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Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier
Journal Pre-proof Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier Jan Sadecki, Wieslaw Marszalek PII:
S0026-2692(19)30074-6
DOI:
https://doi.org/10.1016/j.mejo.2019.104636
Reference:
MEJ 104636
To appear in:
Microelectronics Journal
Received Date: 23 January 2019 Revised Date:
16 August 2019
Accepted Date: 1 October 2019
Please cite this article as: J. Sadecki, W. Marszalek, Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier, Microelectronics Journal (2019), doi: https://doi.org/10.1016/j.mejo.2019.104636. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier Jan Sadeckia , Wieslaw Marszalek1a,b a
b
Opole University of Technology, Institute of Computer Science, 45-758 Opole, Poland Rutgers University, Hill Center for Mathematical Sciences, Piscataway, NJ 08854, USA
Abstract One- and two-parameter bifurcation diagrams of an oscillating circuit (based on a low-pass Sallen-Key topology) with a full-wave memristive rectifier are presented. The diagrams detail peculiar bursting responses of the circuit together with various interactions of chaotic attractors of the circuit when certain parameters vary. The full-wave rectifier has three fingerprints of memristors with a pinched hysteretic current-voltage characteristic. Illustrative examples with both one- and two-parameter bifurcation diagrams are included. A brief description of the parallel computations used to obtain the two-parameter color diagrams is also provided. Keywords: Oscillatory memristive circuits, Bifurcations, Chaos, Two-parameter diagrams PACS: 02.30.Oz, 05.45.-a, 84.30.-r 1. Introduction Memristive elements when added as components of certain linear circuits yield very interesting dynamical responses. Mathematical models of such circuits are in the forms of systems of nonlinear differential equations. Oscillating nonlinear circuits are quite often difficult to be analyzed analytically and, therefore, computationally intense simulations are the ways of getting an insight into the circuits’ dynamics. The cases of determining of multiparameter bifurcation diagrams, when two or more elements of such circuits vary simultaneously, are examples of such computationally intensive tasks. 1
Corresponding author: [email protected], [email protected]
Preprint submitted to Microelectronics Journal
October 1, 2019
Such computations are challenging and the circuits’ responses are quite often surprisingly complex, particularly when the models of circuits’ elements involve transcendental functions, such as the hyperbolic and logarithmic ones [1]-[7]. In this paper we examine an active circuit with a memristive element which is in fact a full-wave bridge rectifier with an inductor L. Such a circuit has been studied by a research group of prof. Bao, see [1],[5]-[7], where both simulation and experimental results have been reported. Using various tools (i.e. one-parameter bifurcation diagrams, the 0–1 test for chaos, time-seiries solutions in Matlab, Lyapunov exponents) the peculiar quasi-periodic, periodic and chaotic bursting phenomena of the circuits have been explained. There have been also results reported in which the oscillating circuits contain full-wave bridge rectifiers with passive RLC [8], LC [9] and RC [10] filters. Similar types of oscillatory circuits are the diode-based jerk circuits [11]. The mathematial models of the active circuits with bridge rectiifiers containing L, LC, RC or RLC passive circuits differ slightly as they may be of order three or four. However, the common property of the full-wave bridge rectiffiers with the passve L, LC, RC and RLC filters is the fact that they all satisfy the three fingerprints of memristors [2],[12],[13]. Those circuits are also characterized by the multistability property (coexisting attractors). Similar properties of the three fingerprints and multistability are exhibited by the memristive oscillatory circuits in [14]-[17]. The rectifier’s dynamical model contains hyperbolic and logarithmic terms, which are the results of using accurate voltage-current diodes’ characteristics. The pinched hysteretic characteristic of the rectifier is particularly interesting as it results in surprising bifurcation diagrams. Time series signals within the circuit are of chaotic or periodic bursting nature, very sensitive to small changes of the values of circuit’s elements. Thus, rather perculiar one- and two-parameter bifurcation diagrams are obtained. The main goal of this paper is to present computationally intensive simulations (done on parallel computers) to obtain various one- and two-parameter (color) bifurcation diagrams illustrating the complicated dynamical nature of the circuit and a variety of its chaotic and periodic bursting responses. None of the color two-parameter diagrams can be in reality obtained using a single processor computing, because of the time and memory requirements. Having two-paramter bifurcation diagrams available may be of importance in certain application aspects of nonlinear oscillatory circuits. For example, a person interested in chaotic cryptography and chaotic number 2
generators would be interested in the chaotic regime of operation of such circuits. On the other hand, when regular (periodic) operation is needed, then one should be able to know how to avoid chaotic responses of the same circuits. In such cases, two-parameter bifurcation diagrams help to quickly identify the range of parameters for which the circuit is safely periodic. Section 2 of this paper presents a summary of the properties of the memristive bridge rectifier used in the oscillating circuit and of a mathematical model of the whole circuit. Then, section 3 follows with examples of oneparameter bifurcation diagrams of the circuit. Section 4 shows the main results with several two-parameter bifurcation diagrams and a discussion of rather unusal dynamical phenomena one can observe in the oscillating circuit. The two-parameter diagrams provide a better and faster insight into the dynamical properties of the nonlinear circuits than the one-parameter diagrams. From the circuit engineers’ point of view the two-parameter bifurcation diagrams may be preferable in the circuit design process than the tedious and lengthy mathematical analysis. Conclusions and further remarks are included in section 5. 2. Circuit with memristive diode bridge rectifier Consider the full-wave bridge rectifier circuit with inductor L shown in Fig.1(a). The bridge with four 1N4148 diodes, each with its reverse saturaC2
+
D3
−v + 2
D1 L
v1 −
R
i D4
iL
+ v1 − i
R
D2
(a) Memristive rectifier.
C1
+ −
U R2
R1
(b) Circuit with memristive rectifier.
Figure 1: (a) Memristive diode bridge rectifier with an inductor. (b) The circuit with memrisive rectifier from Fig. 1(a) having hysteretic pinched characteritic i − v1 .
