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  • P-ISSN1226-0657
  • E-ISSN2287-6081
  • KCI

ISSN : 1226-0657

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Vol.17 No.4

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EXISTENCE OF HOMOCLINIC ORBITS FOR LI¶ENARD TYPE SYSTEMS

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2010, v.17 no.4, pp.389-396
Kim, Yong-In
Kim,, Y. (2010). EXISTENCE OF HOMOCLINIC ORBITS FOR LI¶ENARD TYPE SYSTEMS. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 17(4), 389-396.

Abstract

We investigate the existence of homoclinic orbits of the following systems of <TEX>$Li{\'{e}}nard$</TEX> type: <TEX>$a(x)x^'=h(y)-F(x)$</TEX>, <TEX>$y^'$</TEX>=-a(x)g(x), where <TEX>$h(y)=m{\mid}y{\mid}^{p-2}y$</TEX> with m > 0 and p > 1 and a, F, 9 are continuous functions such that a(x) > 0 for all <TEX>$x{\in}{\mathbb{R}}$</TEX> and F(0)=g(0)=0 and xg(x) > 0 for <TEX>$x{\neq}0$</TEX>. By a series of time and coordinates transformations of the above system, we obtain sufficient conditions for the positive orbits of the above system starting at the points on the curve h(y) = F(x) with x > 0 to approach the origin through only the first quadrant. The method of this paper is new and the results of this paper cover some early results on this topic.

keywords
Lienard type system, homoclinic orbit, phase plane analysis

Reference

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