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Saddle-node bifurcations of multiple homoclinic solutions in ODES
1. | Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
We study codimension 3 degenerate homoclinic bifurcations under periodic perturbations. Assume that among the 3 bifurcation equations, one is due to the homoclinic tangecy along the orbital direction. To the lowest order, the bifurcation equations become 3 quadratic equations. Under generic conditions on perturbations of the normal and tangential directions of the homoclinic orbit, up to 8 homoclinic orbits can be created through saddle-node bifurcations. Our results generate the homoclinic tangency bifurcation in Guckenheimer and Holmes [
References:
[1] |
X. P. Chen and W. S. Wang,
Some conditions for co-diagolizations of two matrixes (in Chinese), J. Zaozhuang University, 22 (2005), 11-13.
|
[2] |
S. N. Chow, J. K. Hale and J. Mallet-Parret,
An example of bifurcation to homoclinic orbits, J. Diff. Equs., 37 (1980), 351-373.
doi: 10.1016/0022-0396(80)90104-7. |
[3] |
T. Dray, The Geometry of Special Relativity CRC Press, Jul 2,2012.
doi: 10.1201/b12293. |
[4] |
M. Fečkan,
Bifurcation from degenerate homoclinics in periodically forced systems, Discr. Cont. Dyn. Systems, 5 (1999), 359-374.
doi: 10.3934/dcds.1999.5.359. |
[5] |
J. R. Gruendler,
Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J.Math. Anal., 23 (1992), 702-721.
doi: 10.1137/0523036. |
[6] |
J. R. Gruendler,
Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbation, J. Diff. Equs., 122 (1995), 1-26.
doi: 10.1006/jdeq.1995.1136. |
[7] |
J. R. Gruendler,
The existence of transverse homoclinic solutions for higher order equations, J. Diff. Equs., 130 (1996), 307-320.
doi: 10.1006/jdeq.1996.0145. |
[8] |
J. R. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. K. Hale, Introduction to dynamic bifurcation, in Bifurcation Theory and Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1057 (1984), 106-151. |
[10] |
J. K. Hale and X.-B. Lin,
Heteroclinic orbits for Retarded functional differential equations, J. Diff. Equs., 65 (1986), 175-202.
doi: 10.1016/0022-0396(86)90032-X. |
[11] |
R. Horn, Topics in Matrix Analysis Cambridge University Press, 1994. |
[12] |
R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[13] |
L. Jaeger and H. Kantz,
Homoclinic tangencies and non-normal Jacobians-effects of noise in nonhyperbolic chaotic systems, Physica D, 105 (1997), 79-96.
doi: 10.1016/S0167-2789(97)00247-9. |
[14] |
J. Knobloch,
Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns., 9 (1997), 427-444.
doi: 10.1007/BF02227489. |
[15] |
J. Li and X.-B. Lin,
Traveling wave solutions for the Painlevé-integrable coupled KDV equations, Electronic. J. Diff. Equs., 2008 (2008), 1-11.
|
[16] |
X.-B. Lin,
Using Melnikov's method to solve Silnikov's problems, Proc. Roy. Soc. Edinburgh, 116 (1990), 295-325.
doi: 10.1017/S0308210500031528. |
[17] |
X.-B. Lin, B. Long and C. Zhu,
Multiple transverse homoclinic solutions near a degenerate homoclinic orbit, J. Diff. Equs., 259 (2015), 1-24.
doi: 10.1016/j.jde.2015.01.046. |
[18] |
J. Mallet-Paret,
Generic periodic solutions of functional differential equations, J. Diff. Equs., 25 (1977), 163-183.
doi: 10.1016/0022-0396(77)90198-X. |
[19] |
V. K. Melnikov,
On the stability of the center for time periodic perturbation, Trans. Moscow Math. Soc., 12 (1963), 1-57.
|
[20] |
K. J. Palmer,
Exponential dichotomies and transversal homoclinic points, J. Diff. Equs., 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[21] |
K. J. Palmer,
Transversal heteroclinic orbits and Cherry'y example of a non-integrable Hamiltonian system, J. Diff. Eqns., 65 (1986), 321-360.