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Figure 2: Pinched tangential hystereses i−v1 obtained from (1) for f = 3 (dash), 10 (dash− dot), 30, (dot), 100 (solid) Hz. The L = 120 mH and each of the four 1N4148 diodes D1 -D4 has the following parameters: Is = 5.84 nA, n = 1.94, VT = 25 mV.
tion current Is , emission coefficient n and thermal voltage VT together with a single inductor L were used by Bao et al. [1] to mathematically analyze the circuit with memristive rectifier as shown in Fig. 1(b). Interesting chaotic bursting responses of the circuit were obtained in simulations and verified by hardware lab experiments in [1] and for other memristive oscillatory circuits in [5]-[11],[14]-[16]. The following model of the rectifier in Fig.1(a) i = (iL + 2Is )tanh(ρv 1) i h h i (iL +2Is )eρv1 iL +2Is diL v1 1 1 = − ln = − ln dt L ρL 2Is cosh(ρv1 ) ρL 2Is cosh(ρv1 )
(1)
with ρ = 1/(2nVT ), used in [2], adequately captures the dynamics of the rectifier with inductor L. In this paper, to complement the previous results [1],[2], the oscillatory circuit in Fig.1(b) is analyzed further and the circuits’ two-parameter bifurcation diagrams are computed. Note that the three fingerprints of memristors, see for example [12], that the rectifier satisfies are detailed in [2]. When the sinusoidal voltage v1 (t) = Vm sin(2πf t) with appropriate values of Vm and f are applied to (1), then the i − v1 characteristics have the pinched hysteretic forms as shown in Fig. 2 for four f values and Vm = 2.7 V. Those characteristics have the three fingerprints of memristors, as follows [2]: • Fingerprint 1: The i − v1 hystereses are pinched at the origin. 4
• Fingerprint 2: The zero current and voltage crossings occur at the same time instants. • Fingerprint 3: With f → ∞, the i − v1 characteristics tend to that of a nonlinear resistor (having a single valued characteristic rather than being a hysteresis). This means that with the increased values of frequency f in v1 = Vmax sin(2πf t) the area A(f ) enclosed by the hysteresis is such that limf →∞ A(f ) = 0. The other feature of the meristive bridge shown in Fig. 1(a) is that the pinched characteristic is tangential rather than self-crossing at (0, 0) [18]-[21], with counterclockwise motion, as indicated by the arrows in Fig. 2. It means that the first derivative di/dv1 has the same (positive) value at (0, 0) every half period and the two hysteresis lobes touch each other at the origin. In other words, if i(t¯) = v1 (t¯) = 0 for certain time instant t¯ and also for half period later, t¯ + 1/(2f ), then di/dv1 |t¯ = di/dv1 |t¯+1/(2f ) > 0, while d2 i/dv12 |t¯ and d2 i/dv12 |t¯+1/(2f ) change their signs at the origin. The circuit in Fig. 1(b) is described by the system [1] x0 = a ln(b cosh(y)) − a ln(x + b) y 0 = ky − z − (x + b)tanh(y) z 0 = (2k + 1)y − 2z
(2)
with x = ρRiL , y = ρv1 , z = ρv2 , C1 = C2 = C, a = R2 C/L, b = 2ρRIs , k = R2 /R1 and the derivatives on the left side in (2) are with respect to τ = t/(RC). The system has one equilibrium at the origin (0, 0, 0) and three parameters a, b and k, whose impact on the oscillatory responses of (2) will be analyzed in the next sections of this paper through various one- and two-parameter bifurcation diagrams. As mentioned earlier, the goal of this paper is to present complex twoparameter bifurcation diagrams of (2) when two parameters (out of the three ones a, b and k) vary. In general, the case of simultaneous varying of two parameters is computationally challenging. Say, the parameters a and b change simultaneously in (2) within the intervals amin ≤ a ≤ amax and bmin ≤ b ≤ bmax with 103 values of each parameter between its minimum and maximum values. Then, 106 solutions of (2) are needed to obtain a single two-parameter bifurcation diagram for the pairs of values of (a, b) [3],[4]. In this paper the two-parameter bifurcation diagrams are complemented by the 5
conventional one-parameter ones to show more details of the complex oscillatory solutions of (2) and the interesting phenomenon of interacting chaotic attractors. The case of simultaneous changes of two parameters makes also sense from a different point of view. Namely, it often happens in nonlinear circuits that varying a single circuit element may yield simultaneous change of two or more parameters (coefficients) in the circuit’s dynamical model. More details about that are given later.
(a) Period−5 oscillations.
(b) Periodic y(t).
(c) Chaotic response.
(d) Chaotic y(t).
(e) Chaotic bursting.
(f) Chaotic bursting y(t).
Figure 3: Various responses of (2) obtained with Matlab’s ode45 solver for the initial condition [0, 1, 0] and parameters a = 0.203, k = 2.25 (Figs. 3(a) and 3(b)), a = 0.215, k = 2.25 (Figs. 3(c) and 3(d)) and a = 0.110, k = 2.23 (Figs. 3(e) and 3(f)). The b was kept constant at b = 2.4082 · 10−5 .
3. One-parameter bifurcation diagrams First, Fig. 3 shows selected three types of steady-state time responses y(t) and the corresponding y versus z plots for (2) obtained with [x(0), y(0), z(0)] = [0, 1, 0]. Those responses show periodic bursting (period−5 in Figs. 3(a) and 3(b)), chaotic response (in Figs. 3(c) and 3(d)) and chaotic bursting (in Figs. 3(e) and 3(f)). Next, several one-parameter bifurcation diagrams of (2) are presented. The diagrams show the maximum values of the variable y in one period of the steady state response of (2). Two initial conditions [x(0), y(0), z(0)] = [0, ±1, 0] were used to illustrate interactions of the co-existing attractors of 6
(a) Bifurcation diagram for 2.07 ≤ k ≤ 2.27. Zoomed-in diagrams in a narrower window are shown in figures (b) and (c) below.
(b) Bifurcation diagram for 2.177 ≤ k ≤ 2.197.
(c) Bifurcation diagram for 2.177 ≤ k ≤ 2.197 (two coexisting attractors). Figure 4: One-parameter bifurcation diagrams for varying k, a = 0.11, b = 2.4082 · 10−5 , initial conditions [0, 1, 0] (black) and [0, −1, 0] (red).
the system. All diagrams obtained for the initial condition [0, 1, 0] are drawn in black, while those for the initial condition [0, −1, 0] are drawn in red. If the ymax values are identical for both the black and red diagrams, then the later is shown. This applies to Figs. 4(c), 5(c), 6(b), 6(c) and 9(c). All oneparameter diagrams were obtained by using the ode45 solver with the time 7
(a) Bifurcation diagram for 0.1 ≤ a ≤ 0.5. Zoomed-in diagrams in a narrower window are shown in figures (b) and (c) below.
(b) Bifurcation diagram for 0.19 ≤ a ≤ 0.23.
