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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 |
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.
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. |










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