doi: 10.1016/0022-0396(86)90023-9. |
show all references
References:
[1] |
X. P. Chen and W. S. Wang,
Some conditions for co-diagolizations of two matrixes (in Chinese), J. Zaozhuang University, 22 (2005), 11-13.
|
[2] |
S. N. Chow, J. K. Hale and J. Mallet-Parret,
An example of bifurcation to homoclinic orbits, J. Diff. Equs., 37 (1980), 351-373.
doi: 10.1016/0022-0396(80)90104-7. |
[3] |
T. Dray, The Geometry of Special Relativity CRC Press, Jul 2,2012.
doi: 10.1201/b12293. |
[4] |
M. Fečkan,
Bifurcation from degenerate homoclinics in periodically forced systems, Discr. Cont. Dyn. Systems, 5 (1999), 359-374.
doi: 10.3934/dcds.1999.5.359. |
[5] |
J. R. Gruendler,
Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J.Math. Anal., 23 (1992), 702-721.
doi: 10.1137/0523036. |
[6] |
J. R. Gruendler,
Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbation, J. Diff. Equs., 122 (1995), 1-26.
doi: 10.1006/jdeq.1995.1136. |
[7] |
J. R. Gruendler,
The existence of transverse homoclinic solutions for higher order equations, J. Diff. Equs., 130 (1996), 307-320.
doi: 10.1006/jdeq.1996.0145. |
[8] |
J. R. Guckenheimer and P. Holmes,
Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. K. Hale, Introduction to dynamic bifurcation, in Bifurcation Theory and Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1057 (1984), 106-151. |
[10] |
J. K. Hale and X.-B. Lin,
Heteroclinic orbits for Retarded functional differential equations, J. Diff. Equs., 65 (1986), 175-202.
doi: 10.1016/0022-0396(86)90032-X. |
[11] |
R. Horn, Topics in Matrix Analysis Cambridge University Press, 1994. |
[12] |
R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, 1985.
doi: 10.1017/CBO9780511810817. |
[13] |
L. Jaeger and H. Kantz,
Homoclinic tangencies and non-normal Jacobians-effects of noise in nonhyperbolic chaotic systems, Physica D, 105 (1997), 79-96.
doi: 10.1016/S0167-2789(97)00247-9. |
[14] |
J. Knobloch,
Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns., 9 (1997), 427-444.
doi: 10.1007/BF02227489. |
[15] |
J. Li and X.-B. Lin,
Traveling wave solutions for the Painlevé-integrable coupled KDV equations, Electronic. J. Diff. Equs., 2008 (2008), 1-11.
|
[16] |
X.-B. Lin,
Using Melnikov's method to solve Silnikov's problems, Proc. Roy. Soc. Edinburgh, 116 (1990), 295-325.
doi: 10.1017/S0308210500031528. |
[17] |
X.-B. Lin, B. Long and C. Zhu,
Multiple transverse homoclinic solutions near a degenerate homoclinic orbit, J. Diff. Equs., 259 (2015), 1-24.
doi: 10.1016/j.jde.2015.01.046. |
[18] |
J. Mallet-Paret,
Generic periodic solutions of functional differential equations, J. Diff. Equs., 25 (1977), 163-183.
doi: 10.1016/0022-0396(77)90198-X. |
[19] |
V. K. Melnikov,
On the stability of the center for time periodic perturbation, Trans. Moscow Math. Soc., 12 (1963), 1-57.
|
[20] |
K. J. Palmer,
Exponential dichotomies and transversal homoclinic points, J. Diff. Equs., 55 (1984), 225-256.
doi: 10.1016/0022-0396(84)90082-2. |
[21] |
K. J. Palmer,
Transversal heteroclinic orbits and Cherry'y example of a non-integrable Hamiltonian system, J. Diff. Eqns., 65 (1986), 321-360.
doi: 10.1016/0022-0396(86)90023-9. |







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