(c) Bifurcation diagram for 0.19 ≤ a ≤ 0.23 (two coexisting attractors). Figure 5: One-parameter bifurcation diagrams for varying a, k = 2.25, b = 2.4082 · 10−5 , initial conditions [0, 1, 0] (black) and [0, −1, 0] (red).
step dt = 0.01, time horizon 0 ≤ t ≤ 1000 seconds and the abserr and relerr errors were both equal 10−8 . The value of parameter b was kept constant and equal b = 2.4082 · 10−5 and parameters a and k varied in certain intervals in such a way that all horizontal axes in the bifurcation diagrams correspond to 600 discrete values of either a or k. 8
(a) Diagram for 0.1 ≤ a ≤ 0.4.
(b) Bifurcation diagram for 0.13 ≤ a ≤ 0.15 with two co-existing attractors.
(c) Diagram for 0.155 ≤ a ≤ 0.165 with two coexisting attractors.
(d) Diagram for 0.205 ≤ a ≤ 0.215.
Figure 6: One-parameter bifurcation diagrams for varying a, b = 2.4082 · 10−5 , k = 2.20, initial condition [0, 1, 0] (black) and [0, −1, 0] (red). Plots (b), (c) and (d) show zoomed-in parts of plot (a).
It is well-known that the oscillating active circuits with memristive wave rectifiers [5] or the diode-base jerk circuits [10],[11] have coexisting attractors.This phenomenon is somtimes called the multistability. Such symmetrical attractors are generated from the special sets of initial conditions of systems. Another way of constructing coexisting attractors is to use two operating configurations, i.e. the incremental memristor and the decremental memristor as discussed in details in [17]. The presence of coexisting attractors is confirmed by analyzing one-parameter bifurcation diagrams in this paper, for example those shown n Figs.4(c), 5(c) and 9(c). With the changing parameters, either k or a, one observes that the attraction basins change slowly and the two different branches of the diagrams (represented by different colors) have discontinuities at certain discrete values of k and a. This happens, for example, at a = 0.2095 in Fig.5(c), the point of discontinuity of the black
9
and red branches. Also, the fixed initial conditions ([0, 1, 0] for the branches represented by the black color and [0, −1, 0] for the the branches marked in red) result in completely different limit cycles for large intervals of a on the left and right hand sides of a = 0.2095. Similar discontinuity of bifurcation branches at certain discrete values of the bifurcation parameter k and different limit cycles are observed in Fig.4(c). The above phenomena falls into the category of multistability in which multiple attractors depend on initial conditions for a particular value of the bifurcation parameter [10],[11],[14]. Also, for the fixed initial conditions, the limit cycles of the period-n solutions and chaotic attractors change significantly when the bifurcation parameter in (2) changes by an infinitely small value, say from k to k ± δk or from a to a ± δa, where δk and δa are of a few orders smaller than k and a, resepctively. In this paper the presence of coexisting attractors is signaled by one-parameter bifurcation diagrams with both black and red branches, but this topic is secondary in comparison to the main topic of this paper, which is the two-parameter bifurcation diagrams and their parallel computation. Analyzing one-parameter bifurcation diagrams in Figs. 4, 5, 6 and 9 one can notice the typical period doubling and halving bifurcations, for example, for a = 0.2045 and a = 0.212 in Fig. 5(b). In many intervals of parameter a there are chaotic responses. A peculiar bifurcation occurs for a = 0.229 (Figs. 5(b) and 5(c)) where a period-9 response for a < 0.229 becomes period-8 at a = 0.229, to switch back to period-9 for a > 0.229. Similar bifurcations happen for a = 0.1575 in Fig. 6(c) with the period-5 →period-4 →period5 changes (black diagram obtained with the initial condition [0, 1, 0]) and also for a = 0.1585 for the red diagram (obtained with the initial condition [0, −1, 0]). Interesting bifurcation happens at a = 0.2095 in Figs. 5(b) and 5(c), with the interacting attractors switching their maximum values of y (compare the black and red diagrams around a = 0.2095). Notice also, that in many intervals of a the initial conditions [0, 1, 0] and [0, −1, 0] yield different responses, as, for example, in the interval 0.1570 ≤ a ≤ 0.1518 in Fig. 6(c) (completely different black and red curves indicating in both cases period-5 responses). However, in other intervals of a, the two different initial conditions yield exactly the same responses, for example, the period-11 responses for 0.144 ≤ a ≤ 0.147 in Fig. 6(b). Notice also that a similar analysis of different limit cycles and coexisting attractors is slightly different in the case of two-parameter bifurcation diagrams, which identify the type of response only, i.e. period-n oscillations and chaotic responses without providing the maximum values of the analyzed 10
time response. We write more about that at the end of the next section. To provide the maximum values in two-parameter rectangular color diagrams would require three dimensional graphs in which two dimensions are used for the two bifurcation parameters, say k and a in (2), and the third dimension is for the meximum values of the analyzed oscillatory response. Such 3D color diagrams will neither be practical nor useful when presented on a flat screen or paper surface, which both are 2D in nature. 4. Two-parameter bifurcation diagrams This section presents two-parameter bifurcation diagrams when both a and k vary simultaneously. Such a change is equivalent to several possible changes of the circuit’s elements. For example, a single change of the value of R changes both a and b, but not k (since a = R2 C/L, b = 2ρRIs and k = R2 /R1 ). Also, by changing any of the pairs of values of (R2 , C1 ), (R2 , L), (R1 , C2 ) or (R1 , L) we change simultaneously a and k. Other scenarios are also possible. While one two-parameter bifurcation diagram with, say 600 × 600 discrete values of a and k in the area (amin ≤ a ≤ amax ; kmin ≤ k ≤ kmax ) is, in some sense, equivalent to 600 bifurcation diagrams with only one varying parameter, either a or k, that relation is not completely true, since from the one-paramter diagrams we not only determine the type of circuit’s response (i.e. period−n or chaotic), but also the values of n maxima in a period can be seen. Moreover, just two one-parameter bifurcation diagrams, each obtained with a different initial condition [x(0), y(0), z(0)] (say [0, 1, 0] and [0, −1, 0]) show interesting interactions of chaotic attractors of (2) that are not seen while using two-parameter bifurcation diagrams. Thus, it is always beneficial when using two-parameter diagrams to supplement them with one-paramters diagrams, particularly that the later are obtained in a tiny fraction of time needed to obtain the former. Details are provided below. 4.1. Software and hardware used Computation of two-parameter bifurcation diagrams was done via parallel computations performed on two processors Inter(R) Xeon(R) CPU ES-2640 v2 at 2.00 GHz. Each processor, made in the 22 nm technology, has 8 cores servicing 16 threads, using the Intel Turbo Boost 2.0. All calculations were done in the C language (compiled by the GCC compiler) using Ubuntu OS, which supports the OpenMP (Open Multi-Processing) [22].
