Periodic solutions of Duffing's equation with forcing term containing first and third harmonics

Abstract

<p>The Duffing equation with damping has been considered when the forcing function is composed of a harmonic term of frequency w and a higher harmonic term of frequency 3w. The existence of subharmonics of order 1/3 and ultra-harmonics of order 2,5,7,9 was shown. The existence of other subharmonics, ultra-harmonics, and ultra-subharmonics when damping is present was left as an open question since the first term, ko, of the expansion of the damping coefficient, k, in powers of epsilon is zero. A procedure was outlined by which the initial conditions could be determined when the other parameters in the buffing equation are known in order to insure the existence of subharmonics of order 1/3. A similar procedure could be used to obtain the initial conditions that would insure the existence of ultra-harmonics or harmonics. The existence of harmonics for the Duffing equation with damping but with only the first harmonic forcing term was shown;Stability conditions were established for the four types of periodic solutions. The special stability conditions were worked out for harmonics, subharmonics of order 1/3, and ultra-harmonies of order 2,5,7,9.</p>