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

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

* 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].

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. ChowJ. 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. LinB. 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. ChowJ. 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. LinB. 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
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
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
Figure 4.  If the asymptotes of $F_1=0$ and $F_2=0$ are alternating, then there always exist exactly two solutions
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
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
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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