Melnikov's method in String Theory

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 ansätze, 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.

37 pages, 5 figures

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• Analytic integrability for strings on $\eta$ and $\lambda$ deformed backgrounds
• The bound on chaos for closed strings in Anti-de Sitter black hole backgrounds
• Anisotropic and frame dependent chaos of suspended strings from a dynamical holographic QCD model with magnetic field
• Chaotic spin chains in AdS/CFT
• Interplay of magnetic field and chemical potential induced anisotropy and frame dependent chaos of a $Q\bar{Q}$ pair in holographic QCD