August 2017, 37(8): 4399-4437. doi: 10.3934/dcds.2017189

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

* Corresponding author: Dongchen Li

Received  February 2017 Revised  February 2017 Published  April 2017

Fund Project: This work was supported by grant RSF 14-41-00044. The authors also acknowledge support by the Royal Society grant IE141468 and EU Marie-Curie IRSES Brazilian-European partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS)

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.

Citation: Dongchen Li, Dmitry V. Turaev. Existence of heterodimensional cycles near Shilnikov loops in systems with a $\mathbb{Z}_2$ symmetry. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4399-4437. doi: 10.3934/dcds.2017189
References:
[1]

V. S. AfraimovichV. 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. AfraimovichV. 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. BarrioA. 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. EvansN. 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. GonchenkoD. 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. GonchenkoL. 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. GonchenkoD. V. TuraevP. 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. AfraimovichV. 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. AfraimovichV. 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. BarrioA. 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. EvansN. 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. GonchenkoD. 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. GonchenkoL. 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. GonchenkoD. V. TuraevP. 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.

Figure 1.  The dashed curves represent the two homoclinic loops and the solid vertical lines represent leaves of the foliation $\mathcal{F}_0$. The coincidence condition for system $X$ is that, for any point of $\Gamma^+$ lying in a leaf $l$, there exists one point of $\Gamma^-$ that also lies in $l$
Figure 2.  For a four-dimensional system, the intersection points $M^+$ and $M^-$ on a small three-dimensional cross-section $\Pi$ belong to the same leaf of the foliation $\mathcal{F}_1$
Figure 3.  As shown in figure (a), we can create an infinite sequence of index-2 point $Q_k^-$ accumulating on $M^-$ while keeping the intersection $W^u(P)\cap W^{ss}(M^-)$ by changing $\mu, \rho$ and $\nu$ together. In figure (b), the intersection $W^u(P)\cap W^s(Q_{k_0}^-)$ is created by changing $\nu$)
Figure 4.  The index is determined by the pair ($\lambda_1+\lambda_2, \lambda_1\lambda_2$)
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