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A memristor-based fractional order circuit derived from Chua’s topology is presented. The dynamic properties of this circuit such as phase trajectories, time evolution characteristics of state variables are analyzed through the approximation method of fractional order operator. In addition, it clearly describes the relationships between the impedance variation of the memristor and the varying mobility of the doped region of the memristor in different circuit parameters. Finally, a periodic memristor-based system driven by another chaotic memristor-based fractional order system is synchronized to chaotic state via the linear error feedback technique.

Since professor Chua predicted the fourth basic element “memristor” [_{2} thin film was implemented in Hewlett-Packard (HP) labs, followed by several other materials and methods [3-5]. And this kind of device may be expected to reform the future computers by using it in place of random access memory (RAM). After this landmark work [

The rest of the paper is organized as follows: in Section 2, a memristor-based fractional order system model is depicted. In Section 3, some illustrative examples and numerical simulation results are presented. Finally, the conclusion is drawn in Section 4.

Memristor is a nonlinear element. It shows the v-i relationship following Ohm’s law, and the equivalent resistance in HP memristor [

where denotes the internal state variable (the width of doped area in the memristor), and denotes the whole thickness of the memristor, is the equivalent resistance of the memristor with respect to the internal variable.

In

The dynamic in the circuit of

the following state equation as Equation (2):

where R_{1}(H_{1}), R_{2}(H_{2}) denote the equivalent resistances of memristors M_{1}, M_{2}, respectively. And, are the window functions. There are different definitions about the window functions, such as the Strukov and collegues’ method [

where is a step function.

This paper aims at studying the memristor-based fractional order system. By one of the famous definitions of fractional order differential equations—Riemann Liouville (RL) definition [

Then Equation (2) based on

Let

the memristor-based fractional order system is transformed into a dimensionless form (5):

where the internal state variables h_{1}, h_{2} of the memristor are still postulated as integer order differential equations.

In this section, by using the approximation of fractional operator [_{1}-x_{2}-x_{3} space of the fractional order system, _{3} vs. x_{2}, _{1}, _{1 }vs. h_{1}. We take the common values for some parameters in all the following simulations: the initial condition and the initial position of the boundary of memristors, the fractional order a = 0.9, R_{ON} = 100, m_{v} = 10^{−10}, g = −0.0019.

In _{1} = 0.02, C_{2} = 0.01, L = 1800, r = 1425, R_{OFF} = 15,000.

_{1}, C_{2}, L, R_{OFF}. And in Figures 3(a)-(d), all the phase trajectories show the periodic dynamic properties of the system.