The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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EXISTENCE OF HOMOCLINIC ORBITS FOR LIENARD TYPE SYSTEMS

  • Received : 2010.08.19
  • Accepted : 2010.11.22
  • Published : 2010.11.30
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Abstract

We investigate the existence of homoclinic orbits of the following systems of $Li{\'{e}}nard$ type: $a(x)x^'=h(y)-F(x)$, $y^'$=-a(x)g(x), where $h(y)=m{\mid}y{\mid}^{p-2}y$ with m > 0 and p > 1 and a, F, 9 are continuous functions such that a(x) > 0 for all $x{\in}{\mathbb{R}}$ and F(0)=g(0)=0 and xg(x) > 0 for $x{\neq}0$. 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

References

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