Memristor, the missing fourth passive circuit element predicted forty years ago by Chua was recognized as a nanoscale device in 2008 by researchers of a H. P. Laboratory. Recently the notion of memristive systems was extended to capacitive and inductive elements, namely, memcapacitor and meminductor whose properties depend on the state and history of the system. In this paper, we use fractional calculus to generalize and provide a mathematical paradigm for describing the behavior of such elements with memory. In this framework, we extend Ohm's law to the generalized Ohm's law and prove it.

Fractional-order financial system introduced by W.-C. Chen (2008) displays chaotic motions at order less than 3. In this paper we have extended the nonlinear feedback control in ODE systems to fractional-order systems, in order to eliminate the chaotic behavior. The results are proved analytically by applying the Lyapunov linearization method and stability condition for fractional system. Moreover numerical simulations are shown to verify the effectiveness of the proposed control scheme.

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In this paper, the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay is investigated. In particular, we introduce a new type of outer synchronization behavior, i.e., mixed outer synchronization (MOS), in which different state variables of the corresponding nodes can evolve into complete synchronization, antisynchronization, and even amplitude death simultaneously for an appropriate choice of the scaling matrix. A novel nonfragile linear state feedback controller is designed to realize the MOS between two networks and proved analytically by using Lyapunov-Krasovskii stability theory. Finally, numerical simulations are provided to demonstrate the feasibility and efficacy of our proposed control approach.

Contrary to integer order derivative, the fractional-order derivative of a nonconstant periodic function is not a periodic function with the same period, as a consequence of this property the time-invariant fractional order system does not have any non-constant periodic solution unless the lower terminal of the derivative is ±∞, which is not practical. This property limits the applicability areas of fractional derivatives and makes it unfavorable, for a * corresponding author Email addresses: medsala3@yahoo.fr (Mohammed-Salah Abdelouahab), n.hamri@centre-univ-mila.dz (Nasr-Eddine Hamri)