Title: Energy Distribution in Intrinsically Coupled Systems: The Spring Pendulum Paradigm,
Abstract: Intrinsically nonlinear coupled systems present different oscillating
components that exchange energy among themselves. We present a new approach to
deal with such energy exchanges and to investigate how it depends on the system
control parameters. The method consists in writing the total energy of the
system, and properly identifying the energy terms for each component and,
especially, their coupling. To illustrate the proposed approach, we work with
the bi-dimensional spring pendulum, which is a paradigm to study nonlinear
coupled systems, and is used as a model for several systems. For the spring
pendulum, we identify three energy components, resembling the spring and
pendulum like motions, and the coupling between them. With these analytical
expressions, we analyze the energy exchange for individual trajectories, and we
also obtain global characteristics of the spring pendulum energy distribution
by calculating spatial and time average energy components for a great number of
trajectories (periodic, quasi-periodic and chaotic) throughout the phase space.
Considering an energy term due to the nonlinear coupling, we identify regions
in the parameter space that correspond to strong and weak coupling. The
presented procedure can be applied to nonlinear coupled systems to reveal how
the coupling mediates internal energy exchanges, and how the energy
distribution varies according to the system parameters.