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On the uniqueness of solution to generalized Chaplygin gas
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, United Kingdom 
2.  Department of Mathematics Imperial College London South Kensington Campus, London SW7 2AZ, UK 
3.  Department of Mathematics Lobachevsky State University of Nizhny Novgorod 23 Prospekt Gagarina, Nizhny Novgorod 603950, Russia 
4.  Joseph Meyerhoff Visiting Professor Weizmann Institute of Science 234 Herzl Street, Rehovot 7610001, Israel 
References:
[1] 
V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov, On the origin and structure of the Lorenz attractor, Akademiia Nauk SSSR Doklady, 234 (1977), 336339. 
[2] 
V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov, On the structurally unstable attracting limit sets of Lorenz attractor type, Tran. Moscow Math. Soc., 2 (1983), 153215. 
[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, L. P. Shilnikov, Kneadings, symbolic dynamics and painting Lorenz chaos, International Journal of Bifurcation and Chaos, 22 (2008), 1230016, 24 pp. 
[5] 
C. Bonatti, S. Crovisier, Center manifolds for partially hyperbolic sets without strong unstable connections, Journal of the Institute of Mathematics of Jussieu, 15 (2016), 785828. 
[6] 
C. Bonatti, L. J. Díaz, Persistent transitive diffeomorphisms, Annals of Mathematics, 143 (1996), 357396. 
[7] 
C. Bonatti, L. J. Díaz, Robust heterodimensional cycles and C1generic dynamics, Journal of the Institute of Mathematics of Jussieu, 7 (2008), 469525. 
[8] 
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. SpringerVerlag, Berlin, 2005. 
[9] 
L. J. Díaz, J. Rocha, Nonconnected heterodimensional cycles: Bifurcation and stability, Nonlinearity, 5 (1992), 13151341. 
[10] 
L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory and Dynamical Systems, 15 (1995), 291315. 
[11] 
L. J. Díaz, Persistence of cycles and nonhyperbolic dynamics at the unfolding of heteroclinic bifurcations, Ergodic Theory and Dynamical Systems, 8 (1995), 693713. 
[12] 
J. W. Evans, N. Fenichel, J. A. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), 219234. 
[13] 
J. A. Feroe, Homoclinic orbits in a parametrized saddlefocus system, Phys. D, 62 (1993), 254262. 
[14] 
P. Gaspard, Generation of a countable set of homoclinic flows through bifurcation, Physics Letters A, 97 (1983), 14. 
[15] 
S. V. Gonchenko, D. V. Turaev, 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), 268273. 
[16] 
S. V. Gonchenko, L. P. Shilnikov, D. V. Turaev, On global bifurcations in threedimensional diffeomorphisms leading to wild Lorenzlike attractors, Reg. Chaot. Dyn., 14 (2009), 137147. 
[17] 
S. V. Gonchenko, D. V. Turaev, P. Gaspard, G. Nicolis, Complexity in the bifurcation structure of homoclinic loops to a saddlefocus, Nonlinearity, 10 (1997), 409423. 
[18] 
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, SpringerLecture Notes on Mathematics, 583, Heidelberg, 1977. 
[19] 
A. J. Homburg, B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, Elsevier, 3 (2010), 379524. 
[20] 
M. Hurley, Attractors: Persistence and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247271. 
[21] 
D. Li, Homoclinic bifurcations that give rise to heterodimensional cycles near a Saddlefocus equilibrium, Nonlinearity, 30 (2017), 173206. 
[22] 
S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 101151. 
[23] 
S. E. Newhouse, J. Palis, Cycles and bifurcation theory, Asterisque, 31 (1976), 43140. 
[24] 
I. M. Ovsyannikov, L. P. Shilnikov, On systems with a saddlefocus homoclinic curve, Math. USSR Sbornik, 58 (1987), 557574. 
[25] 
I. M. Ovsyannikov, L. P. Shilnikov, Systems with a homoclinic curve of multidimensional saddlefocus type, and spiral chaos, Math. USSR Sbornik, 73 (1992), 415443. 
[26] 
J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math, 140 (1994), 207250. 
[27] 
J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335347. 
[28] 
D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137151. 
[29] 
M. V. Shashkov, 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), 525573. 
[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 saddlepoint focus contains denumerable set of periodic motions), (Russian) Dokl. Akad. Nauk SSSR, 160 (1965), 558561. 
[31] 
L. P. Shilnikov, A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddlefocus type, Math. USSR Sbornik, 10 (1970), 91102. 
[32] 
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅰ), 2^{nd} World Sci. Singapore, New Jersey, London, Hong Kong, 2001. 
[33] 
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅱ), 2^{nd} World Sci. Singapore, New Jersey, London, Hong Kong, 2001. 
[34] 
W. Tucker, A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53117. 
[35] 
D. V. Turaev, On dimension of nonlocal bifurcational problems, International Journal of Bifurcation and Chaos, 6 (1996), 919948. 
[36] 
D. V. Turaev, L. P. Shilnikov, An example of a wild strange attractor, Math. USSR Sbornik, 189 (1998), 291314. 
show all references
References:
[1] 
V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov, On the origin and structure of the Lorenz attractor, Akademiia Nauk SSSR Doklady, 234 (1977), 336339. 
