May  2011, 10(3): 817-830. doi: 10.3934/cpaa.2011.10.817

A remark about Sil'nikov saddle-focus homoclinic orbits

1. 

Dipartimento di Scienze Matematiche, Facoltà di Ingegneria, Università, Via Brecce Bianche, 1, 60100 Ancona

2. 

Department of Mathematics, National Taiwan University, Taipei 106

Received  October 2008 Revised  July 2009 Published  December 2010

In this note we study Sil'nikov saddle-focus homoclinic orbits paying particular attention to four and higher dimensions where two additional conditions are needed. We give equivalent conditions in terms of subspaces associated with the variational equation along the orbit. Then we review Rodriguez's construction of a three-dimensional system with Sil'nikov saddle-focus homoclinic orbits and finally show how to construct higher-dimensional systems with such orbits.
Citation: Flaviano Battelli, Ken Palmer. A remark about Sil'nikov saddle-focus homoclinic orbits. Communications on Pure & Applied Analysis, 2011, 10 (3) : 817-830. doi: 10.3934/cpaa.2011.10.817
References:
[1]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: doi:10.1093/imanum/10.3.379.  Google Scholar

[2]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Vol. 629, Springer Verlag, Berlin, 1978.  Google Scholar

[3]

B. Deng, On Sil'nikov's homoclinic-saddle-focus theorem, J. Diff. Equations, 102 (1993), 305-329. doi: doi:10.1006/jdeq.1993.1031.  Google Scholar

[4]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory" 3rd Ed., Springer, 2004.  Google Scholar

[5]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Equations, 55 (1984), 225-256. doi: doi:10.1016/0022-0396(84)90082-2.  Google Scholar

[6]

J. A. Rodriguez, Bifurcation to homoclinic connections of the focus-saddle type, Arch. Rat. Mech. Anal., 93 (1986), 81-90. doi: doi:10.1007/BF00250846.  Google Scholar

[7]

B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynamics Diff. Eqns., 9 (1997), 269-288. doi: doi:10.1007/BF02219223.  Google Scholar

[8]

L. P. Sil'nikov, A case of the existence of a denumerable set of periodic motions, Soviet Math. Doklady, 6 (1965), 163-166.  Google Scholar

[9]

L. P. Sil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik, 10 (1970), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588.  Google Scholar

[10]

L. P. Sil'nikov, A. L. Sil'nikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part I, World Scientific, Singapore, 1998. doi: doi:10.1142/9789812798596.  Google Scholar

[11]

L. P. Sil'nikov, A. L. Sil'nikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part II, World Scientific, Singapore, 2001. doi: doi:10.1142/9789812798558.  Google Scholar

[12]

S. R. Wiggins, "Global Bifurcations and Chaos: Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.  Google Scholar

show all references

References:
[1]

W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 10 (1990), 379-405. doi: doi:10.1093/imanum/10.3.379.  Google Scholar

[2]

W. A. Coppel, "Dichotomies in Stability Theory," Lecture Notes in Math., Vol. 629, Springer Verlag, Berlin, 1978.  Google Scholar

[3]

B. Deng, On Sil'nikov's homoclinic-saddle-focus theorem, J. Diff. Equations, 102 (1993), 305-329. doi: doi:10.1006/jdeq.1993.1031.  Google Scholar

[4]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory" 3rd Ed., Springer, 2004.  Google Scholar

[5]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Diff. Equations, 55 (1984), 225-256. doi: doi:10.1016/0022-0396(84)90082-2.  Google Scholar

[6]

J. A. Rodriguez, Bifurcation to homoclinic connections of the focus-saddle type, Arch. Rat. Mech. Anal., 93 (1986), 81-90. doi: doi:10.1007/BF00250846.  Google Scholar

[7]

B. Sandstede, Constructing dynamical systems having homoclinic bifurcation points of codimension two, J. Dynamics Diff. Eqns., 9 (1997), 269-288. doi: doi:10.1007/BF02219223.  Google Scholar

[8]

L. P. Sil'nikov, A case of the existence of a denumerable set of periodic motions, Soviet Math. Doklady, 6 (1965), 163-166.  Google Scholar

[9]

L. P. Sil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik, 10 (1970), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588.  Google Scholar

[10]

L. P. Sil'nikov, A. L. Sil'nikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part I, World Scientific, Singapore, 1998. doi: doi:10.1142/9789812798596.  Google Scholar

[11]

L. P. Sil'nikov, A. L. Sil'nikov, D. V. Turaev and L. O. Chua, "Methods of Qualitative Theory in Nonlinear Dynamics," Part II, World Scientific, Singapore, 2001. doi: doi:10.1142/9789812798558.  Google Scholar

[12]

S. R. Wiggins, "Global Bifurcations and Chaos: Analytical Methods," Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988.  Google Scholar

[1]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210

[2]

Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1

[3]

Christopher K. R. T. Jones, Siu-Kei Tin. Generalized exchange lemmas and orbits heteroclinic to invariant manifolds. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 967-1023. doi: 10.3934/dcdss.2009.2.967

[4]

Roberto Castelli. Efficient representation of invariant manifolds of periodic orbits in the CRTBP. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 563-586. doi: 10.3934/dcdsb.2018197

[5]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3367-3387. doi: 10.3934/dcds.2020409

[6]

V. Afraimovich, T.R. Young. Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 691-704. doi: 10.3934/dcds.2000.6.691

[7]

Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589

[8]

Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383

[9]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082

[10]

Rovella Alvaro, Vilamajó Francesc, Romero Neptalí. Invariant manifolds for delay endomorphisms. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 35-50. doi: 10.3934/dcds.2001.7.35

[11]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Homoclinic orbits for a class of asymptotically quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2855-2878. doi: 10.3934/cpaa.2019128

[12]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[13]

Martin Wechselberger, Warren Weckesser. Homoclinic clusters and chaos associated with a folded node in a stellate cell model. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 829-850. doi: 10.3934/dcdss.2009.2.829

[14]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[15]

Ricardo Rosa. Approximate inertial manifolds of exponential order. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 421-448. doi: 10.3934/dcds.1995.1.421

[16]

Rossella Bartolo. Periodic orbits on Riemannian manifolds with convex boundary. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 439-450. doi: 10.3934/dcds.1997.3.439

[17]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[18]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778

[19]

Flaviano Battelli. Saddle-node bifurcation of homoclinic orbits in singular systems. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 203-218. doi: 10.3934/dcds.2001.7.203

[20]

Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]