- Email: [email protected]
Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator
Chaos, Solitons and Fractals 109 (2018) 146–153
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Review
Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator B.C. Bao, P.Y. Wu, H. Bao, H.G. Wu, X. Zhang, M. Chen∗ School of Information Science and Engineering, Changzhou University, Changzhou 213164, China
a r t i c l e
i n f o
Article history: Received 22 January 2018 Revised 20 February 2018 Accepted 23 February 2018
Keywords: Symmetric periodic bursting Memristive oscillator Bifurcation sets Bifurcation mechanism
a b s t r a c t This paper presents a novel third-order autonomous memristive diode bridge-based oscillator with fastslow effect. Based on the modeling of the presented memristive oscillator, stability of the equilibrium point is analyzed by using the eigenvalues of the characteristic polynomial, and then symmetric periodic bursting behavior is revealed through bifurcation diagrams, phase plane plots, time sequences, and 0–1 test. Furthermore, bifurcation mechanism of the symmetric periodic bursting behavior is explored by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the variations of the system parameter and slow-scale variable. Consequently, the presented memristive oscillator is always unstable and exhibits complex dynamical behavior of symmetric periodic bursting oscillations with a symmetric fold/Hopf cycle-cycle burster. In addition, experimental measurements are performed by hardware circuit to confirm the numerical simulations. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Based on multifarious memristor emulators using commercially available components, various kinds of memristor- based chaotic circuits have been physically implemented for breadboard experimental studies in recent years [1–10], which effectively promotes the development of the fundamentals of memristive circuits. Generally, the memristor emulators mainly have two types of the implementation circuits with the off-the-shelf discrete components, namely, the op-amps and analog multipliers-based equivalent circuits [1–4] and memristive diode bridges cascaded with L, RC, or LC components [5–10]. Like as the constructing method of memristor-based chaotic circuits reported in [1–10], this paper presents a novel third-order autonomous memristive oscillator, which is extended from the memristive Wien-bridge oscillator [8] or memristive diode bridge-coupled Sallen-Key low-pass filter [9,10]. However, in most of the previous reports except Ref. [8], all the state variables of the memristive circuits are specified on the same time scale. In this paper, the presented third-order memristive oscillator can be considered as two subsystems to have order gap between the two time scales related to the two subsystems [11,12]. Particularly, it should be clarified that the newly presented autonomous memristive oscillator is only third-order but
∗
Corresponding author. E-mail address: [email protected] (M. Chen).
https://doi.org/10.1016/j.chaos.2018.02.031 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
the memristive Wien-bridge oscillator reported in [8] is fourthorder, resulting in the briefness and easiness of the mathematical model and quantitative analyses. Recently, based on some special memristor emulators or pure mathematical memristor models, several memristor-based spiking and bursting neuron circuits have been developed [13–15], which denote that these memristor-based application circuits can exhibit the spiking and bursting firing behaviors with a biologically plausible spike shapes [13,16]. However, the hardware experimental level on a breadboard is hard to fabricate due to the complexity of these memristor emulators or pure mathematical memristor models. The spiking and bursting oscillations are often encountered in a large class of neuronal models [16–21] and nonlinear dynamical systems [11,12,22–26]. These neuronal models and nonlinear dynamical systems usually involve two time scales, leading to the occurrence of the bursting firing behaviors since the fast-scale variables are modulated by the slow-scale variables. Similarly, with the appropriate parameters, the presented third-order memristive oscillator referring to two time scales also can exhibit the novel phenomenon of symmetric periodic bursting oscillations as well, which, to the authors’ knowledge, is rarely appeared in the previously published literatures [8]. In particular, the symmetric periodic bursting phenomenon emerging in such a third-order memristive oscillator is very interesting, which can be taken as a paradigm in mathematical and experimental demonstrations of some special nonlinear dynamics.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
1 d i0 1 = ln(2IS cosh σ v1 ) − ln(i0 + 2IS ) dt σL σL dv1 kv1 (i0 + 2IS ) tanh σ v1 v2 = − − dt RC1 C1 RC1 dv2 kv1 v2 = − dt RC2 RC2
147
(2)
where k = R2 /R1 . By utilizing three dimensionless variables and four dimensionless parameters as
x=
σ Ri0 , y = σ v1 , z = σ v2 , t = RC1 τ ,
a = R2C1 /L, b = 2σ RIS , ε = C1 /C2 , k = R2 /R1 Fig. 1. Third-order autonomous memristive oscillator: (a) memristive diode bridgebased oscillator, (b) memristive diode bridge cascaded with an inductor.
