  Given a dynamical system with $m$ independent conserved quantities, we
construct a multi-parameter family of new systems in which these quantities
evolve monotonically and proportionally, and are replaced by $m-1$ conserved
linear combinations of themselves, with any of the original quantities as
limiting cases. The modification of the dynamics employs an exterior product of
gradients of the original quantities, and often evolves the system towards
asymptotic linear dependence of these gradients in a nontrivial state. The
process may be iterated or adapted to obtain any reduction in the degree of
integrability, and is considerably more general than existing techniques for
selective dissipation in the literature on fluids and plasmas, nonequilibrium
thermodynamics, and nonlinear controls. It may enable selective control or
energy transfer, discovery of extremal states, limit cycles, or solitons, and
the construction of new integrable systems from superintegrable systems. We
briefly illustrate the approach by its application to the cyclic three-body
Toda lattice, driven from an aperiodic orbit towards a limit cycle.
