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April  2021, 41(4): 1875-1895. 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  April 2021 Early access  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, 2021, 41 (4) : 1875-1895. 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. 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

show all references

##### 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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