5 research outputs found
Resonant forcing of nonlinear systems of differential equations
We study resonances of nonlinear systems of differential equations, including
but not limited to the equations of motion of a particle moving in a potential.
We use the calculus of variations to determine the minimal additive forcing
function that induces a desired terminal response, such as an energy in the
case of a physical system. We include the additional constraint that only
select degrees of freedom be forced, corresponding to a very general class of
problems in which not all of the degrees of freedom in an experimental system
are accessible to forcing. We find that certain Lagrange multipliers take on a
fundamental physical role as the effective forcing experienced by the degrees
of freedom which are not forced directly. Furthermore, we find that the product
of the displacement of nearby trajectories and the effective total forcing
function is a conserved quantity. We demonstrate the efficacy of this
methodology with several examples.Comment: 9 pages, 3 figure
Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics
We study resonances of multidimensional chaotic map dynamics. We use the
calculus of variations to determine the additive forcing function that induces
the largest response, that is, the greatest deviation from the unperturbed
dynamics. We include the additional constraint that only select degrees of
freedom be forced, corresponding to a very general class of problems in which
not all of the degrees of freedom in an experimental system are accessible to
forcing. We find that certain Lagrange multipliers take on a fundamental
physical role as the efficiency of the forcing function and the effective
forcing experienced by the degrees of freedom which are not forced directly.
Furthermore, we find that the product of the displacement of nearby
trajectories and the effective total forcing function is a conserved quantity.
We demonstrate the efficacy of this methodology with several examples.Comment: 11 pages, 3 figure
Guiding a self-adjusting system through chaos
Abstract: We study the parametric controls of self-adjusting systems with numerical models. We investigate the situation where the target dynamics changes slowly and passes through a chaotic region. We find that feedback destabilizes controls if the target is chaotic. If the control is unstable the system migrates to the closest non-chaotic target, i.e. it adapts to the edge of chaos. For weak controls the deviation between system dynamics and target is larger, but the system dynamics is less chaotic and therefore more predictable. I