An Improved Memristive Diode Bridge-Based Band Pass Filter Chaotic Circuit

By replacing a series resistor in active band pass filter (BPF) with an improved memristive diode bridge emulator, a third-order memristive BPF chaotic circuit is presented. The improved memristive diode bridge emulator without grounded limitation is equivalently achieved by a diode bridge cascadedwith only one inductor, whose fingerprints of pinchedhysteresis loop are examined by numerical simulations and hardware experiments. The memristive BPF chaotic circuit has only one zero unstable saddle point but causes complex dynamical behaviors including period, chaos, period doubling bifurcation, and coexisting bifurcation modes. Specially, it should be highly significant that two kinds of bifurcation routes are displayed under different initial conditions and the coexistence of three different topological attractors is found in a narrow parameter range. Moreover, hardware circuit using discrete components is fabricated and experimental measurements are performed, upon which the numerical simulations are validated. Notably, the proposed memristive BPF chaotic circuit is only third-order and has simple topological structure.

In [13,19], two memristive Chua's circuits are proposed by bridging ungrounded limitation generalized memristors. Meanwhile, a simplest third-order memristive BPF chaotic circuit is realized by replacing the parallel resistor in classical BPF with a grounded limitation memristor in [38]. Inspired by the advantages of those methodologies in building chaotic circuit, a novel and simple memristive BPF chaotic circuit with complex dynamical behaviors is constructed from a second-order active BPF by replacing a series resistor with our proposed memristive diode bridge emulator. The newly proposed circuit has extremely simple circuit realization, and the dimension is only three. It is significant that the memristive BPF chaotic circuit has only one determined unstable saddle point and can generate complex dynamical behaviors. Particularly, the new finding of coexisting multiple attractors in such a simple memristive BPF circuit has not been previously reported. The rest of the paper is organized as follows. In Section 2, the realization and mathematical model of the improved memristive diode bridge emulator is given and its fingerprints by adding sinusoidal voltage stimuli are illustrated. The schematic structure of the memristive BPF circuit and dimensionless state equations are introduced, and stability of the equilibrium point is investigated in Section 3. Complex dynamical behaviors are revealed numerically in Section 4. Coexisting multiple attractors are performed in Section 5. Some hardware experiments are performed to validate the correctness of the theoretical analyses and numerical simulations in Section 6. Finally, the conclusions are drawn in Section 7.

Improved Memristive Diode Bridge Emulator
Different from the diode bridge-based memristor emulators reported in [16,18,39], an improved memristive diode bridge emulator with much simpler circuit realization is designed as shown in Figure 1(a), where V and represent the voltage and current at the input port 11 , respectively, and V and stand for the voltage and current across the inductor . Consider that the diode bridge is implemented by four unified diodes, where V and represent the voltage across and the current through the diode ( = 1, 2, 3, 4), respectively. The voltage and current relation for can be written as where = 1/(2 ) and , , and are the model parameters of , which stand for the reverse saturation current, emission coefficient, and thermal voltage of the diode, respectively.
According to [39], there are two relations of V 1 = V 3 and V 2 = V 4 . By applying Kirchhoff 's laws, two node current equations are obtained as and two loop voltage equations are yielded as By substituting (1) and (3) into (4), the voltage across 1 can be solved as Then, by combining (1), (2), and (4) with (6), the voltage and current relation at the input port 11 can be expressed as By leading (6) into (5) and using V = d /d , the state equation of inductor is modeled as The mathematical models (7) and (8)   diode bridge emulator, which accords with the defining equations of the class of generalized memristors [4] and it can be implemented by inexpensive off-the-shelf discrete components easily.
In order to illustrate the frequency-dependent pinched hysteresis loops of the improved memristive diode bridge emulator, the circuit element parameters = 10 mH and four 1N4148 diodes with = 5.84 nA, = 1.94, and = 25 mV are selected, and a sinusoidal voltage source is considered as V = sin(2 ), where and are the stimulus amplitude and frequency, respectively. When = 4 V is fixed and is set to 200 Hz, to 1 kHz, and to 3 kHz, respectively, the v-i curves are displayed in Figure 2(a), from which it can be seen that the hysteresis loops are pinched at the origin, and the lobe area decreases with the increase of the frequency, while when f = 200 Hz is fixed and is set to 3 V, to 4 V, and to 5 V, respectively, the v-i curves are plotted in Figure 2(b), which explains that the pinched hysteresis loop is regardless of the stimulus amplitude. The simulation results in Figure 2 show that the improved memristive diode bridge emulator can exhibit three fingerprints for identifying memristors [17].
Also, a hardware level on a breadboard is fabricated and Tektronix AFG 3102C is used to generate a sinusoidal voltage. The experimental results, as shown in Figure 3, are captured by Tektronix TDS 3034C to validate three fingerprints of the memristor emulator, from which the results from hardware circuit are unanimous to those revealed by numerical simulations. It is emphasized that the minor deviations are caused by small difference between numerical simulations and experimental measurements of the inductor parameter.

