Figure 1.
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 $
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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 |
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.
Keywords:
area-preserving Hénon-like map,
symmetry breaking bifurcation,
reversible diffeomorphism,
mixed dynamics,
pitchfork bifurcation.
Mathematics Subject Classification:
Primary: 37G25, 37G40; Secondary: 37C70.
Citation:
Marina Gonchenko, Sergey Gonchenko, Klim Safonov. Reversible perturbations of conservative Hénon-like maps. Discrete and Continuous Dynamical Systems,
2021, 41
(4)
: 1875-1895.
doi: 10.3934/dcds.2020343
![]()
References:
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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. |
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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.
|
| [3] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113.
doi: 10.1090/S0002-9947-1976-0402815-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [24] |
S. E. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [29] |
N. Romero,
Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems., 15 (1995), 735-757.
doi: 10.1017/S0143385700008634. |
| [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. |
| [31] |
D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley Publishing Co., Reading, MA, 1978. |
| [32] |
M. B. Sevryuk, Reversible Systems, Lect. Notes Math., Vol. 1211, Springer-Verlag, Berlin, 1986.
doi: 10.1007/BFb0075877. |
| [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. |
| [34] |
D. Turaev,
Richness of chaos in the absolute Newhouse domain, in Proc. Int. Congr. Math., Hyderabad (India), 3 (2010), 1804-1815.
|
| [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. |
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. |
| [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.
|
| [3] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Am. Math. Soc., 218 (1976), 89-113.
doi: 10.1090/S0002-9947-1976-0402815-3. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [24] |
S. E. Newhouse,
Diffeomorphisms with infinitely many sinks, Topology, 13 (1974), 9-18.
doi: 10.1016/0040-9383(74)90034-2. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [29] |
N. Romero,
Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems., 15 (1995), 735-757.
doi: 10.1017/S0143385700008634. |
| [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. |
| [31] |
D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics, Addison-Wesley Publishing Co., Reading, MA, 1978. |
| [32] |
M. B. Sevryuk, Reversible Systems, Lect. Notes Math., Vol. 1211, Springer-Verlag, Berlin, 1986.
doi: 10.1007/BFb0075877. |
| [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. |
| [34] |
D. Turaev,
Richness of chaos in the absolute Newhouse domain, in Proc. Int. Congr. Math., Hyderabad (India), 3 (2010), 1804-1815.
|
| [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. |
Figure 2.
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 $
Figure 3.
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+} $
Figure 4.
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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