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Exact azimuthal internal waves with an underlying current
Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry
1. | Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK |
2. | Department of Mathematics, Lobachevsky State University of Nizhny Novgorod, 23 Prospekt Gagarina, Nizhny Novgorod 603950, Russia |
3. | Joseph Meyerhoff Visiting Professor, Weizmann Institute of Science, 234 Herzl Street, Rehovot 7610001, Israel |
We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.
References:
[1] |
V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov,
On the origin and structure of the Lorenz attractor, Akademiia Nauk SSSR Doklady, 234 (1977), 336-339.
|
[2] |
V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov,
On the structurally unstable attracting limit sets of Lorenz attractor type, Tran. Moscow Math. Soc., 2 (1983), 153-215.
|
[3] |
D. V. Anosov,
Geodesic flows on closed Riemannian manifolds of negative curvature, (Russian), Trudy Mat. Inst. Steklov., 90 (1967), 209pp.
|
[4] |
R. Barrio, A. L. Shilnikov and L. P. Shilnikov,
Kneadings, symbolic dynamics and painting Lorenz chaos, International Journal of Bifurcation and Chaos, 22 (2008), 1230016, 24 pp.
doi: 10.1142/S0218127412300169. |
[5] |
C. Bonatti and S. Crovisier,
Center manifolds for partially hyperbolic sets without strong unstable connections, Journal of the Institute of Mathematics of Jussieu, 15 (2016), 785-828.
doi: 10.1017/S1474748015000055. |
[6] |
C. Bonatti and L. J. Díaz,
Persistent transitive diffeomorphisms, Annals of Mathematics, 143 (1996), 357-396.
doi: 10.2307/2118647. |
[7] |
C. Bonatti and L. J. Díaz,
Robust heterodimensional cycles and C1-generic dynamics, Journal of the Institute of Mathematics of Jussieu, 7 (2008), 469-525.
doi: 10.1017/S1474748008000030. |
[8] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005. |
[9] |
L. J. Díaz and J. Rocha,
Non-connected heterodimensional cycles: Bifurcation and stability, Nonlinearity, 5 (1992), 1315-1341.
doi: 10.1088/0951-7715/5/6/006. |
[10] |
L. J. Díaz,
Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory and Dynamical Systems, 15 (1995), 291-315.
doi: 10.1017/S0143385700008385. |
[11] |
L. J. Díaz,
Persistence of cycles and nonhyperbolic dynamics at the unfolding of heteroclinic bifurcations, Ergodic Theory and Dynamical Systems, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[12] |
J. W. Evans, N. Fenichel and J. A. Feroe,
Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), 219-234.
doi: 10.1137/0142016. |
[13] |
J. A. Feroe,
Homoclinic orbits in a parametrized saddle-focus system, Phys. D, 62 (1993), 254-262.
doi: 10.1016/0167-2789(93)90285-9. |
[14] |
P. Gaspard,
Generation of a countable set of homoclinic flows through bifurcation, Physics Letters A, 97 (1983), 1-4.
doi: 10.1016/0375-9601(83)90085-3. |
[15] |
S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov,
On the existence of Newhouse regions in a neighbourhood of systems with a structurally unstable homoclinic Poincaré curve (the multidimensional case), Dokl. Akad. Nauk, 47 (1993), 268-273.
|
[16] |
S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev,
On global bifurcations in threedimensional diffeomorphisms leading to wild Lorenz-like attractors, Reg. Chaot. Dyn., 14 (2009), 137-147.
doi: 10.1134/S1560354709010092. |
[17] |
S. V. Gonchenko, D. V. Turaev, P. Gaspard and G. Nicolis,
Complexity in the bifurcation structure of homoclinic loops to a saddle-focus, Nonlinearity, 10 (1997), 409-423.
doi: 10.1088/0951-7715/10/2/006. |
[18] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer-Lecture Notes on Mathematics, 583, Heidelberg, 1977. |
[19] |
A. J. Homburg and B. Sandstede,
Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, Elsevier, 3 (2010), 379-524.
doi: 10.1016/S1874-575X(10)00316-4. |
[20] |
M. Hurley,
Attractors: Persistence and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247-271.
doi: 10.1090/S0002-9947-1982-0637037-7. |
[21] |
D. Li,
Homoclinic bifurcations that give rise to heterodimensional cycles near a Saddle-focus equilibrium, Nonlinearity, 30 (2017), 173-206.
doi: 10.1088/1361-6544/30/1/173. |
[22] |
S. E. Newhouse,
The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 101-151.
