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By coupling a diode bridge-based second-order memristor and an active voltage-controlled memristor with a capacitor, a three-element-based memristive circuit is synthesized and its system model is then built. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle. Besides, the stability distributions of equilibrium points are theoretically and numerically expounded in a 2D parameter plane. The results imply the memristive circuit has a zero unstable saddle focus and a pair of nonzero stable node-foci or unstable saddle-foci depending on the considered parameters. The dynamical behaviors include point attractor, period, chaos, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios, which are explored using some common dynamical methods. Of particular concern, riddled attraction basins and multistability are uncovered under two sets of specified model parameters nearing the tiny neighborhood of crisis scenarios by local attraction basins and phase plane plots. The riddled attraction basins with island-like structure demonstrate that their dynamical behaviors are extremely sensitive to the initial conditions, resulting in the coexistence of limit cycles with period-2 and period-6, as well as the coexistence of period-1 limit cycles and single-scroll chaotic attractors. Moreover, a feasible on-breadboard hardware circuit is manually made and the experimental measurements are executed, upon which phase plane trajectories for some discrete model parameters are captured to further confirm the numerically simulated ones.

Chaos has attracted appreciable attention due to its potential applications in weather forecasting, aircraft control, and secure communications [

The most vital experience to construct a memristor-based chaotic circuit is to lead one memristor or more with different nonlinearities into an existing linear or nonlinear electronic circuit [

Interestingly, the novel dynamical behaviors including riddled attraction basin [

Considering that a simple memristive chaotic circuit can serve as an elegant paradigm for better understanding of bifurcation and chaotic dynamics, it is a significant research topic to simplify memristive chaotic circuits by minimizing the number of dynamic elements and physical components [

The aim of this work is to reveal the unknown features in the proposed memristive circuit. Toward this purpose, the remainder of this paper is organized as follows. The mathematical model in a dimensionless form is built, and then the equilibrium points and their stabilities are explored, as well as the boundedness is proved in

The proposed three-element-based memristive circuit contains only a capacitor _{1}, a passive diode bridge-based second-order memristor _{1} [_{2} [_{1} and _{2} are illustrated at the left- and right-hand sides, respectively, with dotted rectangles in Figure _{3} is implemented by an NIC [

Circuit schematic of the proposed three-element-based memristive circuit.

For a passive diode bridge-based second-order memristor _{1} [_{1} are the voltage and current at input terminal, respectively; _{1} can be described as_{T}), _{S}, _{T} are three diode model parameters standing for the reverse saturation current, emission coefficient, and thermal voltage, respectively.

The active voltage-controlled memristor _{2} [_{1} to realize a voltage follower for restraining load effect, _{2} to implement an integrator with _{2} to avoid DC voltage integral drift, two multipliers _{3} and _{4}, and a current inverter. Denoting the integral capacitor voltage as _{4} can be deduced as_{3} and _{4}. Thus, the current _{2} at the input terminal of the memristor _{2} can be obtained and simplified as

By employing Kirchhoff's current law to the inverting input terminal of _{2}, we can obtain

Consequently, the mathematical model for the voltage-controlled memristor can be described as_{2} and

Applying Kirchhoff's current law to node A and allying (

The parameters of the linear circuit elements are determined as _{1} = 2 kΩ, _{2} = 4 kΩ, _{3} = 1.5 kΩ, _{0} = _{1} = ^{–2}, and the model parameters of four 1N4148 diodes utilized in the diode bridge are assigned as _{S} = 5.84 nA, _{T} = 25 mV. These circuit parameters are collectively named as the typical circuit parameters in the following sections.

Denote

Equation (

Thus, the mathematical model of the proposed memristive circuit in dimensionless form can be built.

Herein, we discuss the boundedness of the solutions of system (

(see _{1}, _{2} ∈

We use the supremum metric denoted by ^{T} and taking

Fixed point method [

Let _{0} is the Lipschitz constant), and let

For a given

We use the supremum metric and define the complete metric space

For

Then,

Therefore, by the contraction mapping principle, there is a unique fixed point

The normalized typical model parameters in (

In the following investigations, the model parameters

Making the left-hand side of (

According to (

The characteristic equation of (

The model parameters _{0}) and two nonzero equilibrium points (marked as _{±}) being symmetric about the origin. The distributions for _{±} are changed with the variation of the model parameter _{0} and _{±} by applying different model parameters

For the zero equilibrium point _{0}, (_{0} is an unstable saddle focus. For the nonzero equilibrium points _{±}, the stabilities for them are the same. Thus, only the corresponding stability distributions for the equilibrium point _{+} are shown in Figure

The values of _{±} in the

As aforementioned, the model parameters

To fully demonstrate the dynamical behaviors, two-dimensional (2D) bifurcation plots depicted by the periodicities of the variable ^{−9}, 10^{−9}) are utilized.

2D dynamical behaviors in the

In Figure

It is clearly uncovered that the dynamical behaviors exhibited by the 2D bifurcation plots in Figure

To further brighten the dynamical behaviors in (^{−9}, 10^{−9}) and the trajectories colored in blue are those starting from the initial conditions (−0.1, 0.1, 10^{−9}, 10^{−9}) in the bifurcation plots, whereas the initial conditions (0.1, 0.1, 10^{−9}, 10^{−9}) are employed in the calculation process of finite-time Lyapunov exponents.

