The motion of a periodically forced, damped anharmonic oscillator governed by the equation of motion, x¨+vẋ=${\mathit{A}}_2$x+${\mathit{A}}_4$${\mathit{x}}^3$+${\mathit{A}}_6$${\mathit{x}}^5$+F cos(Ωt), has been studied. The analysis of the response function of this equation when treated analytically, and later, numerically, uncovers a hysteresis-type phenomenon. The stability and response of the system and the onset of period doubling have been observed through an analytical approach, and they are corroborated with a numerical analysis for different values of F and Ω (Ω is the frequency of the periodic forcing system). Two different methods have been used. In the first, the damped system is converted into an undamped one by making an ansatz for ẋ of the form ẋ=R(x), a polynomial in x. The second approach, however, studies the system directly. It has been observed that there exists a wide difference between these two systems. Furthermore, period doubling may be predicted through the use of the harmonic balance technique and Mathieu equation. Lastly, a numerical integration in phase space clearly indicates orbits corresponding to the initial period, then to double the initial period, and subsequently to higher multiples.

• Received 27 August 1990

DOI:https://doi.org/10.1103/PhysRevA.44.1049