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The conventional fourth-order explicit Runge-Kutta (RK-4) method was used to solve (2) when computing the color two-parameter bifurcation diagrams. The time horizon was the same as in computing one-parameter diagrams (0 ≤ t ≤ 1000), while the time step in the RK-4 method was fixed at dt = 0.0001. The absolute error used in distinguishing two maximum values of variable y was 0.01, that is the identified maximum values with difference less than 0.01 were considered identical. Also, the maximum number n in the period−n type of oscillations was 64. To visualize those various period−n responses (n = 1, 2, . . . , 64) we used a continuous colorbar with colors ranging from dark brown (n = 1) to white (n = 64). All periodic responses with n ≥ 64 and chaotic ones were grouped together and assigned the white color (see the colorbars in each diagram in Figs. 7 and 8). Finally, all 360, 000 runs of the RK-4 solver needed to obtain each two-parameter bifurcation diagrams used the initial condition [0, 1, 0]. 4.2. Results of numerical experiments Figs. 7(a)-8(b) show four two-parameter bifurcation diagrams, each obtained with the 600 × 600 discrete values of parameters (a, k). Fig. 7(b) is a zoomed-in version of the diagram in rectangle A in Fig. 7(a). Similarly, Fig. 8(b) is the zoomed-in version of the diagram in rectangle B in Fig. 8(a). Fig. 5(a) showing a one-parameter bifurcation diagram corresponds to the line p in Fig. 7(a) for k = 2.25. Also, Fig. 6(a) shows a one-parameter bifurcation diagram along the line q in Fig. 7(a) for k = 2.20. The rather narrow intervals of parameter a with periodic responses (white gaps in the plots in Figs. 5(a) and 6(a)) correspond to the intersection of lines p and q, respectively, with the brown areas (of various intensity) in Fig. 7(a). The single line for approximately a > 3.4 in Fig. 6(a) indicates period−1 oscillations. This is confirm in Fig. 7(a) where the right end of line r falls onto the dark brown area of the two-parameter diagram. Diagrams shown in Figs. 6(c) and 6(d) can also be linked to the narrow intervals of parameter a along the line r in Fig. 7(a). Similarly, the one-parameter diagrams in Figs. 9(a) and 9(b) were taken along the lines r (for k = 2.40) and s (for k = 2.335), respectively (see Fig. 8(a)). Fig. 6(d) is worth commenting on, too. It shows period−4 response of the circuit for the middle range of paramter a and chaos elsewhere. Transition to and from the periodic responses is gradual, and not rapid (instant) as it occurs typically when a crisis of a chaotic attractor happens.
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(a) Two-parameter 600 × 600 diagram.
(b) Rectangle A in Fig.7(a); 600 × 600 points. Figure 7: Two-parameter bifurcation diagrams obtained with the RK-4 solver, time step 0.0001.
Although all two-parameter diagrams in Figs. 7 and 8 were obtained by using the initial condition [0, 1, 0], one would obtain exactly the same color diagrams for the initial condition [0, −1, 0], as the two-parameter diagrams show only the type of response of (2) and not the values of maximum of y, This is not the case for one-parameter diagrams as the maximum values of y are quite often different for the initial condition [0, 1, 0] than for [0, −1, 0], see Figs. 4(b), 5(c), 6(c) and 9(b).
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(a) Two-parameter 600 × 600 diagram.
(b) Rectangle B in Fig.8(a); 600 × 600 points. Figure 8: Two-parameter bifurcation diagrams obtained with the RK-4 solver, time step 0.0001.
5. Conclusions When a memristive bridge rectifier is added to a circuit based on the Sallen-Key topology, then interesting periodic and chaotic responses are obtained. Bifurcation diagrams (both one- and two-parameter ones) of the circuit show complicated dynamical responses that are sensitive to the changes of the circuit’s parameters. The use of two numerical solvers, the ode45 to compute one-parameter diagrams, and RK-4 to obtain two-parameter ones was done to show that the former complements and confirms the results obtained by the later. The one-parameter diagrams were obtained through a 14
(a) Bifurcation diagram for k = 2.40 (along line r in Fig. 8(a)).
(b) Bifurcation diagram for k = 2.335 (along line s in Fig. 8(a)).
(c) Bifurcation diagram for k = 2.335 (along line s in Fig. 8(a)) with different initial conditions. Figure 9: One-parameter bifurcation diagrams along the lines r and s in Fig. 8(a) with initial condition [0, 1, 0] (black) and [0, −1, 0] (red).
single processor sequential calculations, while the two-parameter color diagrams are the results of parallel computations. Distinguishing between periodic and chaotic responses is quite important in practice, as nonlinear oscillating circuits find more and more applications in, for example, secure transmission of information (data), designing random number generators, 15
chaotic cryptography and other areas. The main contribution of this paper are the two-parameter color bifurcation diagrams and the analysis of parallel computations through which the diagrams were obtained. One such two-parameter diagram provides a quick inside into the dynamics of system (2) with two changing parameters, instead of using tens or hundreds of diagrams with one varying parameter. It is very likely that other memristive emulators [23]-[26] or fractional memristors [27] when used as a replacement of the memristive diode bridge may also result in interesting bifurcations of the circuit in Fig. 1(b). Conversely, the same may happen if the memristive diode bridge in Fig. 1(a) is used as a memristive component in other chaotic circuits, for example, those discussed in [14],[28]-[30]. Computing bifurcation diagrams with one varying parameter is very common nowadays, and perhaps less common in the cases with two varying parameters. Significant increase in difficulty one may expect when three parameters vary simultaneously. Presenting the threeparameter bifurcation diagrams visually is one such difficulty and solving an ODE system, say, (103 )3 (billion) times would also be a problem. Finally, another topic worth examining is the use of the 0-1 test for chaos to be implemented when two parameters of a dynamical system vary. No such analysis and results for two-parameter bifurcations for the 0-1 test have been reported in the literature. However, the results for the 0-1 test for chaos applied to one-parameter bifurcations are encouraging [31]. Acknowledgements The authors would like to thank the four annonymous reviewers for their helpful and constructive comments on our paper. [1] B.-C. Bao, P. Wu, H. Bao, M. Chen and Q. Xu, “Chaotic bursting in memristive diode bridge-coupled Sallen-Key lowpass filter,” Elect. Lett., vol. 53, pp. 1104–1105, 2017. [2] J. Sadecki and W. Marszalek, “Analysis of a memristive diode bridge rectifier,” Elect. Lett., DOI: 10.1049/el.2018.6921, 2018. [3] W. Marszalek and H. Podhaisky, “2D bifurcations and Newtonian properties of memristive Chua’s circuits,” Europhysics Lett. (EPL), vol. 113, 10005, 2016.