The paper content is constructed as follows. In Sect. 2, based on a simplified first-order memristive diode bridge, a novel thirdorder autonomous memristive oscillator is presented and its corresponding three-dimensional system model is then built. Moreover, with the eigenvalues of the characteristic polynomial, stability of the equilibrium point is analyzed. In Sect. 3, by using several conventional dynamical methods, period oscillations and symmetric periodic bursting behaviors are revealed. In the following Sect. 4, based on the fast-scale and slow-scale subsystems divided from the three-dimensional system model, the evolutions of the equilibrium point and its stability of the fast-scale subsystem are discussed and its bifurcation mechanism is explored through constructing the fold and Hopf bifurcation sets of the system parameter and slow variable. In Sect. 5, hardware circuit experiments are performed to confirm the numerical simulations. The conclusions are summarized in the last section.
(3)
A normalized system model is thus built from (2) as
x˙ = a ln(b cosh y ) − a ln(x + b) y˙ = ky − (x + b) tanh y − z z˙ = ε (ky − z )
(4)
Therefore, the newly presented autonomous memristive oscillator is a three-dimensional nonlinear dynamical system, which has much simpler algebraic equations than the fourth-order memristive Wien-bridge oscillator reported in [8], more suitable for the quantitative analyses. The desired circuit parameters of linear elements in Fig. 1 are chosen as L = 20 mH, C1 = 1 nF, C2 = 33 nF, R = 30 k, R1 = 2 k, and R2 = 6 k, and the considered system parameters of (4) are then calculated by (3) as
a = 45, b = 0.0036, ε = 0.0303, and k = 3
(5)
Note that the inductor L and capacitors C1 and C2 considered in Fig. 1 are three differentiable elements, which are completely different from the non-differentiable elements reported in [28].
2. Third-order autonomous memristive oscillator
2.2. Stability for the equilibrium point
A novel third-order autonomous memristive oscillator is presented, as shown in Fig. 1(a), by introducing a first- order memristive diode bridge emulator only cascaded with an inductor, as shown in Fig. 1(b), to substitute the resistor of the parallel RC network in Wien-bridge oscillator [27]. Also, the proposed memristive oscillator can be derived from the fourth-order memristive Wien-bridge oscillator reported in [8] through simplifying the LC network in second-order memristive diode bridge emulator by an inductor or the third-order memristive diode bridge-coupled Sallen-Key low-pass filter reported in [9,10] through removing the grounded resistor. It is remarked that the third-order memristive diode bridge-based oscillator has the same topological structure as the traditional Wien-bridge oscillator.
The system (4) has only one original equilibrium point (0, 0, 0). The characteristic polynomial of the system Jacobian at (0, 0, 0) is given by
2.1. State equations and normalized model For the emulator input voltage v1 and current i along with the inductor L current i0 , as shown in Fig. 1(b), the memristive diode bridge emulator is modeled by [9,10]
i = W (i0 , v1 )v = (i0 + 2IS ) tanh σ v1 1 d i0 1 = ln(2IS cosh ρv1 ) − ln(i0 + 2IS ) dt σL σL
(1)
where σ = 1/(2nVT ). Three considered model parameters of 1N4148 diode are assigned as IS = 5.84 nA, n = 1.94, and VT = 25 mV, respectively. With the two capacitor voltages v1 and v2 and one inductor current i0 in Fig. 1, the circuit equations of the third-order autonomous memristive diode bridge-based oscillator are established as
P (λ ) = (λ + a/b)[λ2 − (k − b − ε )λ + bε ] = 0
(6)
Thus, three eigenvalues are yielded as
λ1 = 0.5(k − b − ε ) + 0.5 (k − b − ε )2 − 4bε λ2 = 0.5(k − b − ε ) − 0.5 (k − b − ε )2 − 4bε λ3 = −a/b
(7)
As k is considerably larger than b and ɛ, the roots of λ1 and λ2 are positive real constants, whereas the root of λ3 is negative real constant, which indicate that the original equilibrium point (0, 0, 0) of (4) is an unstable saddle point. Considering the typical system parameters given in (5) as an example, three eigenvalues in (7) are calculated as λ1 = 2.9661, λ2 = 0.0 0 0 04, and λ3 = − 12,50 0. Consequently, the third-order autonomous memristive diode bridgebased oscillator is always unstable. 3. Symmetric periodic bursting behavior By utilizing MATLAB software tool, ODE23S algorithm is considered to draw bifurcation diagrams, phase plane plots, time sequences, and 0–1 test in our next works. Due to the existence of complex nonlinearities in system (4), pure mathematical analysis methods, for instance, integral transform methods proposed in [29–32], are inapplicable to the solutions of system (4), leading to that some computer-aided simulation tools should be utilized.
