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.Graphical abstract
Highlights
In [27] it is shown that electronic circuits which operate at high frequency and having a high degree of freedom are of great advantage in cryptography and can contribute to the generation of random or pseudo-random bit streams with much reasonable speeds, and at low cost
Two complementary types of diagrams were presented in order to characterize the stability of dynamic systems, namely: the stability diagram based on twoparameters Lyapunov exponents and Isospike diagram based on counting the number of spikes contained in one period of the periodic oscillations [29–35]
We have studied the fractional-order version of a two-component circuit operating at high frequency using the analytic and numerical methods
Summary
Pinning control and synchronization of complex networks with coupling terms were studied due to its theoretical importance and practical application [48–52] In these works, firstly the authors used all the directions of the node to insure the synchronization of complex networks which is a very drawback of the practical view point [53], secondly these authors used the usual form of the adaptive feedback control law which is not robust for the synchronization in presence of perturbation and the uncertainties. Firstly the authors used all the directions of the node to insure the synchronization of complex networks which is a very drawback of the practical view point [53], secondly these authors used the usual form of the adaptive feedback control law which is not robust for the synchronization in presence of perturbation and the uncertainties These works did not present any stability discussion for the sliding mode dynamics.
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