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S0026-2692(19)30074-6
DOI:
https://doi.org/10.1016/j.mejo.2019.104636
Reference:
MEJ 104636
To appear in:
Microelectronics Journal
Received Date: 23 January 2019 Revised Date:
16 August 2019
Accepted Date: 1 October 2019
Please cite this article as: J. Sadecki, W. Marszalek, Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier, Microelectronics Journal (2019), doi: https://doi.org/10.1016/j.mejo.2019.104636. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.
Complex oscillations and two-parameter bifurcations of a memristive circuit with diode bridge rectifier Jan Sadeckia , Wieslaw Marszalek1a,b a
b
Opole University of Technology, Institute of Computer Science, 45-758 Opole, Poland Rutgers University, Hill Center for Mathematical Sciences, Piscataway, NJ 08854, USA
Abstract One- and two-parameter bifurcation diagrams of an oscillating circuit (based on a low-pass Sallen-Key topology) with a full-wave memristive rectifier are presented. The diagrams detail peculiar bursting responses of the circuit together with various interactions of chaotic attractors of the circuit when certain parameters vary. The full-wave rectifier has three fingerprints of memristors with a pinched hysteretic current-voltage characteristic. Illustrative examples with both one- and two-parameter bifurcation diagrams are included. A brief description of the parallel computations used to obtain the two-parameter color diagrams is also provided. Keywords: Oscillatory memristive circuits, Bifurcations, Chaos, Two-parameter diagrams PACS: 02.30.Oz, 05.45.-a, 84.30.-r 1. Introduction Memristive elements when added as components of certain linear circuits yield very interesting dynamical responses. Mathematical models of such circuits are in the forms of systems of nonlinear differential equations. Oscillating nonlinear circuits are quite often difficult to be analyzed analytically and, therefore, computationally intense simulations are the ways of getting an insight into the circuits’ dynamics. The cases of determining of multiparameter bifurcation diagrams, when two or more elements of such circuits vary simultaneously, are examples of such computationally intensive tasks. 1
Corresponding author: [email protected], [email protected]
Preprint submitted to Microelectronics Journal
October 1, 2019
Such computations are challenging and the circuits’ responses are quite often surprisingly complex, particularly when the models of circuits’ elements involve transcendental functions, such as the hyperbolic and logarithmic ones [1]-[7]. In this paper we examine an active circuit with a memristive element which is in fact a full-wave bridge rectifier with an inductor L. Such a circuit has been studied by a research group of prof. Bao, see [1],[5]-[7], where both simulation and experimental results have been reported. Using various tools (i.e. one-parameter bifurcation diagrams, the 0–1 test for chaos, time-seiries solutions in Matlab, Lyapunov exponents) the peculiar quasi-periodic, periodic and chaotic bursting phenomena of the circuits have been explained. There have been also results reported in which the oscillating circuits contain full-wave bridge rectifiers with passive RLC [8], LC [9] and RC [10] filters. Similar types of oscillatory circuits are the diode-based jerk circuits [11]. The mathematial models of the active circuits with bridge rectiifiers containing L, LC, RC or RLC passive circuits differ slightly as they may be of order three or four. However, the common property of the full-wave bridge rectiffiers with the passve L, LC, RC and RLC filters is the fact that they all satisfy the three fingerprints of memristors [2],[12],[13]. Those circuits are also characterized by the multistability property (coexisting attractors). Similar properties of the three fingerprints and multistability are exhibited by the memristive oscillatory circuits in [14]-[17]. The rectifier’s dynamical model contains hyperbolic and logarithmic terms, which are the results of using accurate voltage-current diodes’ characteristics. The pinched hysteretic characteristic of the rectifier is particularly interesting as it results in surprising bifurcation diagrams. Time series signals within the circuit are of chaotic or periodic bursting nature, very sensitive to small changes of the values of circuit’s elements. Thus, rather perculiar one- and two-parameter bifurcation diagrams are obtained. The main goal of this paper is to present computationally intensive simulations (done on parallel computers) to obtain various one- and two-parameter (color) bifurcation diagrams illustrating the complicated dynamical nature of the circuit and a variety of its chaotic and periodic bursting responses. None of the color two-parameter diagrams can be in reality obtained using a single processor computing, because of the time and memory requirements. Having two-paramter bifurcation diagrams available may be of importance in certain application aspects of nonlinear oscillatory circuits. For example, a person interested in chaotic cryptography and chaotic number 2
generators would be interested in the chaotic regime of operation of such circuits. On the other hand, when regular (periodic) operation is needed, then one should be able to know how to avoid chaotic responses of the same circuits. In such cases, two-parameter bifurcation diagrams help to quickly identify the range of parameters for which the circuit is safely periodic. Section 2 of this paper presents a summary of the properties of the memristive bridge rectifier used in the oscillating circuit and of a mathematical model of the whole circuit. Then, section 3 follows with examples of oneparameter bifurcation diagrams of the circuit. Section 4 shows the main results with several two-parameter bifurcation diagrams and a discussion of rather unusal dynamical phenomena one can observe in the oscillating circuit. The two-parameter diagrams provide a better and faster insight into the dynamical properties of the nonlinear circuits than the one-parameter diagrams. From the circuit engineers’ point of view the two-parameter bifurcation diagrams may be preferable in the circuit design process than the tedious and lengthy mathematical analysis. Conclusions and further remarks are included in section 5. 2. Circuit with memristive diode bridge rectifier Consider the full-wave bridge rectifier circuit with inductor L shown in Fig.1(a). The bridge with four 1N4148 diodes, each with its reverse saturaC2
+
D3
−v + 2
D1 L
v1 −
R
i D4
iL
+ v1 − i
R
D2
(a) Memristive rectifier.
C1
+ −
U R2
R1
(b) Circuit with memristive rectifier.
Figure 1: (a) Memristive diode bridge rectifier with an inductor. (b) The circuit with memrisive rectifier from Fig. 1(a) having hysteretic pinched characteritic i − v1 .
