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Saddle-node bifurcations of multiple homoclinic solutions in ODES

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

    * Corresponding author: Changrong Zhu
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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  • 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].

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


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  • 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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