Radiophysics and Quantum Electronics, Vol. 44, No. 8, 2001 CHAOTIC OSCILLATIONS IN A NONLINEAR LUMPED-PARAMETER TRANSMISSION LINE A. A. Balyakin and N. M. Ryskin UDC 537.86/87:530.182 We present the results of numerical simulation of the complex dynamics of a nonlinear radio- technical line having reflections at the boundaries and excited by an external harmonic signal. It is shown that, with increase in the amplitude of the input signal, periodic oscillations at the external-forcing frequency become unstable and are changed to more complex regimes, either quasiperiodic or chaotic. The main scenarios of transition to chaos are studied. The influence of the modulation instability and soliton formation on the complex dynamics is discussed. Considerable attention has recently been given to a study of deterministic chaos in distributed sys- tems, i.e., systems with an infinite number of degrees of freedom [1, 2]. The importance of such studies is stipulated by their obvious relation to the fundamental problem of the origin of turbulence. In particular, the problem of chaotization of stimulated oscillations of a distributed resonator formed by a section of a nonlinear medium with the reflecting boundaries is of interest. Since a nonlinear oscillator affected by an external forcing is a standard model for the nonlinear dynamics of systems with a finite number of degrees of freedom [1–3], one can expect that this problem will play the same role for distributed systems. For studying nonlinear wave phenomena, lumped-parameter transmission lines containing nonlinear elements are widely used in radiophysics as the model media. Such lines permit modeling of processes in media with different types of nonlinearity, dispersion, and dissipation [2, 4, 5]. Moreover, they can be easily realized in experiments. In particular, the excitation of deterministic (i.e., stipulated by the complex dynamics of the system itself rather than the amplification of fluctuations in it) chaotic oscillations in such systems was experimentally observed in [6–8]. In the present paper, we consider the nonlinear dy- Fig. 1. Equivalent scheme of the nonlinear line. namics of the simplest lumped-parameter chain composed of inductances L 0 and nonlinear capacitances with square- law dependence of the charge Q on the applied voltage V : Q(V )= C 0 V - C 2 V 2 . (1) On the one end, the chain is excited by a harmonic signal of constant amplitude A, and on the other end, the chain is loaded by ohmic resistance R (Fig. 1). The fundamental possibility of transition to chaos in this system with increase in the signal amplitude was shown earlier in [9]. We also note that in [6] Ezersky et al. performed theoretical and experimental studies of chaotic oscillations for the case of parametric excitation where the external signal was applied to each cell of the chain. The dynamics of the chain shown in Fig. 1 is described by a system of nonlinear differential-difference equations: L 0 ˙ I n = V n-1 - V n , ˙ Q n = I n - I n+1 , (2) N. G. Chernyshevsky State University of Saratov, Saratov, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 44, No. 8, pp. 691–699, August, 2001. Original article submitted De- cember 25, 2000. 0033-8443/01/4408-0637$25.00 c  2001 Plenum Publishing Corporation 637