3
Figure 2: Pinched tangential hystereses i−v1 obtained from (1) for f = 3 (dash), 10 (dash− dot), 30, (dot), 100 (solid) Hz. The L = 120 mH and each of the four 1N4148 diodes D1 -D4 has the following parameters: Is = 5.84 nA, n = 1.94, VT = 25 mV.
tion current Is , emission coefficient n and thermal voltage VT together with a single inductor L were used by Bao et al. [1] to mathematically analyze the circuit with memristive rectifier as shown in Fig. 1(b). Interesting chaotic bursting responses of the circuit were obtained in simulations and verified by hardware lab experiments in [1] and for other memristive oscillatory circuits in [5]-[11],[14]-[16]. The following model of the rectifier in Fig.1(a) i = (iL + 2Is )tanh(ρv 1) i h h i (iL +2Is )eρv1 iL +2Is diL v1 1 1 = − ln = − ln dt L ρL 2Is cosh(ρv1 ) ρL 2Is cosh(ρv1 )
(1)
with ρ = 1/(2nVT ), used in [2], adequately captures the dynamics of the rectifier with inductor L. In this paper, to complement the previous results [1],[2], the oscillatory circuit in Fig.1(b) is analyzed further and the circuits’ two-parameter bifurcation diagrams are computed. Note that the three fingerprints of memristors, see for example [12], that the rectifier satisfies are detailed in [2]. When the sinusoidal voltage v1 (t) = Vm sin(2πf t) with appropriate values of Vm and f are applied to (1), then the i − v1 characteristics have the pinched hysteretic forms as shown in Fig. 2 for four f values and Vm = 2.7 V. Those characteristics have the three fingerprints of memristors, as follows [2]: • Fingerprint 1: The i − v1 hystereses are pinched at the origin. 4
• Fingerprint 2: The zero current and voltage crossings occur at the same time instants. • Fingerprint 3: With f → ∞, the i − v1 characteristics tend to that of a nonlinear resistor (having a single valued characteristic rather than being a hysteresis). This means that with the increased values of frequency f in v1 = Vmax sin(2πf t) the area A(f ) enclosed by the hysteresis is such that limf →∞ A(f ) = 0. The other feature of the meristive bridge shown in Fig. 1(a) is that the pinched characteristic is tangential rather than self-crossing at (0, 0) [18]-[21], with counterclockwise motion, as indicated by the arrows in Fig. 2. It means that the first derivative di/dv1 has the same (positive) value at (0, 0) every half period and the two hysteresis lobes touch each other at the origin. In other words, if i(t¯) = v1 (t¯) = 0 for certain time instant t¯ and also for half period later, t¯ + 1/(2f ), then di/dv1 |t¯ = di/dv1 |t¯+1/(2f ) > 0, while d2 i/dv12 |t¯ and d2 i/dv12 |t¯+1/(2f ) change their signs at the origin. The circuit in Fig. 1(b) is described by the system [1] x0 = a ln(b cosh(y)) − a ln(x + b) y 0 = ky − z − (x + b)tanh(y) z 0 = (2k + 1)y − 2z
(2)
with x = ρRiL , y = ρv1 , z = ρv2 , C1 = C2 = C, a = R2 C/L, b = 2ρRIs , k = R2 /R1 and the derivatives on the left side in (2) are with respect to τ = t/(RC). The system has one equilibrium at the origin (0, 0, 0) and three parameters a, b and k, whose impact on the oscillatory responses of (2) will be analyzed in the next sections of this paper through various one- and two-parameter bifurcation diagrams. As mentioned earlier, the goal of this paper is to present complex twoparameter bifurcation diagrams of (2) when two parameters (out of the three ones a, b and k) vary. In general, the case of simultaneous varying of two parameters is computationally challenging. Say, the parameters a and b change simultaneously in (2) within the intervals amin ≤ a ≤ amax and bmin ≤ b ≤ bmax with 103 values of each parameter between its minimum and maximum values. Then, 106 solutions of (2) are needed to obtain a single two-parameter bifurcation diagram for the pairs of values of (a, b) [3],[4]. In this paper the two-parameter bifurcation diagrams are complemented by the 5
conventional one-parameter ones to show more details of the complex oscillatory solutions of (2) and the interesting phenomenon of interacting chaotic attractors. The case of simultaneous changes of two parameters makes also sense from a different point of view. Namely, it often happens in nonlinear circuits that varying a single circuit element may yield simultaneous change of two or more parameters (coefficients) in the circuit’s dynamical model. More details about that are given later.
(a) Period−5 oscillations.
(b) Periodic y(t).
(c) Chaotic response.
(d) Chaotic y(t).
(e) Chaotic bursting.
(f) Chaotic bursting y(t).
Figure 3: Various responses of (2) obtained with Matlab’s ode45 solver for the initial condition [0, 1, 0] and parameters a = 0.203, k = 2.25 (Figs. 3(a) and 3(b)), a = 0.215, k = 2.25 (Figs. 3(c) and 3(d)) and a = 0.110, k = 2.23 (Figs. 3(e) and 3(f)). The b was kept constant at b = 2.4082 · 10−5 .
3. One-parameter bifurcation diagrams First, Fig. 3 shows selected three types of steady-state time responses y(t) and the corresponding y versus z plots for (2) obtained with [x(0), y(0), z(0)] = [0, 1, 0]. Those responses show periodic bursting (period−5 in Figs. 3(a) and 3(b)), chaotic response (in Figs. 3(c) and 3(d)) and chaotic bursting (in Figs. 3(e) and 3(f)). Next, several one-parameter bifurcation diagrams of (2) are presented. The diagrams show the maximum values of the variable y in one period of the steady state response of (2). Two initial conditions [x(0), y(0), z(0)] = [0, ±1, 0] were used to illustrate interactions of the co-existing attractors of 6
(a) Bifurcation diagram for 2.07 ≤ k ≤ 2.27. Zoomed-in diagrams in a narrower window are shown in figures (b) and (c) below.
(b) Bifurcation diagram for 2.177 ≤ k ≤ 2.197.
(c) Bifurcation diagram for 2.177 ≤ k ≤ 2.197 (two coexisting attractors). Figure 4: One-parameter bifurcation diagrams for varying k, a = 0.11, b = 2.4082 · 10−5 , initial conditions [0, 1, 0] (black) and [0, −1, 0] (red).
the system. All diagrams obtained for the initial condition [0, 1, 0] are drawn in black, while those for the initial condition [0, −1, 0] are drawn in red. If the ymax values are identical for both the black and red diagrams, then the later is shown. This applies to Figs. 4(c), 5(c), 6(b), 6(c) and 9(c). All oneparameter diagrams were obtained by using the ode45 solver with the time 7
(a) Bifurcation diagram for 0.1 ≤ a ≤ 0.5. Zoomed-in diagrams in a narrower window are shown in figures (b) and (c) below.
(b) Bifurcation diagram for 0.19 ≤ a ≤ 0.23.
