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 & 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.   Google Scholar

[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.  Google Scholar

[3]

T. Dray, The Geometry of Special Relativity CRC Press, Jul 2,2012. doi: 10.1201/b12293.  Google Scholar

[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.  Google Scholar

[5]

J. R. Gruendler, Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J.Math. Anal., 23 (1992), 702-721.  doi: 10.1137/0523036.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. K. Hale, Introduction to dynamic bifurcation, in Bifurcation Theory and Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1057 (1984), 106-151. Google Scholar

[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.  Google Scholar

[11]

R. Horn, Topics in Matrix Analysis Cambridge University Press, 1994.  Google Scholar

[12]

R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[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.  Google Scholar

[14]

J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns., 9 (1997), 427-444.  doi: 10.1007/BF02227489.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. K. Melnikov, On the stability of the center for time periodic perturbation, Trans. Moscow Math. Soc., 12 (1963), 1-57.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[3]

T. Dray, The Geometry of Special Relativity CRC Press, Jul 2,2012. doi: 10.1201/b12293.  Google Scholar

[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.  Google Scholar

[5]

J. R. Gruendler, Homoclinic solutions for autonomous systems in arbitrary dimension, SIAM J.Math. Anal., 23 (1992), 702-721.  doi: 10.1137/0523036.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. K. Hale, Introduction to dynamic bifurcation, in Bifurcation Theory and Applications, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1057 (1984), 106-151. Google Scholar

[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.  Google Scholar

[11]

R. Horn, Topics in Matrix Analysis Cambridge University Press, 1994.  Google Scholar

[12]

R. Horn and C. Johnson, Matrix Analysis Cambridge University Press, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[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.  Google Scholar

[14]

J. Knobloch, Bifurcation of degenerate homoclinics in reversible and conservative systems, J. Dyn. Diff. Eqns., 9 (1997), 427-444.  doi: 10.1007/BF02227489.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

V. K. Melnikov, On the stability of the center for time periodic perturbation, Trans. Moscow Math. Soc., 12 (1963), 1-57.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
[1]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[2]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[3]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[4]

Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381

[5]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[6]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029

[7]

Djamel Aaid, Amel Noui, Özen Özer. Piecewise quadratic bounding functions for finding real roots of polynomials. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 63-73. doi: 10.3934/naco.2020015

[8]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[9]

Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342

[10]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[11]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[12]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460

[13]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[14]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108

[15]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

[16]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (60)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]