Saddle-node bifurcations of multiple homoclinic solutions in ODES

Xiao-Biao Lin; Changrong Zhu

doi:10.3934/dcdsb.2017069

Article Contents

2017, Volume 22, Issue 4: 1435-1460. Doi: 10.3934/dcdsb.2017069

This issue Previous Article Feedback controllability for blowup points of semilinear heat equations Next Article Global existence for a thin film equation with subcritical mass

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

Author Bio: E-mail address: xblin@ncsu.edu; E-mail address: zhuchangrong126@126.com
* Corresponding author: Changrong Zhu
* Corresponding author: Changrong Zhu

Received: July 31, 2015

Revised: November 30, 2016

Published: August 2017

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.

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Abstract

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].

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Mathematics Subject Classification: Primary:34C23, 34C25;Secondary:34C45, 34C40.

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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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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.