Effect of the shape of periodic forces and second periodic forces on   horseshoe chaos in Duffing oscillator
  The effect of the shape of six different periodic forces and second periodic forces on the onset of horseshoe chaos are studied both analytically and numerically in a Duffing oscillator. The external periodic forces considered are sine wave, square wave, symmetric saw-tooth wave, asymmetric saw-tooth wave, rectified sine wave, and modulus of sine wave. An analytical threshold condition for the onset of horseshoe chaos is obtained in the Duffing oscillator driven by various periodic forces using the Melnikov method. Melnikov threshold curve is drawn in a parameter space. For all the forces except modulus of sine wave, the onset of cross-well asymptotic chaos is observed just above the Melnikov threshold curve for onset of horseshoe chaos. For the modulus of sine wave long time transient motion followed by a periodic attractor is realized. The possibility of controlling of horseshoe and asymptotic chaos in the Duffing oscillator by an addition of second periodic force is then analyzed. Parametric regimes where suppression of horseshoe chaos occurs are predicted. Analytical prediction is demonstrated through direct numerical simulations. Starting from asymptotic chaos we show the recovery of periodic motion for a range of values of amplitude and phase of the second periodic force. Interestingly, suppression of chaos is found in the parametric regimes where the Melnikov function does not change sign. 
V. Ravichandran, V. Chinnathambi, S. Rajasekar, Choy Heng Lai