mixed dynamics of two-dimensional reversible diffeomorphisms with symmetric non-transversal heteroclinic cycles, Izv. Ross. Akad. Nauk Ser. Mat., 84 (2020), 27-59.  doi: 10.4213/im8786.  Google Scholar [18] A. O. Kazakov, On the appearance of mixed dynamics as a result of collision of strange attractors and repellers in reversible systems, Radiophysics and Quantum Electronics, 61 (2019), 650-658.  doi: 10.1007/s11141-019-09925-6.  Google Scholar [19] A. O. Kazakov, Merger of a Hénon-like attractor with a Hńon-like repeller in a model of vortex dynamics, Chaos, 30 (2020), 011105. doi: 10.1063/1.5144144.  Google Scholar [20] J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar [21] 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 [22] L. M. Lerman and D. V. Turaev, Breakdown of symmetry in reversible systems, Reg. Chaotic Dyn., 17 (2012), 318-336.  doi: 10.1134/S1560354712030082.  Google Scholar [23] S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101-151.   Google Scholar [24] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar [25] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. 2, 140 (1994), 207-250.  doi: 10.2307/2118546.  Google Scholar [26] A. Politi, G. L. Oppo and R. Badii, Coexistence of conservative and dissipative behaviour in reversible dynamicla systems, Phys. Rev. A, 33 (1986), 4055-4060.   Google Scholar [27] T. Post, H. W. Capel, G. R. W. Quispel and J. R. van der Weele, Bifurcations in two-dimensional reversible maps, Phys. A, 164 (1990), 625-662.  doi: 10.1016/0378-4371(90)90226-I.  Google Scholar [28] J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep., 216 (1992), 63-177.  doi: 10.1016/0370-1573(92)90163-T.  Google Scholar [29] N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems., 15 (1995), 735-757.  doi: 10.1017/S0143385700008634.  Google Scholar [30] D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137-151.  doi: 10.1007/BF01206949.  Google Scholar [31] D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley Publishing Co., Reading, MA, 1978.  Google Scholar [32] M. B. Sevryuk, Reversible Systems, Lect. Notes Math., Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar [33] C. Simó and A. Vieiro, Resonant zones, inner and outer splitting in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245.  doi: 10.1088/0951-7715/22/5/012.  Google Scholar [34] D. Turaev, Richness of chaos in the absolute Newhouse domain, in Proc. Int. Congr. Math., Hyderabad (India), 3 (2010), 1804-1815.   Google Scholar [35] D. Turaev, Maps close to identity and universal maps in the Newhouse domain, Commun. Math. Phys., 335 (2015), 1235-1277.  doi: 10.1007/s00220-015-2338-4.  Google Scholar

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##### References:
 [1] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, 2$^nd$ edition, Springer-Verlag, NY, 1996. doi: 10.1007/978-1-4612-1037-5.  Google Scholar [2] V. S. Biragov, Bifurcations in a two-parameter family of conservative mappings that are close to the Hénon mapping, Selecta Math. Soviet, 9 (1990), 273-282.   Google Scholar [3] R. L. Devaney, Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113.  doi: 10.1090/S0002-9947-1976-0402815-3.  Google Scholar [4] A. Delshams, S. V. Gonchenko, V. S. Gonchenko, J. T. Lazaro and O. Sten'kin, Abundance of attracting, repelling and elliptic orbits in two-dimensional reversible maps, Nonlinearity, 26 (2013), 1-33.  doi: 10.1088/0951-7715/26/1/1.  Google Scholar [5] A. Delshams, M. Gonchenko, S. V. Gonchenko and and J. T. Lazaro, Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies, Discrete Contin. Dyn. Syst., 38 (2018), 4483-4507.  doi: 10.3934/dcds.2018196.  Google Scholar [6] A. A. Emelianova, V. I. Nekorkin, On the intersection of a chaotic attractor and a chaotic repeller in the system of two adaptively coupled phase oscillators, Chaos, 29 (2019), 111102. doi: 10.1063/1.5130994.  Google Scholar [7] A. A. Emelianova, V. I. Nekorkin, The third type of chaos in a system of two adaptively coupled phase oscillators, Chaos, 30 (2020), 051105. doi: 10.1063/5.0009525.  Google Scholar [8] A. S. Gonchenko, S. V. Gonchenko, A. O. Kazakov and D. V. Turaev, On the phenomenon of mixed dynamics in Pikovsky-Topaj system of coupled rotators, Phys. D, 350 (2017), 45-57.  doi: 10.1016/j.physd.2017.02.002.  Google Scholar [9] M. Gonchenko, S. Gonchenko and I. Ovsyannikov, Bifurcations of cubic homoclinic tangencies in two-dimensional symplectic maps, Math. Model. Nat. Phenom., 12 (2017), 41-61.  doi: 10.1051/mmnp/201712104.  Google Scholar [10] S. Gonchenko, Reversible mixed dynamics: A concept and examples, Discontinuity, Nonlinearity, and Complexity, 5 (2016), 345-354.  doi: 10.5890/DNC.2016.12.003.  Google Scholar [11] M. S. Gonchenko, A. O. Kazakov, E. A. Samylina and A. I. Shyhmamedov, On the 1: 3 resonance under reversible perturbations of conservative cubic Hénon maps, preprint, 2020. Google Scholar [12] S. V. Gonchenko and D. V. Turaev, On three types of dynamics and the notion of attractor, Proc. Steklov Inst. Math., 297 (2017), 116-137.  