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

A remark about Sil'nikov saddle-focus homoclinic orbits

Flaviano Battelli 1, and  Ken Palmer 2,

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.
Keywords: exponential dichotomy., homoclinic orbits, invariant manifolds, Sil'nikov chaos, transversality.
Mathematics Subject Classification: Primary: 34C27; 37G20; Secondary: 34C2.
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

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