Third-Order Memristive BPF Chaotic Circuit
A second-order active BPF circuit has a simple circuit topology, which contains only one amplifier, two capacitors, and four resistors [40]. In this paper, by replacing a series resistor with the proposed memristive diode bridge emulator, a third-order memristive BPF circuit is proposed, as shown in Figure 1(b). Therefore, the proposed memristive chaotic circuit is much simpler and more intuitive in practical realization than other memristive chaotic circuits reported in [13][14][15][16].

Mathematical Model.
The proposed circuit has three dynamic elements of capacitor 1 , capacitor 2 , and memristor , respectively, corresponding to three state variables of V 1 , V 2 , and . According to Kirchhoff 's circuit laws and constitutive relationships of basic circuit elements, a state equation set is written as where V = V 2 − V 1 and = / .
Equation (9) Thus the parameter amount of the dimensionless equation (11) will decrease to three. The circuit parameters shown in Figure 1 are selected as 1 = 2 = 20 nF, = 10 mH, = 50 Ω, = 50 Ω, = 1 kΩ, and four 1N4148 diodes with = 5.84 nA, = 1.94, = 25 mV, which are used as typical circuit parameters. Therefore, the normalized parameters are calculated by (10) as In our following work, the parameters given in (12) will be taken as typical system parameters to reveal dynamical behaviors in the third-order memristive BPF circuit.

Stability
Analysis. Obviously, system (11) has only one zero equilibrium point S(0, 0, 0). By linearizing (11) It is notable that the values of 1,2 and the symbol of 3 at S are considered to remain unchanged with c = 6.02 × 10 -6 and positive . For the typical system parameters, the eigenvalues at S are calculated as 1 = 0.0499, 2 = 0.0001, 3 = -830.5648 (16) which implies that S is always an unstable saddle.

Typical Chaotic
Attractor. For the typical system parameters of (12) and the initial conditions of (0, 0.01, 0), phase portraits of the typical chaotic attractor in three different planes are numerically simulated by solving system (11) and shown in Figures 4(a), 4(b), and 4(c), respectively, and Poincaré mapping on = 0 section is depicted in Figure 4(d). It is noted that the proposed third-order memristive BPF circuit can generate chaos indeed.

Dynamical Behaviors in Memristive BPF Chaotic Circuit
Consider that the parameter increases from 0.001 to 0.1 and the other parameters are selected as given in (12). Bifurcation diagrams of the system variable and first two Lyapunov exponents calculated by Wolf 's method [41] are presented as shown in Figure 5. Two sets of initial states, positive initial conditions (0, 0.01, 0) colored in red and negative initial conditions (0, −0.01, 0) colored in blue, are utilized in Figure 5  determined by the Lyapunov exponents are consistent well with those revealed by the bifurcation diagrams. Note that the bifurcation diagrams in the narrow region of 0.0145 < < 0.017 have imperfect bifurcation structures, that is, without bifurcation route from period to chaos. Therefore, there must exist an attractive basin with special initial conditions to be located [9]. Under these special initial conditions, a period doubling bifurcation route can be found in system (11), which leads to the coexistence of multiple attractors. When 0.001 ≤ ≤ 0.004, system (11) shows periodic behavior, while when 0.004 < ≤ 0.015, the first Lyapunov exponent is zero or positive alternately, which indicates the occurrences of periodic and chaotic behaviors. With the intervals of in 0.015 < ≤ 0.025 and 0.056 ≤ ≤ 0.096, the occurrences of period doubling bifurcation, reverse period doubling, and coexisting bifurcation modes are discovered. In 0.025 < < 0.056, system (11) locates in the region of chaos with a larger periodic windows near = 0.03.
For different values of , phase portraits of system (11) in the -plane are numerically simulated, as shown in Figure 6, where the initial conditions of the red and blue trajectories are the same as those used in Figure 5(a). These results just emulate the dynamical behaviors of period, chaos, period doubling bifurcation, and coexisting bifurcation modes emerging from system (11). Note that the chaotic attractor in Figure 6(e) is spiral structure, similar to that revealed in the delay system [42].