148
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
Fig. 2. Dynamics versus the resistance R: (a) bifurcation diagram, (b) partial enlarged view of (a), where ɛ = 0.0303 and k = 3 are fixed, while a and b are synchronously adjusted with the variation of the resistance R.
Fig. 3. Numerically periodic behaviors by using time sequences of y and z for different values of R: (a) R = 23 k, (b) R = 25 k, (c) R = 25.92 k, (d) R = 25.96 k.
Taking the resistance R as a bifurcation parameter, i.e., the system parameters a and b being synchronously adjustable, the bifurcation diagrams of the local maxima X of the state variable x of system (4) are plotted in Figs. 2(a) and 2(b), respectively, where Fig. 2(b) is the partial enlarged view of Fig. 2(a). It should be mentioned that the first Lyapunov exponent is always less than or equal to zero, indicating the inexistence of chaotic behavior in the third-order memristive diode bridge-based oscillator. As the adjusting resistance R is changed from 20 k to 50 k, the system orbit starting from the period 1 oscillation successively enters into multiple types of periodic oscillations with period 4, 6, and 8 at R = 24.06 k, 25.90 k, and 25.94 k, respectively. Beyond the region of the periodic oscillations, the system orbit suddenly shifts into symmetric periodic bursting oscillations with uncountable periodicities at R = 26.01 k, and then the periodicities and dynamic amplitudes of the system orbit will gradually shrink with the increase of the resistance R in a wider parameter region.
In the region of 20 k ≤ R < 26.01 k, only some simple periodically oscillating behaviors with period-adding routes are displayed in Fig. 2. Correspondingly, four types of time sequences of the variables y and z for four different values of R are depicted in Fig. 3, which demonstrate that the symmetric block structured dynamics exists in this parameter adjustable region [33]. Additionally, the sequence of the variable y in Fig. 3(a) is analogous to the bipolar spike firing, whereas the sequences of y in Figs. 3(b), 3(c), and 3(d) are similar to the bipolar burst firing [18]. However, in the region of 26.01 k ≤ R ≤ 50 k, complex symmetric periodic bursting behaviors with the fast-slow effects are exhibited in Fig. 2. When R = 30 k, an example of the symmetric periodic bursting dynamics is illustrated in Fig. 4, where Figs. 4(a) and 4(b) display two phase plane plots in the z − y and y − x planes, respectively, Fig. 4(c) shows the overlapped sequences of the fast variable y and slow variable z, and Fig. 4(d) given the p − q dynamics that behaves a grotesquely flywheel-shaped torus with
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
149
Fig. 4. Symmetric periodic bursting behavior by MATLAB numerical simulations at R = 30 k, where a = 45, b = 0.0036, ɛ = 0.0303, and k = 3: (a) phase plane plot in the z−y plane, (b) phase plane plot in the y−x plane, (c) time sequences of y and z, (d) p–q periodic bursting dynamics using 0–1 test.
Since ɛ = 0.0303, the system (4) can be considered as two subsystems with the order gap between the two time scales. The fastscale subsystem is described by the first two equations of (4) as
x˙ = a ln(b cosh y ) − a ln(x + b) (8a)
and the slow-scale subsystem is expressed by the third equation of (4) as
z˙ = ε (ky − z )
(8b)
4.1. Evolutions of the equilibrium point and its stability For the fast-scale subsystem in (8a), the equilibrium points are determined as EFS = (b cosh y¯ − b, y¯ ), in which y¯ can be numerically solved by
G1 = ky¯ − b sinh y¯ − z = 0
*
h2 = z = 19.2455 h2 = z = 0 * *
h1 *
h2 = z = –19.2455 *
* y
Fig. 5. Evolution of the equilibrium points of the fast-scale subsystem illustrated by the crossing points of two curves.