|
[23] |
S. E. Newhouse and J. Palis,
Cycles and bifurcation theory, Asterisque, 31 (1976), 43-140.
|
[24] |
I. M. Ovsyannikov and L. P. Shilnikov,
On systems with a saddle-focus homoclinic curve, Math. USSR Sbornik, 58 (1987), 557-574.
doi: 10.1070/SM1987v058n02ABEH003120. |
[25] |
I. M. Ovsyannikov and L. P. Shilnikov,
Systems with a homoclinic curve of multidimensional saddle-focus type, and spiral chaos, Math. USSR Sbornik, 73 (1992), 415-443.
|
[26] |
J. Palis and M. Viana,
High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math, 140 (1994), 207-250.
doi: 10.2307/2118546. |
[27] |
J. Palis,
A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.
|
[28] |
D. Ruelle,
Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137-151.
doi: 10.1007/BF01206949. |
[29] |
M. V. Shashkov and D. V. Turaev,
An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop, J. Nonlinear Sci., 9 (1999), 525-573.
doi: 10.1007/s003329900078. |
[30] |
L. P. Shilnikov,
A case of the existence of a countable number of periodic motions (Point mapping proof of existence theorem showing neighbourhood of trajectory which departs from and returns to saddle-point focus contains denumerable set of periodic motions), (Russian), Dokl. Akad. Nauk SSSR, 160 (1965), 558-561.
|
[31] |
L. P. Shilnikov,
A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik, 10 (1970), 91-102.
|
[32] |
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅰ), 2nd World Sci. -Singapore, New Jersey, London, Hong Kong, 2001.
doi: 10.1142/9789812798558_0001. |
[33] |
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅱ), 2nd World Sci. -Singapore, New Jersey, London, Hong Kong, 2001.
doi: 10.1142/9789812798558_0001. |
[34] |
W. Tucker,
A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
[35] |
D. V. Turaev,
On dimension of non-local bifurcational problems, International Journal of Bifurcation and Chaos, 6 (1996), 919-948.
doi: 10.1142/S0218127496000515. |
[36] |
D. V. Turaev and L. P. Shilnikov,
An example of a wild strange attractor, Math. USSR Sbornik, 189 (1998), 291-314.
doi: 10.1070/SM1998v189n02ABEH000300. |
show all references
References:
[1] |
V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov,
On the origin and structure of the Lorenz attractor, Akademiia Nauk SSSR Doklady, 234 (1977), 336-339.
|
[2] |
V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov,
On the structurally unstable attracting limit sets of Lorenz attractor type, Tran. Moscow Math. Soc., 2 (1983), 153-215.
|
[3] |
D. V. Anosov,
Geodesic flows on closed Riemannian manifolds of negative curvature, (Russian), Trudy Mat. Inst. Steklov., 90 (1967), 209pp.
|
[4] |
R. Barrio, A. L. Shilnikov and L. P. Shilnikov,
Kneadings, symbolic dynamics and painting Lorenz chaos, International Journal of Bifurcation and Chaos, 22 (2008), 1230016, 24 pp.
doi: 10.1142/S0218127412300169. |
[5] |
C. Bonatti and S. Crovisier,
Center manifolds for partially hyperbolic sets without strong unstable connections, Journal of the Institute of Mathematics of Jussieu, 15 (2016), 785-828.
doi: 10.1017/S1474748015000055. |
[6] |
C. Bonatti and L. J. Díaz,
Persistent transitive diffeomorphisms, Annals of Mathematics, 143 (1996), 357-396.
doi: 10.2307/2118647. |
[7] |
C. Bonatti and L. J. Díaz,
Robust heterodimensional cycles and C1-generic dynamics, Journal of the Institute of Mathematics of Jussieu, 7 (2008), 469-525.
doi: 10.1017/S1474748008000030. |
[8] |
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005. |
[9] |
L. J. Díaz and J. Rocha,
Non-connected heterodimensional cycles: Bifurcation and stability, Nonlinearity, 5 (1992), 1315-1341.
doi: 10.1088/0951-7715/5/6/006. |
[10] |
L. J. Díaz,
Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory and Dynamical Systems, 15 (1995), 291-315.
doi: 10.1017/S0143385700008385. |
[11] |
L. J. Díaz,
Persistence of cycles and nonhyperbolic dynamics at the unfolding of heteroclinic bifurcations, Ergodic Theory and Dynamical Systems, 8 (1995), 693-713.
doi: 10.1088/0951-7715/8/5/003. |
[12] |
J. W. Evans, N. Fenichel and J. A. Feroe,
Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), 219-234.