[2] 
V. S. Afraimovich, V. V. Bykov, L. P. Shilnikov, On the structurally unstable attracting limit sets of Lorenz attractor type, Tran. Moscow Math. Soc., 2 (1983), 153215. 
[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, L. P. Shilnikov, Kneadings, symbolic dynamics and painting Lorenz chaos, International Journal of Bifurcation and Chaos, 22 (2008), 1230016, 24 pp. 
[5] 
C. Bonatti, S. Crovisier, Center manifolds for partially hyperbolic sets without strong unstable connections, Journal of the Institute of Mathematics of Jussieu, 15 (2016), 785828. 
[6] 
C. Bonatti, L. J. Díaz, Persistent transitive diffeomorphisms, Annals of Mathematics, 143 (1996), 357396. 
[7] 
C. Bonatti, L. J. Díaz, Robust heterodimensional cycles and C1generic dynamics, Journal of the Institute of Mathematics of Jussieu, 7 (2008), 469525. 
[8] 
C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. SpringerVerlag, Berlin, 2005. 
[9] 
L. J. Díaz, J. Rocha, Nonconnected heterodimensional cycles: Bifurcation and stability, Nonlinearity, 5 (1992), 13151341. 
[10] 
L. J. Díaz, Robust nonhyperbolic dynamics and heterodimensional cycles, Ergodic Theory and Dynamical Systems, 15 (1995), 291315. 
[11] 
L. J. Díaz, Persistence of cycles and nonhyperbolic dynamics at the unfolding of heteroclinic bifurcations, Ergodic Theory and Dynamical Systems, 8 (1995), 693713. 
[12] 
J. W. Evans, N. Fenichel, J. A. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math., 42 (1982), 219234. 
[13] 
J. A. Feroe, Homoclinic orbits in a parametrized saddlefocus system, Phys. D, 62 (1993), 254262. 
[14] 
P. Gaspard, Generation of a countable set of homoclinic flows through bifurcation, Physics Letters A, 97 (1983), 14. 
[15] 
S. V. Gonchenko, D. V. Turaev, 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), 268273. 
[16] 
S. V. Gonchenko, L. P. Shilnikov, D. V. Turaev, On global bifurcations in threedimensional diffeomorphisms leading to wild Lorenzlike attractors, Reg. Chaot. Dyn., 14 (2009), 137147. 
[17] 
S. V. Gonchenko, D. V. Turaev, P. Gaspard, G. Nicolis, Complexity in the bifurcation structure of homoclinic loops to a saddlefocus, Nonlinearity, 10 (1997), 409423. 
[18] 
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, SpringerLecture Notes on Mathematics, 583, Heidelberg, 1977. 
[19] 
A. J. Homburg, B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, in Handbook of Dynamical Systems, Elsevier, 3 (2010), 379524. 
[20] 
M. Hurley, Attractors: Persistence and density of their basins, Trans. Amer. Math. Soc., 269 (1982), 247271. 
[21] 
D. Li, Homoclinic bifurcations that give rise to heterodimensional cycles near a Saddlefocus equilibrium, Nonlinearity, 30 (2017), 173206. 
[22] 
S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math, 50 (1979), 101151. 
[23] 
S. E. Newhouse, J. Palis, Cycles and bifurcation theory, Asterisque, 31 (1976), 43140. 
[24] 
I. M. Ovsyannikov, L. P. Shilnikov, On systems with a saddlefocus homoclinic curve, Math. USSR Sbornik, 58 (1987), 557574. 
[25] 
I. M. Ovsyannikov, L. P. Shilnikov, Systems with a homoclinic curve of multidimensional saddlefocus type, and spiral chaos, Math. USSR Sbornik, 73 (1992), 415443. 
[26] 
J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math, 140 (1994), 207250. 
[27] 
J. Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque, 261 (2000), 335347. 
[28] 
D. Ruelle, Small random perturbations of dynamical systems and the definition of attractors, Comm. Math. Phys., 82 (1981), 137151. 
[29] 
M. V. Shashkov, 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), 525573. 
[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 saddlepoint focus contains denumerable set of periodic motions), (Russian) Dokl. Akad. Nauk SSSR, 160 (1965), 558561. 
[31] 
L. P. Shilnikov, A contribution to the problem of the structure of an extended neighbourhood of a rough equilibrium state of saddlefocus type, Math. USSR Sbornik, 10 (1970), 91102. 
[32] 
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅰ), 2^{nd} World Sci. Singapore, New Jersey, London, Hong Kong, 2001. 
[33] 
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev and L. O. Chua, Methods Of Qualitative Theory In Nonlinear Dynamics (Part Ⅱ), 2^{nd} World Sci. Singapore, New Jersey, London, Hong Kong, 2001. 
[34] 
W. Tucker, A rigorous ODE solver and Smale's 14th problem, Foundations of Computational Mathematics, 2 (2002), 53117. 
[35] 
D. V. Turaev, On dimension of nonlocal bifurcational problems, International Journal of Bifurcation and Chaos, 6 (1996), 919948. 
[36] 
D. V. Turaev, L. P. Shilnikov, An example of a wild strange attractor, Math. USSR Sbornik, 189 (1998), 291314. 
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