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June  2017, 22(4): 1435-1460. doi: 10.3934/dcdsb.2017069

Saddle-node bifurcations of multiple homoclinic solutions in ODES

Xiao-Biao Lin 1, and  Changrong Zhu 2,,

1. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA
2. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Changrong Zhu

Received  August 2015 Revised  December 2016 Published  February 2017

Fund Project: XBL was partially supported by NSF grant DMS-1211070. ZCR was partially supported by NSFC grant 11671058 and NCET-12-0586. The authors would like to thank Moody Chu for helpful discussion on the codiagonalization of quadratic forms.

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 [8].

Keywords: Degenerate homoclinic bifurcation, Lyapunov-Schmidt reduction, exponential dichotomy, codiagonalization of quadratic forms.
Mathematics Subject Classification: Primary:34C23, 34C25;Secondary:34C45, 34C40.
Citation: Xiao-Biao Lin, Changrong Zhu. Saddle-node bifurcations of multiple homoclinic solutions in ODES. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1435-1460. doi: 10.3934/dcdsb.2017069
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.

Figure 1.  Graphs for the (HH) type systems
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Figure 2.  In case (4.6-1), if $h_1>0$ and $r_1<r_2$, or in case (4.6-2), if $h_1<0$ and $r_1<r_2$, then the system has $4$ solutions
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Figure 3.  In case (4.6-3), if $h_1<0$ and $r_1>r_2$, or in case (4.6-4), if $h_1>0$ and $r_1>r_2$, then the system has $4$ solutions
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Figure 4.  If the asymptotes of $F_1=0$ and $F_2=0$ are alternating, then there always exist exactly two solutions
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Figure 5.  Figure (1) is about the (EE) case. Figure (2) is about the (HE) case. The (HE) case where the hyperbola has $y$ intercept is not shown. Figure (3) is about the (LE) case where the lines have $x$-intercepts. The (LE) case where the lines have $y$ intercepts is not shown
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Figure 6.  Figure (1) is about the (HH) case where one curve has $x$ and the other has $y$ intercepts. Figure (2) is about the (HH) case where both graphs have $x$-intercepts. The (HH) case where both have $y$ intercepts in not shown
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Figure 7.  Figure (1) is about the (LH) case where the lines have $x$-intercepts and the hyperbola has $y$-intercept. Figure (2) is about the (LH) case where both the lines and the hyperbola have $y$-intercepts. The two (LH) cases where the hyperbola has $x$-intercepts are not shown
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