Multiple Attractors Depending on the Initial Conditions
In this section, the special phenomenon of coexisting multiple attractors in system (11) is mainly concerned. The bifurcation diagrams of and Lyapunov exponents in the region of 0.0145 ≤ a ≤ 0.017 under different initial conditions are presented in Figure 7 to highlight the phenomenon of multiple attractors. The initial conditions are specified as (0, ±0.01, 0) and (0.01, ±0.01, 0) and the color settings are marked in Figure 7(a), respectively. Note that there exists different bifurcation route in concerned parameter region for different initial conditions, which leads to the existence of multiple attractors. In Figure 7(a), the narrow parameter range of can be divided into four different regions of I, II, III, and IV. In region I, two kinds of coexisting attractors, including chaotic attractor and limit cycle, are revealed. Coexisting limit cycles with different periods are given in region III, while, in regions II and IV, coexisting twin limit cycles and limit cycle are observed. Within two regions, phase portraits of coexisting multiple attractor with different topological structures are plotted in Figure 8.

Experimental Verifications
A hardware level on a breadboard is fabricated to validate the complex dynamics of the proposed memristive BPF chaotic circuit. The experimental prototype for the memristive BPF chaotic circuit is photographed and shown in Figure 9, where the passive elements of precision potentiometer and monolithic ceramic capacitor and manually winding inductor as well as the active devices of operational amplifier AD711KN with ±15 V DC power supplies are chosen in our experiment. Note that two auxiliary gadget circuits are hired in experimental measurements to obtain the terminal voltages of capacitors 1 and 2 . Additionally, the experimental results are captured by a Tektronix TDS 3034C digital oscilloscope in mode. Phase portraits in different planes under typical circuit parameters are easily observed, as shown in Figure 10.     Figure 11: Experimentally measured phase portraits in the V 2 -V 1 plane with different 1 and 2 , where = 50 Ω, = 50 Ω, = 1 kΩ, and = 10 mH. (a) Period-1 limit cycle at 1 = 2 = 56 nF; (b) coexisting period-1 limit cycles at 1 = 2 = 72 nF; (c) coexisting period-2 limit cycles at 1 = 2 = 94 nF; (d) coexisting chaotic attractors at 1 = 2 = 105 nF; (e) chaotic attractor at 1 = 2 = 115 nF; (f) period-3 limit cycle at 1 = 2 = 126 nF. convenience, the two capacitors 1 and 2 are adjusted to meet the variation of dimensionless parameter in our experimental measurements. When the two capacitors 1 and 2 turned as six different values, phase portraits in the V 2 -V 1 plane are measured, as shown in Figure 11, where the phase portraits of coexisting attractors in Figures 11(b)-11(d) are captured separately and handed with after-treatment. It is emphasized that the desired different initial capacitor voltages and inductor current are difficult to assign in hardware circuit, which are randomly sensed through turning on the hardware circuit power supplies again [9,13]. Ignoring the minor deviations caused by parasitic parameters, the experimental results shown in Figures 10 and 11 match well with the results of numerical simulations in Figures 4 and 6.

Conclusion
In this paper, a third-order memristive BPF chaotic circuit is presented, which is constructed by replacing a resistor in second-order active BPF with an improved memristive diode bridge emulator. Numerical simulations of the mathematical model and the corresponding hardware experiments are performed, which show that the memristive BPF chaotic circuit has only one zero unstable saddle and generates complex dynamical behaviors of period, chaos, period doubling bifurcation, and coexisting bifurcation modes. The most significant feature of the proposed memristive chaotic circuit is that the stability depends on the initial conditions of dynamic elements, thereby leading to the occurrence of coexisting multiple attractors. Besides, the proposed memristor is much simpler in practical circuit realization and the constructing memristive BPF chaotic circuit is realized with less discrete components.

Conflicts of Interest
The authors declare that they have no conflicts of interest.