Quenching Lorenzian Chaos

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, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE’s) ẋ = ay − ax, ẏ = −xzq + bx − y and ż = xyq − cz (where (x, y, z) are time(t)-dependent dimensionless variables and ẋ = 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 F-region, which disappears altogether when e = 1 leaving persistent solutions that are steady throughout the entire parameter space.