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Following tho analytical results based on the position of the roots of tho characteristic equation, sinusoidal oscillations in inherently relaxation type of oscillators like n multivibrat.or have been established hero by optimizing tho margin for harmonic modo of operation and by introducing an instantaneous gain defining mechanism. Implementation of the latter requirement has been obtained simply by converting a straight coupled pair of stages to 11cathode coupled one. Sinusoidal oscillations with waveform distortion of about -40 dB and frequency stability of few parts in 10" have boon obtuinecl hero quite easily.When largo variations of circuit parameters arc introduced, the results of linear steady-state analysis, howovor, arc found to be iriadoqunto. A non-linear theory of oscillations has therefore, been developed for such oscillations, the results of which lend; more precisely not only to expressions for frequency and amplitude hut also to. conditions for freedom from relaxation and for optimum frequency stability. Corroboration of thcorcbical conclusions with prect.icel resulte show that a now class of oscillatorswith circuit of low seloot.ivif.y ami having tho values of Van del' Pol non-linear parameter e lying within 0< £<2 exists which may havo some of the combined attributes of both multivibrator and sinusoidal oscillator. IntroductionHarmonie oscillutions and relaxation oscillations are two important modes of operation in a self-excited regenerative system. A two-stage BO amplifier fed back to itself is a simple form of one of the oldest and most widely used regenerative systems. It is charaeterized by its switching mode of operations and was studied thoroughly by van del' Pol (1926) under the general class of relaxation oseillations. The possibility of establishing the harmonic mode of operation in such a simple eireuit as this, for the generation of stable sinusoidal oscillations, has always been eonsidered to be impracticable and until now very few attempts appear to have been made to investigate the feasibility (Kundu 1\)51). In the absence of a rigorous approach for ascertaining the limiting factors quantitatively, no fruitful attempt eould have been made for the practical rcalization of the desired objective.'With this end in view an investigation based on the position of the roots on the eomplex frequeney plano of the charaeteristic equation describing the operation of such aBO eoupled system has been made for obtaining eonditions under which the gain margin between the harmonic and relaxation mode may be maximized. The maximum value is, however, found to be limited because of the inherently low seleetivity of thc eoupling circuits. This gives rise to a class of oscillators which are characterized by low gain margin. Stable sinusoidal

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Self-excited periodic, quasiperiodic and chaotic oscillations have many significant applications in engineering devices and processes. In the present paper a centralized nonlinear controller is proposed to artificially generate and control self-excited periodic, quasiperiodic, chaotic and hyper-chaotic oscillations of required characteristics in a fully-actuated n-DOF spring-mass-damper mechanical system. The analytical relations among the amplitude, frequency and controller parameters for minimum control energy have been obtained using the method of two-time scale. It is shown that the proposed control can generate modal and nonmodal self-excited periodic and quasiperiodic oscillations of desired amplitude and frequency for minimum control energy. The analytical results have been verified numerically with MATLAB SIMULINK. Bifurcation analysis and extensive numerical simulations reveal a region of multistability in the plane of control parameters, where system responses may be periodic, quasiperiodic, chaotic and hyper-chaotic depending on initial conditions. However, it has been shown that the probability of obtaining chaotic and hyper-chaotic oscillations are very high for a wide range of controller parameters. The procedures of controlling the amplitude, frequency and characteristics of chaotic oscillations are also discussed. The results of the present paper is expected to find applications in various macro and micro mechanical systems and applications.