How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here we extend previous work on ``nonlinear quenching'' (Hide, R., Nonlinear Processes in Geophysics 4, 201-205, 1997) by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) x'=ay-ax, and z'=xyq-cz (where (x,y,z) are time(t)-dependent dimensionless variables and x'=dx/dt, etc.) in representative cases when q, the ``quenching function'', satisfies q=1-e+ey with 0<= e <= 1. Control parameter space based on a,b and c can be divided into two ``regions'', an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The ``Hopf boundary'' between the two regions is located where b=bH (a,c;e) (say), with the much studied point (a,b,c) = (10,28,8/3), where the persistent ``Lorenzian'' chaos that arises in the case when e=0 was first found lying close to b=bH (a,c;0). As e increases from zero the S-region expands in total ``volume'' at the expense of the F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the whole of parameter space.
Chaos, bifurcations, nonlinear dynamics, quenching.