Amplitude Bounds on Periodic and Aperiodic Oscillations: Second-order Systems

The global behaviour of the control systems described by thepair of differential equations x{dot} = –f(x)±g(y)+p(t), y{dot} = –f(x)±g(y)+p(t) has been investigated. Here f(x), g(y), h(x) and k(y) are polynomialsof odd degree with leading coefficients positive and p(t) andq(t) are bounded functions of time. Sufficient conditions havebeen found under which the trajectories of the above systemmay eventually be confined in a subset of (x, y, t)-space, thusgiving bounds on the amplitude of periodic as well as aperiodicoscillations. Further bounds on the amplitude of oscillationshave been investigated by finding regions in (x,y,t)-space fromwhich all trajectories eventually leave and into which no trajectoriesenter. Thus sufficient conditions have been derived for theexistence of an annulus in which oscillatory behaviour may beconfined.