Hopf bifurcation in fractional two-stage Colpitts oscillator: analytical and numerical investigations

This paper deals with the problem of determining the critical point of appearance of fractional Hopf bifurcation in two-stage Colpitts system in the absence of delay. To determine the condition of appearance of the Hopf bifurcation in the system, we use the methods of Cardan, Ferrari and Newton-Raphson when the order of the derivative is taken as a control parameter and another parameter of the system is fixed as a bifurcation parameter. Using the analytical method, we show that when the order of the derivative is less than 0.5, the system does not have critical points of Hopf bifurcation whatever the values taken by the control parameter. However, when the value of the order of the derivative is greater than 0.5, the system has the Hopf bifurcation. Moreover, it should be pointed out that when the fractional order of the derivative decreases, the critical value of the control parameter increases and thus the stability domain grows, while the instability domain shrinks. Numerical resolution using the Adam-Moulton-Baschford and Benettin-Wolf methods confirmed the analytical results. Finally, by plotting Lyapunov spectrum, we showed that the oscillations that appear in the system are periodic or chaotic but not hyperchaotic. The results obtained in this paper can be extended to all systems of order 4.