(c) Bifurcation diagram for 0.19 ≤ a ≤ 0.23 (two coexisting attractors). Figure 5: One-parameter bifurcation diagrams for varying a, k = 2.25, b = 2.4082 · 10−5 , initial conditions [0, 1, 0] (black) and [0, −1, 0] (red).
step dt = 0.01, time horizon 0 ≤ t ≤ 1000 seconds and the abserr and relerr errors were both equal 10−8 . The value of parameter b was kept constant and equal b = 2.4082 · 10−5 and parameters a and k varied in certain intervals in such a way that all horizontal axes in the bifurcation diagrams correspond to 600 discrete values of either a or k. 8
(a) Diagram for 0.1 ≤ a ≤ 0.4.
(b) Bifurcation diagram for 0.13 ≤ a ≤ 0.15 with two co-existing attractors.
(c) Diagram for 0.155 ≤ a ≤ 0.165 with two coexisting attractors.
(d) Diagram for 0.205 ≤ a ≤ 0.215.
Figure 6: One-parameter bifurcation diagrams for varying a, b = 2.4082 · 10−5 , k = 2.20, initial condition [0, 1, 0] (black) and [0, −1, 0] (red). Plots (b), (c) and (d) show zoomed-in parts of plot (a).
It is well-known that the oscillating active circuits with memristive wave rectifiers [5] or the diode-base jerk circuits [10],[11] have coexisting attractors.This phenomenon is somtimes called the multistability. Such symmetrical attractors are generated from the special sets of initial conditions of systems. Another way of constructing coexisting attractors is to use two operating configurations, i.e. the incremental memristor and the decremental memristor as discussed in details in [17]. The presence of coexisting attractors is confirmed by analyzing one-parameter bifurcation diagrams in this paper, for example those shown n Figs.4(c), 5(c) and 9(c). With the changing parameters, either k or a, one observes that the attraction basins change slowly and the two different branches of the diagrams (represented by different colors) have discontinuities at certain discrete values of k and a. This happens, for example, at a = 0.2095 in Fig.5(c), the point of discontinuity of the black
9
and red branches. Also, the fixed initial conditions ([0, 1, 0] for the branches represented by the black color and [0, −1, 0] for the the branches marked in red) result in completely different limit cycles for large intervals of a on the left and right hand sides of a = 0.2095. Similar discontinuity of bifurcation branches at certain discrete values of the bifurcation parameter k and different limit cycles are observed in Fig.4(c). The above phenomena falls into the category of multistability in which multiple attractors depend on initial conditions for a particular value of the bifurcation parameter [10],[11],[14]. Also, for the fixed initial conditions, the limit cycles of the period-n solutions and chaotic attractors change significantly when the bifurcation parameter in (2) changes by an infinitely small value, say from k to k ± δk or from a to a ± δa, where δk and δa are of a few orders smaller than k and a, resepctively. In this paper the presence of coexisting attractors is signaled by one-parameter bifurcation diagrams with both black and red branches, but this topic is secondary in comparison to the main topic of this paper, which is the two-parameter bifurcation diagrams and their parallel computation. Analyzing one-parameter bifurcation diagrams in Figs. 4, 5, 6 and 9 one can notice the typical period doubling and halving bifurcations, for example, for a = 0.2045 and a = 0.212 in Fig. 5(b). In many intervals of parameter a there are chaotic responses. A peculiar bifurcation occurs for a = 0.229 (Figs. 5(b) and 5(c)) where a period-9 response for a < 0.229 becomes period-8 at a = 0.229, to switch back to period-9 for a > 0.229. Similar bifurcations happen for a = 0.1575 in Fig. 6(c) with the period-5 →period-4 →period5 changes (black diagram obtained with the initial condition [0, 1, 0]) and also for a = 0.1585 for the red diagram (obtained with the initial condition [0, −1, 0]). Interesting bifurcation happens at a = 0.2095 in Figs. 5(b) and 5(c), with the interacting attractors switching their maximum values of y (compare the black and red diagrams around a = 0.2095). Notice also, that in many intervals of a the initial conditions [0, 1, 0] and [0, −1, 0] yield different responses, as, for example, in the interval 0.1570 ≤ a ≤ 0.1518 in Fig. 6(c) (completely different black and red curves indicating in both cases period-5 responses). However, in other intervals of a, the two different initial conditions yield exactly the same responses, for example, the period-11 responses for 0.144 ≤ a ≤ 0.147 in Fig. 6(b). Notice also that a similar analysis of different limit cycles and coexisting attractors is slightly different in the case of two-parameter bifurcation diagrams, which identify the type of response only, i.e. period-n oscillations and chaotic responses without providing the maximum values of the analyzed 10
time response. We write more about that at the end of the next section. To provide the maximum values in two-parameter rectangular color diagrams would require three dimensional graphs in which two dimensions are used for the two bifurcation parameters, say k and a in (2), and the third dimension is for the meximum values of the analyzed oscillatory response. Such 3D color diagrams will neither be practical nor useful when presented on a flat screen or paper surface, which both are 2D in nature. 4. Two-parameter bifurcation diagrams This section presents two-parameter bifurcation diagrams when both a and k vary simultaneously. Such a change is equivalent to several possible changes of the circuit’s elements. For example, a single change of the value of R changes both a and b, but not k (since a = R2 C/L, b = 2ρRIs and k = R2 /R1 ). Also, by changing any of the pairs of values of (R2 , C1 ), (R2 , L), (R1 , C2 ) or (R1 , L) we change simultaneously a and k. Other scenarios are also possible. While one two-parameter bifurcation diagram with, say 600 × 600 discrete values of a and k in the area (amin ≤ a ≤ amax ; kmin ≤ k ≤ kmax ) is, in some sense, equivalent to 600 bifurcation diagrams with only one varying parameter, either a or k, that relation is not completely true, since from the one-paramter diagrams we not only determine the type of circuit’s response (i.e. period−n or chaotic), but also the values of n maxima in a period can be seen. Moreover, just two one-parameter bifurcation diagrams, each obtained with a different initial condition [x(0), y(0), z(0)] (say [0, 1, 0] and [0, −1, 0]) show interesting interactions of chaotic attractors of (2) that are not seen while using two-parameter bifurcation diagrams. Thus, it is always beneficial when using two-parameter diagrams to supplement them with one-paramters diagrams, particularly that the later are obtained in a tiny fraction of time needed to obtain the former. Details are provided below. 4.1. Software and hardware used Computation of two-parameter bifurcation diagrams was done via parallel computations performed on two processors Inter(R) Xeon(R) CPU ES-2640 v2 at 2.00 GHz. Each processor, made in the 22 nm technology, has 8 cores servicing 16 threads, using the Intel Turbo Boost 2.0. All calculations were done in the C language (compiled by the GCC compiler) using Ubuntu OS, which supports the OpenMP (Open Multi-Processing) [22].
