EXISTENCE OF HOMOCLINIC ORBITS FOR LI¶ENARD TYPE SYSTEMS
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We investigate the existence of homoclinic orbits of the following sys-tems of Lienard type: a(x)x0 = h(y)¡F(x); y0 = ¡a(x)g(x), where h(y) = mjyjp¡2y with m > 0 and p > 1 and a; F; g are continuous functions such that a(x) > 0 for all x 2 R and F(0) = g(0) = 0 and xg(x) > 0 for x 6= 0. By a series of time and coor-dinates transformations of the above system, we obtain su±cient 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 ¯rst quadrant. The method of this paper is new and the results of this paper cover some early results on this topic.
Abstract
We investigate the existence of homoclinic orbits of the following sys-tems of Lienard type: a(x)x0 = h(y)¡F(x); y0 = ¡a(x)g(x), where h(y) = mjyjp¡2y with m > 0 and p > 1 and a; F; g are continuous functions such that a(x) > 0 for all x 2 R and F(0) = g(0) = 0 and xg(x) > 0 for x 6= 0. By a series of time and coor-dinates transformations of the above system, we obtain su±cient 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 ¯rst quadrant. The method of this paper is new and the results of this paper cover some early results on this topic.
참고문헌 (10)
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