Numerically simulated 1D bifurcation plots of the maxima of the variable

In Figure

When the parameter

As shown in Figure

With respect to some discrete

Phase plane plots in the

When _{1} = 0.0011, LE_{2} = –0.0046, LE_{3} = –0.0515, and LE_{4} = –0.1044; when _{1} = 0.0011, LE_{2} = –0.0283, LE_{3} = –0.0394, and LE_{4} = –8.8207; when _{1} = 0.1339, LE_{2} = 0.0001, LE_{3} = –0.4610, and LE_{4} = –8.6482; when _{1} = 0.1660, LE_{2} = 0.0006, LE_{3} = –0.4701, and LE_{4} = –11.1751.

It is interesting that the occurrence of crisis scenarios leads to the appearance of imperfect bifurcation routes in the 1D bifurcation plots, as shown in Figure

Numerically simulated 1D bifurcation plots of the maxima of the variable

With the generality and comparison, four sets of typical model parameters _{0}, _{0}, 10^{−9}, 10^{−9}) for the four sets of model parameters are plotted in Figures _{0} and _{0} are all scanned in the region of [–15, 15]. Remark that the local attraction basins in Figure

Local attraction basins for four different sets of the model parameters _{0}–_{0} plane. (a)

For comparison, two sets of the model parameters

By numerical simulations, the coexistence of limit cycles with period-2 and period-6 are demonstrated corresponding to the dynamical behaviors revealed in the riddled attraction basins, as shown in Figure ^{−9}, 10^{−9}) for period-2 and (±3, ^{−9}, 10^{−9}) for period-6 located in different adjacent islands are employed. Note that the initial conditions are symmetrical about the origin and arbitrarily tiny perturbation in the initial conditions, which lead to the fact that the periodic behaviors change from period-2 to period-6, or vice versa. For ^{−9}, 10^{−9}) for limit cycles with period-1 and (±5, ^{−9}, 10^{−9}) for the single-scroll chaotic attractors are utilized. The numerical simulations show that the riddled attraction basin and multistability emerge in such a simple memristive circuit under specified model parameters near the crisis scenarios happening.

Numerically simulated phase plane plots in the ^{−9}, 10^{−9}) and (±3, ^{−9}, 10^{−9}), respectively for ^{−9}, 10^{−9}) and (±5, ^{−9}, 10^{−9}), respectively, for

A hardware experimental circuit is manually welded on a breadboard using commercially available discrete components, upon which the circuit running trajectories can be captured to verify the MATLAB numerical simulations. The hardware experimental circuit employs seven potentiometers (two of the seven potentiometers are utilized to adjust the gain of multiplier _{4}), a manually winding inductor, three monolithic capacitors, four N4148 diodes as well as three operational amplifiers TL082CP, and two multipliers AD633 with bipolar ±15 V supply, as shown in Figure

The hardware experimental circuit of the three-element-based circuit. (a) An overviewed graph. (b) An enlargement of on-breadboard circuit.

By keeping the typical circuit parameters and tuning the potentiometer _{2} and the inductor _{2} and _{2} = 20 kΩ, 5.71 kΩ, and 4 kΩ with _{2} = 4 kΩ with _{2} = 5.57 kΩ with _{2} = 4 kΩ with

Experimentally captured trajectories in the _{2} and _{2} = 20 kΩ and _{2} = 5.71 kΩ and _{2} = 4 kΩ and _{2} = 4 kΩ and

Experimentally captured trajectories in the _{2} and _{2} = 5.57 kΩ and _{2} = 4 kΩ and

This paper mainly presents a comprehensive investigation of the dynamical behaviors in theoretical, numerical, and experimental surveys in a three-element-based memristive circuit. The memristive circuit having fewer circuit elements with a simply connected topology contains only two memristors and one capacitor, which can be regarded as two memristor-capacitor oscillating units with a shared capacitor. The boundedness of the three-element-based memristive circuit is theoretically proved by employing the contraction mapping principle, which implies that the trajectories of the memristive circuit are all confined in a limited phase space. The memristive circuit has three equilibrium points including one zero equilibrium point and two nonzero ones. The zero equilibrium point is always an unstable saddle focus, but the stability is stable node-foci or unstable saddle-foci for the two nonzero equilibrium points with respect to two considered model parameters, which lead to the occurrence of rich dynamical behaviors in the proposed memristive circuit. Dynamical behaviors associated with the two adjustable model parameters including point attractor, limit cycles with different periodicities, single-scroll and double-scroll chaotic attractors, coexisting bifurcation mode, period-doubling bifurcation route, and crisis scenarios are numerically simulated by MATLAB-based programs for 2D/1D bifurcation plots, dynamical maps, finite-time Lyapunov exponents, and phase plane plots. Particularly, under two sets of specified model parameters near crisis scenarios, the phenomena of riddled attraction basins and multistability are numerically discovered by local attraction basins. This is the first time to demonstrate the riddled attraction basins and multistability in such a simple memristive circuit. Moreover, an on-breadboard circuit is manually made by the off-the-shelf components and hardware experiments are executed, from which the memristive circuit running trajectories under some discrete model parameters are captured to further validate the numerically simulated ones. Besides, the analog implementation of memristive circuit can effectively promote the integrated circuit design. These revealed results demonstrate that the presented three-element-based memristive circuit has a great potential for the application in real chaos-based engineering. Moreover, this memristive circuit is one of the simplest chaotic circuits in the literature, which also can provide powerful experimental and analytical platforms for people to understand the dynamical behaviors in electronic engineering and neurology. Moreover, the work done lets us conjecture that there are still some unknown features of this three-element-based memristive circuit, which are to be uncovered in our future research.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant nos. 61801054, 51777016, 61601062, and 61504013, the Natural Science Foundation of Jiangsu Province, China, under Grant nos. BK20160282 and BK20191451, and the Postgraduate Research and Practice Innovation Program of Jiangsu Province, China, under Grant no. KYCX19_1768.