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A chaotic oscillator based on the memristor is analyzed from a chaos theory viewpoint. Sensitivity to initial conditions is studied by considering a nonlinear model of the system, and also a new chaos analysis methodology based on the energy distribution is presented using the Discrete Wavelet Transform (DWT). Then, using Advance Design System (ADS) software, implementation of chaotic oscillator based on the memristor is considered. Simulation results are provided to show the main points of the paper.

The study of nonlinear systems is an important research topic for scientists and researchers. One of the complex types of behavior that can be observed in these systems is the chaotic behavior. Chaos is widely available in engineering and the natural systems. Chaos phenomenon is completely deterministic and specific to nonlinear systems. In fact, chaos theory is a branch of mathematics and physics related to systems that their dynamic represents a very sensitive behavior to changes in the initial values so that their future behavior is not predictable anymore; these systems are called chaotic systems which are of nonlinear ones. Nowadays, there are examples of potential benefits of the chaotic behavior which make a lot of engineers and researchers attend it. Bianca and Rondoni proposed the analytical and numerical investigations of Ehrenfest gas, which is a billiard model with an electric field and a Gaussian thermostat [

One of these nonlinear systems occurs in that chaos is sinusoidal oscillator circuits [

In the theoretical circuit studies, there are certain fundamental elements in the circuit, including resistance, capacitance, and inductance. Resistance is the relationship between the current and the voltage, capacitance is the relationship between the charge and the voltage, and also inductance is the relationship between the magnetic charge and the current. These elements have two inactive terminals used in different circuits and it is known that no information is saved in them. This point may seem unimportant, but in fact it is considered as a fundamental principle in circuits which is indicative of a charge pattern in electronics [

Memristor (memory resistor) is a microscopic tool that can keep its previous electrical conditions and can therefore preserve the temporary memory even after a power cut. According to the definition, memristor is determined by

It is readily observed that the value of the memductance at

The physical structure of the memristor consists of a thin-layer film of two-layer titanium oxide TiO_{2} (size

One of the layers is doped with oxygen vacancies with a high dopant and converted to a semiconductor with low resistance. The second well-known undoped region (doped with a low dopant concentration) has high resistance. The physical model of memristor is shown in Figure

Physical model of memristor.

Overall resistance of the elements is equal to two series resistance that both are dependent on the width

The voltage versus current relation of a charge-controlled memristor is given as follows:

The speed of the movement of the boundary between the doped and undoped regions depends on the resistance of doped area, on the passing current, and on other factors according to the following state equation [^{2}s^{−1}V^{−1} is the so-called dopant mobility and

The schematic of the Advance Design System (ADS) software implementation is presented in Figure

Schematic of the memristor ADS implementation.

A typical characteristic of the memristor is the hysteresis behavior followed by current-voltage domains as plotted in Figure

Memristor current versus voltage (hysteresis loop).

Figure

The schematic of Chua's circuit.

Simulation results indicate that the circuit has a chaotic behavior. Limit cycles for different initial conditions are shown in Figure

Transient chaotic attractor in different initial conditions.

3D Chaotic attractor in different initial conditions.

The chaotic oscillator based on the memristor proposed in Figure

An effective tool in the study of chaotic behavior is the frequency domain periodic analysis. In periodic signals, the energy is focused on some special frequencies, while in the chaotic behavior the energy in different frequency values is nonzero. Therefore chaotic signals are wideband signals. In deterministic systems, a spectrum having a wideband represents the sign of starting a chaotic behavior [

The self-power density spectrum chart of the chaotic oscillator.

In the wavelet analysis, similar to the short-time Fourier transform, a desired signal is multiplied by a wavelet function that plays the role of a window function. In addition, the wavelet transform is also performed separately on different time segments of the signal.

Wavelet is a given function with zero mean and the expansion is conducted in terms of transitions and dilations of the function. Unlike trigonometric polynomials, wavelets in the space are considered locally and thus there is a closer relationship between some functions and their coefficients in the wavelet analysis.

The continuous wavelet transform is defined as follows:

The transfer concept is exactly similar to a time transfer concept in short-time Fourier transform that clarifies window displacement value and includes transform time information, but, unlike the short-time wavelet transform, there are no frequency parameter in the wavelet transform directly. Instead, there is a scale parameter that is connected conversely with the frequency. And in wavelet instead of frequency transform, the scale parameter is available. In Discrete Wavelet Transform method the signal can be decomposed into different frequency bands. In this method two sets of coefficients are computed: approximation coefficients and detail coefficients. The approximation coefficients are obtained by convolving the signal with the low-pass filter and detail coefficients are obtained by convolving the signal with the high-pass filter for details.

Scaling as a mathematical operator shrinks or expands the signal. Thus, in high scales in which the signal is expanded, we will have details; and in low scales in which the signal is shrunk, we will have generality [

In order to analyze more precisely, details and generalities of the signal can be extracted considering the Discrete Wavelet transform. By extracting the coefficient of the signal details using the wavelet transform, energy can be calculated for any detail coefficient as follows:

To illustrate irregular energy distribution of the signal details in each step, entropy is applied. Entropy is the irregularity degree in a system with energy or data. The less is a system regular, the more is the entropy. It is obvious that chaos is a wideband signal. In other words, it can be said that the energy distribution of signal details includes irregular changes in chaos signal [

Energy distributions in different initial condition of the chaotic oscillator.

Chua’s circuit can be realized in variety of ways using standard or custom-made electronic components. Since all of the linear elements (capacitor, resistor, and inductor) are ready available as two-terminal devices, our principal concern here will be with circuitry to realize the negative resistor.

A two-terminal negative resistance convertor can now be produced by connecting three positive resistances around an operation amplifier. An operational amplifier (op-amp) provides us with a real approximation to a voltage-controlled voltage source. Figure

(a) Negative resistance convertor; (b)

Figure

(a) ADS implementation of chaotic oscillator based on memristor; (b) phase plane

This paper studies a new structure of the chaotic oscillator based on thememristor. For nonlinear analysis of the memristor chaotic circuit, the Lyapunov exponent method and a new method based on the energy distribution are presented using the Discrete Wavelet Transform (DWT). In this paper, the physical model of memristive element such as the memristor and chaotic circuit based on memristor are implemented using the ADS software.

Since circuit parameters can be continuously tuned in wide ranges, the number of orthogonal chaotic signals can be infinite. This analysis provides a rigorous proof of chaos for memristor circuits.

To summarize, this work attempts to approach nonlinear circuits implemented by the use of memristors. This analysis provides a rigorous proof of chaos for memristor circuits.

The authors declare that there is no conflict of interests regarding the publication of this paper.