Analytic structure and chaotic dynamics of the damped driven Toda oscillator

Abstract

The singularity structure exhibited by the solution of the damped driven Toda oscillator in the complex time (t-) plane is investigated through Painlev\'e(P-) analysis. We find that there exists a specific parametric choice for which the free but damped Toda oscillator possesses the P- property and hence is likely to be integrable. We present the exact solution corresponding to this integrable choice. In the nonintegrable regime, we show that the singularities exhibit locally a complicated, clustered, two-armed infinite-sheeted Riemann structure in the complex t- plane. Further, we have analyzed numerically the global singularity structure in the complex t- plane (i.e., analytic structure) corresponding to the real time chaotic dynamics exhibited by the system. From the investigations, we observe that the global singularity structure exhibits a ``chimneylike'' pattern in which the width at the bottom of the chimney decreases and the singularities tend to cluster at the top of the chimney, in the complex t- plane corresponding to the real-time chaotic dynamics exhibited by the system, as the control parameter is varied.