Melnikov's method in String Theory

TL;DR

This paper applies Melnikov's method to a specific supergravity background, demonstrating the analytical existence of chaos in coupled pendulum-oscillator systems derived from string theory backgrounds.

Contribution

It extends Melnikov's method to a new class of supergravity solutions, enabling analytical detection of chaos in string theory-related dynamical systems.

Findings

Analytical proof of chaos in supergravity background

Application of Melnikov's method to string theory models

Demonstration of chaos in coupled oscillator systems

Abstract

Melnikov's method is an analytical way to show the existence of classical chaos generated by a Smale horseshoe. It is a powerful technique, though its applicability is somewhat limited. In this paper, we present a solution of type IIB supergravity to which Melnikov's method is applicable. This is a brane-wave type deformation of the AdS$_5\times$S$^5$ background. By employing two reduction ans\"atze, we study two types of coupled pendulum-oscillator systems. Then the Melnikov function is computed for each of the systems by following the standard way of Holmes and Marsden and the existence of chaos is shown analytically.