On modulated complex non-linear dynamical systems

This paper is concerned with the development of an approximate analytical m ethod to investigate periodic solutions and their stability in the case of modulated non-linear dynamical systems whose equation of motion is describe d by: z + w(2)(t)z + epsilon f(z, (z) over bar, (z) over dot, (z) over bar) g(nu t) = 0, epsilon much less than 1, where z(t) is complex, w(t) and g(nu t) are periodic functions of t and f is a non-linear function. Such differ ential equations appear, for example, in problems of colliding particle bea ms in high-energy accelerators or one-mass systems with two or more degrees of freedom, e.g., rotors. The significance of periodic solutions lies on t he fact that all non-periodic responses, if convergent, would approach to p eriodic solutions at the steady-state conditions. Our example shows a good agreement between numerical and analytical results for small values of epsi lon. The effect of the periodic modulation on the stability of the 2 pi-per iodic solutions is discussed.