Chaotic orbits by Melnikov's method: A survey of applications

Melnikov [1963] introduced a method for proving the existence of chaotic orbits in dynamlcal systems. This was used by Arnold [1964], the paper in which "Arnold diffusion" was discovered. The method lay relatively dormant until it was revived by Chirikov [1979], Holmes [1980], and Chow, et al. [19801. In this lecture I will survey a number of recent applications of Melnikov's method to a variety of interesting physical situations. There are many modifications of the basic technique possible depending on the dimension of the system, whether or not dissipation or forcing are included or whether or not the system is autonomous. However to get the basic idea, it is useful to begin with a conservative but externally forced one degree of freedom Hamiltonian system. After presenting the basic theorem for this case in section 2, we shall discuss the generalizations of the theory in section 3 and the various applications in section 4. For additional background and applications, see Lichtenberg and Lieberman [1983] and Guckenheimer and Holmes [1983]