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Stability analysis and robust synchronisation of fractional-order modified Colpitts oscillators

Published Online:pp 52-79

Based on the stability theory of the fractional order system, the dynamic behaviours of the uncertain Colpitts oscillator with fractional order-derivative is studied. Furthermore, based on the extended bounded real lemma, the robust controller is obtained using the drive-response synchronisation concept together with the Lyapunov stability theory formulated using the fractional Lyapunov direct method where the fractional-order q belongs to 0 < q < 1. In order to bring out the dynamic behaviour of this system, their phase portraits, the bifurcation diagrams and the Lyapunov exponent are simulated. Moreover, in this work, an approximated solution for both systems to show that the solution of such a system can be represented as a simple power-series function is provided. This study equally provides a systematic procedure to highlight the simplicity and flexibility of the suggested control approach. Simulations with both parameters uncertainty and external disturbance show the applicability and the efficiency of the proposed scheme.


Lyapunov exponent, bifurcation, chaos, fractional modified Colpitts oscillator, H? synchronisation, robust controller