This thesis is concerned with the analysis of the multimodal behaviour and transition characteristics of the jump phenomenon in nonlinear coupled circuits. The work is based on earlier work done in studying second order nonlinear systems whereby a graphical method based on the describing function and integral curve representation is employed to estimate transients. The analysis of a simple series nonlinear L.C.R. cirucit is carried further to include a shorted secondary coil. A further step in the analysis is reached when a voltage in series with the shorted secondary is considered and. the analysis carried out to determine the effect of this secondary voltage on the behaviour of the system. This is done for different phases and amplitudes of the secondary voltage when applied to the system at different instants of time. These two cases are fully analysed both in the steady state and transiently. The overall transformer coupled circuit is then considered and due to the complexity of the circuit the analysis is done by assuming that the transformer coupled circuit be replaced by an equivalent T-network of inductances. This representation results in three second order nonlinear differential equations for the system which can be solved for the steady state. With some approximations a transient transpose analysis even is possible and the simulation of the system gives more insight into its transitional behaviour. In all these systems considerable use is made of the digital computer to plot the integral curves and the transients derived from them. The analogue computer is also extensively used to verify the theoretical results. Methods whereby integral curves can be related to the time varying quantities in the systems are given and it is shown how initial conditions leading to a given mode of response can be estimated. The whole work in this thesis is directed mostly to the question of the transition of the current (or flux) by means of a mutually coupled circuit so that it can be applied to the development of an amplitude oontrol scheme for the application of osoillating motors to bobbin winders.