Chaotic Dynamics of One-Dimensional Systems with Periodic Boundary Conditions

TL;DR

This paper introduces a novel method for analyzing chaos in one-dimensional systems with periodic boundaries, demonstrated on a plasma model, enabling precise Lyapunov exponent calculations and dynamic behavior insights.

Contribution

It presents a new approach for defining phase-space distances and calculating Lyapunov exponents in periodic systems, with applications to plasma dynamics.

Findings

Derived exact analytic expressions for electric field and potential.

Successfully calculated Lyapunov exponents and other properties in simulations.

Revealed chaotic behavior in small and large plasma systems.

Abstract

We provide appropriate tools for the analysis of dynamics and chaos for one-dimensional systems with periodic boundary conditions. Our approach allows for the investigation of the dependence of the largest Lyapunov exponent on various initial conditions of the system. The method employs an effective approach for defining the phase-space distance appropriate for systems with periodic boundary and allows for an unambiguous test-orbit rescaling in the phase space required to calculate the Lyapunov exponents. We elucidate our technique by applying it to investigate the chaotic dynamics of a one-dimensional plasma with periodic boundary. Exact analytic expressions are derived for the electric field and potential using Ewald sums thereby making it possible to follow the time-evolution of the plasma in simulation without any special treatment of the boundary. By employing a set of event-driven algorithms, we calculate the largest Lyapunov exponent, the radial distribution function and the kinetic pressure by following the evolution of the system in phase space without resorting to numerical manipulation of the equations of motion. Simulation results are presented and analyzed for the one-dimensional plasma with a view to examining the dynamical and chaotic behavior exhibited by small and large versions of the system.