An external magnetic field surrounding a magnetic wire restricts the forced vibrations of a wire to a single plane. Alternatively, we could fasten the ends of the wire in such a way as to constrain the motion to planar oscillations. In either case, the nonlinear equation of motion governing the single-mode planar vibrations of a string is the Duffing equation ,
where equation (3.14) is calculated from equation (3.7) by assuming that the string's motion is confined to the x-z plane in Figure 3.3. The forcing term in equation (3.7) is assumed to be a periodic excitation of the form
where the constant A is the forcing amplitude and 
is the
forcing frequency. The literature studying the Duffing
equation is extensive, and it is well known that the solutions to equation
(3.14) are already complicated enough to exhibit multiple periodic
solutions, quasiperiodic orbits, and
chaos. A good guide to the nonchaotic properties of the Duffing equation is the
book by Nayfeh and Mook [10]. Highly recommended as
a pioneering work in nonlinear dynamics
is the book by Hayashi, which deals almost exclusively with the
Duffing equation [11].
