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The effect of the shape of six different periodic forces and second periodicforces on the onset of horseshoe chaos are studied both analytically andnumerically in a Duffing oscillator. The external periodic forces consideredare sine wave, square wave, symmetric saw-tooth wave, asymmetric saw-toothwave, rectified sine wave, and modulus of sine wave. An analytical thresholdcondition for the onset of horseshoe chaos is obtained in the Duffingoscillator driven by various periodic forces using the Melnikov method.Melnikov threshold curve is drawn in a parameter space. For all the forcesexcept modulus of sine wave, the onset of cross-well asymptotic chaos isobserved just above the Melnikov threshold curve for onset of horseshoe chaos.For the modulus of sine wave long time transient motion followed by a periodicattractor is realized. The possibility of controlling of horseshoe andasymptotic chaos in the Duffing oscillator by an addition of second periodicforce is then analyzed. Parametric regimes where suppression of horseshoe chaosoccurs are predicted. Analytical prediction is demonstrated through directnumerical simulations. Starting from asymptotic chaos we show the recovery ofperiodic motion for a range of values of amplitude and phase of the secondperiodic force. Interestingly, suppression of chaos is found in the parametricregimes where the Melnikov function does not change sign.

Effect of the shape of periodic forces and second periodic forces on horseshoe chaos in Duffing oscillator