A Two-Dimensional Map of Arbitrary Period

TL;DR

This paper presents a two-dimensional discrete dynamical system modeling angle and angular velocity, analyzing its periodicity, stability, and relation to pendulum dynamics through simulations and approximate invariants.

Contribution

It introduces a novel 2D map combining fixed-angle evolution with nonlinear angular velocity dynamics, linking it to pendulum behavior and providing analysis tools.

Findings

System exhibits periodic solutions for specific parameters and initial conditions.

In the limit, the system approximates simple pendulum dynamics.

Simulations demonstrate diverse behaviors depending on parameters.

Abstract

We introduce a two-dimensional discrete-time dynamical system which represents the evolution of an angle and angular velocity. While the angle evolves by a fixed amount in every step, the evolution of the angular velocity is governed by a nonlinear map. We study the periodicity and stability of solutions to the system for a range of parameter values and initial conditions. The coupled system is shown to be periodic for certain parameter choices and initial conditions. In the limit, when the change in the angle tends to zero, the map is equivalent to the dynamics of a simple pendulum. Based on the integral of motion of the pendulum, an approximate invariant for the system is obtained. Simulations showing the behavior of the system for different parameter values and initial conditions are presented.