Chaotic dynamics of one-dimensional systems with periodic boundary conditions

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 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.

Bifurcation and Chaos in Real Dynamics of a Two-Parameter Family Arising from Generating Function of Generalized Apostol-Type Polynomials

“…For sine families λsinx and λsin 2 x, chaotic behaviour in the real dynamics can be seen in [7]. The chaotic behaviour in the dynamics of the one-dimensional families of maps corresponds to Fibonacci-generating functions associated with the golden-, the silver-and the bronze mean is explored in [8]; and in [9], it is described with periodic boundary conditions. The bifurcation and chaotic behaviour in the real dynamics of one-parameter families of transcendental functions are found in [10][11][12].…”

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

“…One dimensional systems are of great interest to physicists in terms of their intrinsic properties and as a starting point in the analysis of their more-complicated higher-dimensional counterparts (see [8][9][10][11][12] and references therein). Not only can one-dimensional systems map the behaviors of experimental results [13], one-dimensional interactions have, in fact, been recently emulated in the laboratory [14].…”

“…In the analysis of large systems considered in plasma and gravitational physics, periodic boundary conditions are preferred [15][16][17][18] and have been utilized in the study of one-dimensional Coulombic and gravitational systems [12,[19][20][21]. Such studies often rely on numerical simulations to validate the predictions made by theory.…”

“…Such studies often rely on numerical simulations to validate the predictions made by theory. Moreover, in cases where theoretical relations have not been mathematically formulated, numerical simulations serve as a powerful approach in characterizing the dynamical behaviors and thermodynamic properties [12].…”

“…In general, however, if the time evolution of each particles' position and velocity can be followed for a given system, the largest LCE may be calculated by finding the rate of divergence between a reference trajectory and a nearby test trajectory obtained by pertubing the former [26]. This numerical approach was extended to the case of systems with periodic boundary conditions by Kumar and Miller and was applied to find the largest LCE for a spatially-periodic one-dimensional Coulombic system [12].…”