Complex Dynamical Behavior of a Two-Stage Colpitts Oscillator with Magnetically Coupled Inductors

A five-dimensional (5D) controlled two-stage Colpitts oscillator is introduced and analyzed. This new electronic oscillator is constructed by considering the well-known two-stage Colpitts oscillator with two further elements (coupled inductors and variable resistor). In contrast to current approaches based on piecewise linear (PWL) model, we propose a smooth mathematical model (with exponential nonlinearity) to investigate the dynamics of the oscillator. Several issues, such as the basic dynamical behaviour, bifurcation diagrams, Lyapunov exponents, and frequency spectra of the oscillator, are investigated theoretically and numerically by varying a single control resistor. It is found that the oscillator moves from the state of fixed point motion to chaos via the usual paths of period-doubling and interior crisis routes as the single control resistor is monitored. Furthermore, an experimental study of controlled Colpitts oscillator is carried out. An appropriate electronic circuit is proposed for the investigations of the complex dynamics behaviour of the system. A very good qualitative agreement is obtained between the theoretical/numerical and experimental results. 1. Introduction During the last three decades, a tremendous attention has been devoted to design chaotic electronic oscillators. The focus on this interesting research field comes mainly from two facts: first, one can observe chaos and can also control the dynamics of the oscillator by simply changing the physically accessible parameters of the oscillator, for example, linear resistor, linear capacitor, voltage levels, coupled inductors, and so forth; second, there are a multitude of applications of chaotic electronic oscillators starting from chaotic electronic secure communication to cryptography [1]. In this regard, the classical Colpitts oscillator with single transistor was investigated at 1？KHz frequency [2], high (3–300？MHz) frequencies [3], and ultrahigh (300–1000？MHz) frequencies [4] using both numerical and experimental methods. The interest devoted to this oscillator is motivated by its simple physical realization and low power requirement. Nevertheless, the main limitation of the classical Colpitts oscillator is its incapacity to exhibit higher fundamental frequencies in chaotic regime [5]. In order to solve this problem, alternatives to this standard version of the Colpitts oscillator, namely, the two-stage and improved version, were reported in [6, 7]. In comparison to a single stage Colpitts oscillator, the two-stage Colpitts oscillator presents better spectral properties