Differentiating G1 with respect to y¯ , there yields
4. Bifurcation mechanism of the fast-scale subsystem
y˙ = ky − (x + b) tanh y − z
*
h1, h2
a 0–1 test median value close zero [34–36]. Additionally, three Lyapunov exponents are calculated as L1 = 0.0, L2 = − 0.5164, and L3 = − 13.6290, respectively, which further declare that the moving trajectory of the symmetric periodic bursting behavior shown in Fig. 4 is in a periodic pattern with large number of periodicities. The numerical results shown in Fig. 4 denote that the symmetric periodic bursting behavior appears in such a third-order autonomous memristive oscillator. Furthermore, it is noted that the variables y and z in system (4) have the order difference between the two non-dimensional time scales due to the appearance of the very small controlling strength ɛ in third equation, leading to that the three-dimensional system of (4) may be considered as the coupling of two subsystems of the two-dimensional fast-scale subsystem and one-dimensional slow-scale subsystem.
(9)
G2 = k − b cosh y¯ = 0
(10)
which has two roots, obtained as
y¯ 1 = arccosh kb , y¯ 2 = −arccosh kb
(11)
At these roots y¯ 1 and y¯ 2 , G1 has two extreme values, which can be given by
G1 (y¯ 1 ) = k arccosh
k b
− b sinh(arccosh kb ) − z
G1 (y¯ 2 ) = b sinh(arccosh kb ) − k arccosh
k b
−z
(12)
respectively. Take the specified system parameters listed in (5) as an example. Considering that
h1 = ky¯ − b sinh y¯ , h2 = z
(13)
the values of y¯ can be then determined by the crossing points of two function curves of (13), as shown in Fig. 5. When z > 19.2455 or z < − 19.2455, i.e., G1 (y¯ 1 ) < 0orG1 (y¯ 2 ) > 0, only one crossing point exists; when − 19.2455 < z < 19.2455, i.e., G1 (y¯ 2 ) < 0 <
150
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
R = 45 kΩ *H H*4 F4* 3 HB2 HB1 FB1 FB2
R, kΩ
*F 3
*F
1
H*2
*H 1
GND
R
F2*
U
R2
R = 30 kΩ
R1
C2
z
C1 Second-order autonomous oscillator
Fig. 6. Fold and Hopf bifurcation sets of the fast-scale subsystem distributed in the z – R phase plane.
G1 (y¯ 1 ), three crossing points appear; while when z = 19.2455 or z = −19.2455, i.e., G1 (y¯ 1 ) = 0orG1 (y¯ 2 ) = 0, one tangent point along with one crossing point occur, leading to the generation of fold bifurcation [11]. To estimate the stability of the equilibrium point EFS , the Jacobian matrix at EFS is derived and simplified as
JFS
− ab sech y¯ = − tanh y¯
a tanh y¯ k − bsech y¯
(14)
The corresponding characteristic polynomial is deduces as
P (λ ) = λ2 + αλ + β = 0
(15)
y¯ − k] and β = a − y¯ . With the where α = [(b + characteristic polynomial (15), bifurcation analyses of the fast-scale subsystem can be performed. It should be remarkable that the coefficients α and β of the characteristic polynomial (15) are only determined by the solutions of the equilibrium point Eq. (9), whose values are closely associated with the slow variable z. Therefore, the stability of the fastscale subsystem is dependent to the slow-scale subsystem. a )sech b
ka sech b
4.2. Fold and Hopf bifurcation sets
D4
D1
D3
D2
L Memristive diode bridge Fig. 8. Screen capture of the experimental prototype for the presented third-order autonomous memristive diode bridge-based oscillator, the left is a global graph of the oscilloscope linking to the circuit breadboard and the right is the partial enlarged drawing of the circuit breadboard.