doi: 10.1137/0142016. |
[13] |
J. A. Feroe,
Homoclinic orbits in a parametrized saddle-focus system, Phys. D, 62 (1993), 254-262.
doi: 10.1016/0167-2789(93)90285-9. |
[14] |
P. Gaspard,
Generation of a countable set of homoclinic flows through bifurcation, Physics Letters A, 97 (1983), 1-4.
doi: 10.1016/0375-9601(83)90085-3. |
[15] |
S. V. Gonchenko, D. V. Turaev and L. P. Shilnikov,
On the existence of Newhouse regions in a neighbourhood of systems with a structurally unstable homoclinic Poincaré curve (the multidimensional case), Dokl. Akad. Nauk, 47 (1993), 268-273.
|
[16] |
S. V. Gonchenko, L. P. Shilnikov and D. V. Turaev,
On global bifurcations in threedimensional diffeomorphisms leading to wild Lorenz-like attractors, Reg. Chaot. Dyn., 14 (2009), 137-147.
doi: 10.1134/S1560354709010092. |
[17] |
S. V. Gonchenko, D. V. Turaev, P. Gaspard and G. Nicolis,
Complexity in the bifurcation structure of homoclinic loops to a saddle-focus, Nonlinearity, 10 (1997), 409-423.
doi: 10.1088/0951-7715/10/2/006. |
[18] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Springer-Lecture Notes on Mathematics, 583, Heidelberg, 1977. |
[19] |
A. J. Homburg and B. Sandstede,
Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, Elsevier, 3 (2010), 379-524.
doi: 10.1016/S1874-575X(10)00316-4. |
[20] |
M. Hurley,
Attractors: Persistence and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247-271.
doi: 10.1090/S0002-9947-1982-0637037-7. |
[21] |
D. Li,
Homoclinic bifurcations that give rise to heterodimensional cycles near a Saddle-focus equilibrium, Nonlinearity, 30 (2017), 173-206.
doi: 10.1088/1361-6544/30/1/173. |
[22] |
S. E. Newhouse,
The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 101-151.
|
[23] |
S. E. Newhouse and J. Palis,
Cycles and bifurcation theory, Asterisque, 31 (1976), 43-140.
|
[24] |
I. M. Ovsyannikov and L. P. Shilnikov,
On systems with a saddle-focus homoclinic curve, Math. USSR Sbornik, 58 (1987), 557-574.
doi: 10.1070/SM1987v058n02ABEH003120. |
[25] |
I. M. Ovsyannikov and L. P. Shilnikov,
Systems with a homoclinic curve of multidimensional saddle-focus type, and spiral chaos, Math. USSR Sbornik, 73 (1992), 415-443.
|
[26] |
J. Palis and M. Viana,
High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math, 140 (1994), 207-250.
doi: 10.2307/2118546. |
[27] |
J. Palis,
A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335-347.
|
[28] |
D. Ruelle,
Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137-151.
doi: 10.1007/BF01206949. |
[29] |
M. V. Shashkov and D. V. Turaev,
An existence theorem of smooth nonlocal center manifolds for systems close to a system with a homoclinic loop, J. Nonlinear Sci., 9 (1999), 525-573.
doi: 10.1007/s003329900078. |
[30] |
L. P. Shilnikov,
A case of the existence of a countable number of periodic motions (Point mapping proof of existence theorem showing neighbourhood of trajectory which departs from and returns to saddle-point focus contains denumerable set of periodic motions), (Russian), Dokl. Akad. Nauk SSSR, 160 (1965), 558-561.
|
[31] |
L. P. Shilnikov,
A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik, 10 (1970), 91-102.
|
[32] |
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅰ), 2nd World Sci. -Singapore, New Jersey, London, Hong Kong, 2001.
doi: 10.1142/9789812798558_0001. |
[33] |
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅱ), 2nd World Sci. -Singapore, New Jersey, London, Hong Kong, 2001.
doi: 10.1142/9789812798558_0001. |
[34] |
W. Tucker,
A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53-117.
doi: 10.1007/s002080010018. |
[35] |
D. V. Turaev,
On dimension of non-local bifurcational problems, International Journal of Bifurcation and Chaos, 6 (1996), 919-948.
doi: 10.1142/S0218127496000515. |
[36] |
D. V. Turaev and L. P. Shilnikov,
An example of a wild strange attractor, Math. USSR Sbornik, 189 (1998), 291-314.
doi: 10.1070/SM1998v189n02ABEH000300. |




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