doi: 10.1134/S0371968517020078.  Google Scholar [13] S. V. Gonchenko, A. S. Gonchenko and A. O. Kazakov, Richness of chaotic dynamics in nonholonomic models of a Celtic stone, Regu. Chaotic Dyn., 18 (2013), 521-538.  doi: 10.1134/S1560354713050055.  Google Scholar [14] S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov, On the existence of Newhouse domains in a neighborhood of systems with a structurally unstable Poincare homoclinic curve (the higher-dimensional case), Dokl. Math., 47 (1993), 268-273.   Google Scholar [15] S. V. Gonchenko, D. V. Turaev and L. P. Shil'nikov, On Newhouse domains of two-dimensional diffeomorphisms that are close to a diffeomorphism with a structurally unstable heteroclinic contour, Proc. Steklov Inst. Math., 216 (1997), 70-118.   Google Scholar [16] S. V. Gonchenko, J. S. V. Lèmb, I. Rios and D. Turaev, Attractors and repellers in the neighborhood of elliptic points of reversible systems, Dokl. Math., 89 (2014), 65-67.   Google Scholar [17] S. V. Gonchenko, M. S. Gonchenko and I. O. Sinitsky, On mixed dynamics of two-dimensional reversible diffeomorphisms with symmetric non-transversal heteroclinic cycles, Izv. Ross. Akad. Nauk Ser. Mat., 84 (2020), 27-59.  doi: 10.4213/im8786.  Google Scholar [18] A. O. Kazakov, On the appearance of mixed dynamics as a result of collision of strange attractors and repellers in reversible systems, Radiophysics and Quantum Electronics, 61 (2019), 650-658.  doi: 10.1007/s11141-019-09925-6.  Google Scholar [19] A. O. Kazakov, Merger of a Hénon-like attractor with a Hńon-like repeller in a model of vortex dynamics, Chaos, 30 (2020), 011105. doi: 10.1063/1.5144144.  Google Scholar [20] J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar [21] 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 [22] L. M. Lerman and D. V. Turaev, Breakdown of symmetry in reversible systems, Reg. Chaotic Dyn., 17 (2012), 318-336.  doi: 10.1134/S1560354712030082.  Google Scholar [23] S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101-151.   Google Scholar [24] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.  doi: 10.1016/0040-9383(74)90034-2.  Google Scholar [25] J. Palis and M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. 2, 140 (1994), 207-250.  doi: 10.2307/2118546.  Google Scholar [26] A. Politi, G. L. Oppo and R. Badii, Coexistence of conservative and dissipative behaviour in reversible dynamicla systems, Phys. Rev. A, 33 (1986), 4055-4060.   Google Scholar [27] T. Post, H. W. Capel, G. R. W. Quispel and J. R. van der Weele, Bifurcations in two-dimensional reversible maps, Phys. A, 164 (1990), 625-662.  doi: 10.1016/0378-4371(90)90226-I.  Google Scholar [28] J. A. G. Roberts and G. R. W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep., 216 (1992), 63-177.  doi: 10.1016/0370-1573(92)90163-T.  Google Scholar [29] N. Romero, Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems., 15 (1995), 735-757.  doi: 10.1017/S0143385700008634.  Google Scholar [30] D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137-151.  doi: 10.1007/BF01206949.  Google Scholar [31] D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley Publishing Co., Reading, MA, 1978.  Google Scholar [32] M. B. Sevryuk, Reversible Systems, Lect. Notes Math., Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar [33] C. Simó and A. Vieiro, Resonant zones, inner and outer splitting in generic and low order resonances of area preserving maps, Nonlinearity, 22 (2009), 1191-1245.  doi: 10.1088/0951-7715/22/5/012.  Google Scholar [34] D. Turaev, Richness of chaos in the absolute Newhouse domain, in Proc. Int. Congr. Math., Hyderabad (India), 3 (2010), 1804-1815.   Google Scholar [35] D. Turaev, Maps close to identity and universal maps in the Newhouse domain, Commun. Math. Phys., 335 (2015), 1235-1277.  doi: 10.1007/s00220-015-2338-4.  Google Scholar
Elements of the bifurcation diagram in the $(b,M)$-parameter plane for the maps (a) $T_2$ and (b) $T_{2\mu}$ with small fixed $\mu$
Main bifurcations in maps (29) and (30) when $\mu$ is small and fixed and $M$ changes. The points $Q_1$ and $Q_2$ are symmetric elliptic 2-periodic orbits, while the points $S_1$ and $S_2$ are nonorientable saddle fixed points that compose a symmetric couple of points. The points $S_1$ and $S_2$ are conservative with the Jacobian $-1$ for $\mu = 0$ and non-conservative with the Jacobians $-1<J_1<0$ and $J_2<-1$, respectively, for $\mu>0$
A schematic tree for the bifurcation scenario of the appearance of a symmetric couple of 6-periodic orbits in the third power of Hénon map $H_{1+}$
Phase portraits of 3- and 6-periodic orbits for the Hénon map $H_{+1}$: the bottom plots are magnifications of some important details of the top plots
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