Oscillatory control parameters in nonlinear chaotic systems

For a better understanding of the role of control parameters in nonlinear dynamical systems, we examined numerically the effect of the oscillatory control parameter D = C₀ cos(ω₁t) on the Lorenz model for a dynamo, where C₀ and ω₁ are the driving amplitude and frequency, respectively. Despite the absence of the mean value of D, finite amplitude solutions are found for sufficiently large C₀ and small ω₁. Overall, smaller C₀ and higher ω₁ are less efficient in generating finite amplitude solutions, the bifurcation occurring at a larger value of C₀ and a smaller value of ω₁. Furthermore, we find a linear relationship between C₀ and ω₁ for the transition between finite amplitude and damping solutions.