Regular Article

Fractional-order two-component oscillator: stability and network synchronization using a reduced number of control signals

¹ Research Group on Experimental and Applied Physics for Sustainable Development, Faculty of Science, Department of Physics, University of Dschang, P.O. Box 412, Dschang, Cameroon
² Laboratory of Electronics and Signal Processing Faculty of Science, Department of Physics, University of Dschang, P.O. Box 67, Dschang, Cameroon
³ Institute of Surface Chemistry and Catalysis, University of Ulm, Albert-Einstein-Allee 47, 89081 Ulm, Germany
⁴ Lublin University of Technology, Faculty of Mechanical Engineering, Nadbystrzycka 36, 20-618 Lublin, Poland
⁵ AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics, Department of Process Control, Mickiewicza 30, 30-059 Krakow, Poland
⁶ Laboratoire de Mécanique et de Modélisation des Systèmes, L2MS, Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, P.O. Box 67, Dschang, Cameroon
⁷ College of Physics and Electronics, Hunan Institute of Science and Technology Yueyang, Hunan 414006, P.R. China

^a e-mail: g.litak@pollub.pl

Received: 30 May 2018
Received in final form: 20 August 2018
Published online: 5 December 2018

Abstract

In this paper, a fractional-order version of a chaotic circuit made simply of two non-idealized components operating at high frequency is presented. The fractional-order version of the Hopf bifurcation is found when the bias voltage source and the fractional-order of the system increase. Using Adams–Bashforth–Moulton predictor–corrector scheme, dynamic behaviors are displayed in two complementary types of stability diagrams, namely the two-parameter Lyapunov exponents and the isospike diagrams. The latest being a more fruitful type of stability diagrams based on counting the number of spikes contained in one period of the periodic oscillations. These two complementary types of stability diagrams are reported for the first time in the fractional-order dynamical systems. Furthermore, a new fractional-order adaptive sliding mode controller using a reduced number of control signals was built for the stabilization of a fractional-order complex dynamical network. Two examples are shown on a fractional-order complex dynamical network where the nodes are made of fractional-order two-component circuits. Firstly, we consider an ideal channel, and secondly, a non ideal one. In each case, increasing of the coupling strength leads to the phase transition in the fractional-order complex network.