11
The conventional fourth-order explicit Runge-Kutta (RK-4) method was used to solve (2) when computing the color two-parameter bifurcation diagrams. The time horizon was the same as in computing one-parameter diagrams (0 ≤ t ≤ 1000), while the time step in the RK-4 method was fixed at dt = 0.0001. The absolute error used in distinguishing two maximum values of variable y was 0.01, that is the identified maximum values with difference less than 0.01 were considered identical. Also, the maximum number n in the period−n type of oscillations was 64. To visualize those various period−n responses (n = 1, 2, . . . , 64) we used a continuous colorbar with colors ranging from dark brown (n = 1) to white (n = 64). All periodic responses with n ≥ 64 and chaotic ones were grouped together and assigned the white color (see the colorbars in each diagram in Figs. 7 and 8). Finally, all 360, 000 runs of the RK-4 solver needed to obtain each two-parameter bifurcation diagrams used the initial condition [0, 1, 0]. 4.2. Results of numerical experiments Figs. 7(a)-8(b) show four two-parameter bifurcation diagrams, each obtained with the 600 × 600 discrete values of parameters (a, k). Fig. 7(b) is a zoomed-in version of the diagram in rectangle A in Fig. 7(a). Similarly, Fig. 8(b) is the zoomed-in version of the diagram in rectangle B in Fig. 8(a). Fig. 5(a) showing a one-parameter bifurcation diagram corresponds to the line p in Fig. 7(a) for k = 2.25. Also, Fig. 6(a) shows a one-parameter bifurcation diagram along the line q in Fig. 7(a) for k = 2.20. The rather narrow intervals of parameter a with periodic responses (white gaps in the plots in Figs. 5(a) and 6(a)) correspond to the intersection of lines p and q, respectively, with the brown areas (of various intensity) in Fig. 7(a). The single line for approximately a > 3.4 in Fig. 6(a) indicates period−1 oscillations. This is confirm in Fig. 7(a) where the right end of line r falls onto the dark brown area of the two-parameter diagram. Diagrams shown in Figs. 6(c) and 6(d) can also be linked to the narrow intervals of parameter a along the line r in Fig. 7(a). Similarly, the one-parameter diagrams in Figs. 9(a) and 9(b) were taken along the lines r (for k = 2.40) and s (for k = 2.335), respectively (see Fig. 8(a)). Fig. 6(d) is worth commenting on, too. It shows period−4 response of the circuit for the middle range of paramter a and chaos elsewhere. Transition to and from the periodic responses is gradual, and not rapid (instant) as it occurs typically when a crisis of a chaotic attractor happens.
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(a) Two-parameter 600 × 600 diagram.
(b) Rectangle A in Fig.7(a); 600 × 600 points. Figure 7: Two-parameter bifurcation diagrams obtained with the RK-4 solver, time step 0.0001.
Although all two-parameter diagrams in Figs. 7 and 8 were obtained by using the initial condition [0, 1, 0], one would obtain exactly the same color diagrams for the initial condition [0, −1, 0], as the two-parameter diagrams show only the type of response of (2) and not the values of maximum of y, This is not the case for one-parameter diagrams as the maximum values of y are quite often different for the initial condition [0, 1, 0] than for [0, −1, 0], see Figs. 4(b), 5(c), 6(c) and 9(b).
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(a) Two-parameter 600 × 600 diagram.
(b) Rectangle B in Fig.8(a); 600 × 600 points. Figure 8: Two-parameter bifurcation diagrams obtained with the RK-4 solver, time step 0.0001.
5. Conclusions When a memristive bridge rectifier is added to a circuit based on the Sallen-Key topology, then interesting periodic and chaotic responses are obtained. Bifurcation diagrams (both one- and two-parameter ones) of the circuit show complicated dynamical responses that are sensitive to the changes of the circuit’s parameters. The use of two numerical solvers, the ode45 to compute one-parameter diagrams, and RK-4 to obtain two-parameter ones was done to show that the former complements and confirms the results obtained by the later. The one-parameter diagrams were obtained through a 14
(a) Bifurcation diagram for k = 2.40 (along line r in Fig. 8(a)).
(b) Bifurcation diagram for k = 2.335 (along line s in Fig. 8(a)).
(c) Bifurcation diagram for k = 2.335 (along line s in Fig. 8(a)) with different initial conditions. Figure 9: One-parameter bifurcation diagrams along the lines r and s in Fig. 8(a) with initial condition [0, 1, 0] (black) and [0, −1, 0] (red).
single processor sequential calculations, while the two-parameter color diagrams are the results of parallel computations. Distinguishing between periodic and chaotic responses is quite important in practice, as nonlinear oscillating circuits find more and more applications in, for example, secure transmission of information (data), designing random number generators, 15
chaotic cryptography and other areas. The main contribution of this paper are the two-parameter color bifurcation diagrams and the analysis of parallel computations through which the diagrams were obtained. One such two-parameter diagram provides a quick inside into the dynamics of system (2) with two changing parameters, instead of using tens or hundreds of diagrams with one varying parameter. It is very likely that other memristive emulators [23]-[26] or fractional memristors [27] when used as a replacement of the memristive diode bridge may also result in interesting bifurcations of the circuit in Fig. 1(b). Conversely, the same may happen if the memristive diode bridge in Fig. 1(a) is used as a memristive component in other chaotic circuits, for example, those discussed in [14],[28]-[30]. Computing bifurcation diagrams with one varying parameter is very common nowadays, and perhaps less common in the cases with two varying parameters. Significant increase in difficulty one may expect when three parameters vary simultaneously. Presenting the threeparameter bifurcation diagrams visually is one such difficulty and solving an ODE system, say, (103 )3 (billion) times would also be a problem. Finally, another topic worth examining is the use of the 0-1 test for chaos to be implemented when two parameters of a dynamical system vary. No such analysis and results for two-parameter bifurcations for the 0-1 test have been reported in the literature. However, the results for the 0-1 test for chaos applied to one-parameter bifurcations are encouraging [31]. Acknowledgements The authors would like to thank the four annonymous reviewers for their helpful and constructive comments on our paper. [1] B.-C. Bao, P. Wu, H. Bao, M. Chen and Q. Xu, “Chaotic bursting in memristive diode bridge-coupled Sallen-Key lowpass filter,” Elect. Lett., vol. 53, pp. 1104–1105, 2017. [2] J. Sadecki and W. Marszalek, “Analysis of a memristive diode bridge rectifier,” Elect. Lett., DOI: 10.1049/el.2018.6921, 2018. [3] W. Marszalek and H. Podhaisky, “2D bifurcations and Newtonian properties of memristive Chua’s circuits,” Europhysics Lett. (EPL), vol. 113, 10005, 2016.
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