i.e., β = 0. Therefore, with the Eq. (9), the fold bifurcation set can be deduced as
ky¯ − b sinh y¯ − z = 0 a − ka sech y¯ = 0 b
FB :
and simplified as
FB : ± k arccosh
k b
(17)
− b sinh arccosh
k b
−z=0
(18)
The Hopf bifurcation is associated with the appearance of a pair of pure imaginary eigenvalues. Then, (15) can be expressed as
PHB (λ ) = λ2 + β = 0
Observed from Fig. 5, the equilibrium points of the fast-scale subsystem have a transition from three to one with the variation of the slow variable z. At the critical condition of G1 (y¯ 1 ) = 0orG1 (y¯ 2 ) = 0, a small perturbation of z may cause the tangent point to disappear or to split into two crossing points, resulting in the occurrence of fold bifurcation. Based on the critical condition, the fold bifurcation set can be easily derived. Alternatively, in consideration of that the fold bifurcation is referred to one zero eigenvalue, (14) can be thereby rewritten as
PFB (λ ) = λ(λ + α ) = 0
+15 V
−15 V
(16)
(19)
i.e., α = 0. Therefore, with the Eq. (9), the Hopf bifurcation set can be gotten as
HB :
k y¯ − b sinh y¯ − z = 0 (b + ab )sech y¯ − k = 0
and simplified as
HB : ± k arccosh
a/b+b k
(20)
− b sinh arccosh
a/b+b k
−z=0
(21)
Consider that the resistance R is adjusted in the region [20 k, 50 k]. By MATLAB numerical simulations, the fold bifurcation
Fig. 7. Bifurcation mechanism of periodic bursters at (a) R = 30 k and (b) R = 45 k, where F1 , F2 , F3 , and F4 stand for the fold bifurcation points, H1 , H2 , H3 , and H4 represent the Hopf bifurcation points, and EP1 and EP2 are the equilibrium point curves of the fast-scale subsystem with the z variation.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
151
Fig. 9. Experimentally periodic behaviors by using time sequences of v1 (t) and v2 (t) for different R: (a) R = 21.5 k, (b) R = 21.8 k.
Fig. 10. Symmetric periodic bursting behaviors by hardware circuit experiments: (a1) phase plane plot in the v1 (t) − v2 (t) plane and (a2) time sequences of v1 (t) and v2 (t) at R = 26.6 k, (b1) phase plane plot in the v1 (t) − v2 (t) plane and (b2) time sequences of v1 (t) and v2 (t) at R = 41.5 k.
set of (18) (marked as FB1 and FB2 ) and Hopf bifurcation set of (21) (marked as HB1 and HB2 ) in the z – R phase plane can be together drawn, as shown in Fig. 6. The bifurcation sets in Fig. 6 are the possible bifurcations of the fast-scale subsystem, which may be correspondence to the turning points between the quiescent state and spiking state when periodic bursting oscillation takes place in the system (4).
4.3. Bifurcation mechanism of periodic bursters Let R = 30 k and R = 45 k be two examples using for explaining the mechanism of the periodic bursters. Referring to Fig. 6, when R = 30 k, two fold bifurcation points locate at z = ± 19.2455 V, marked as F2 and F1 , whereas two Hopf bifurcation points are at z = ± 12.0738 V, marked as H2 and H1 . The phase plane plot of system (4) and the equilibrium point curve EP1
152
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
of the fast-scale subsystem dotted by the four bifurcation points are depicted in Fig. 7(a). The moving trajectory of the system (4) beginning at H1 gradually enters into the quiescent state with the decrease of the oscillating amplitude. The quiescent state ends once the trajectory moves at the fold bifurcation point F1 , leading to that the trajectory rises to the upper and the system (4) goes into the spiking stat The spiking oscillation around the equilibrium point curve EP1 is maintained with the large oscillating amplitude. After the Hopf bifurcation occurs at H2 , the moving trajectory re-enters into the quiescent state and dives to the lower via the fold bifurcation route at F2 . The spiking state appears again and ends once the trajectory moves at the Hopf bifurcation point H1 , which initiates next period of the periodic burster. Due to the small oscillation of the quiescent state, the periodic burster can be regarded as a symmetric fold/Hopf cycle-cycle burster [16]. Similarly, when R = 45 k, two fold bifurcation points locate at z = ± 18.0291, marked as F4 and F 3 , whereas two Hopf bifurcation points are at z = ± 5.4598, marked as H3 and H4 . The phase plane plot of the system (4) and the equilibrium point curve EP2 of the fast-scale subsystem dotted by the four bifurcation points are demonstrated in Fig. 7(b). By utilizing the bifurcation analysis in the above section, analogous bifurcation details of EP2 are encountered for this case, i.e., the system (4) has a transition from the quiescent state to spiking state and back via the fold and Hopf bifurcation routes. It should be addressed that the spanning time length of the quiescent oscillation increases with the resistance R.
6. Conclusion
5. Hardware experimental confirmations
References
According to the circuit designed in Fig. 1, an experimental prototype of the presented third-order autonomous memristive diode bridge-based oscillator is photographed and its screen capture is shown in Fig. 8. Three precision potentiometers, two monolithic ceramic capacitors, four 1N4148 diodes, a manually winding inductor, and an AD711KN op-amp with ± 15 V power supply are chosen. The linear element parameters used during numerical simulations are selected in hardware experiments and the experimental phase plane orbits and time sequences are measured by a 4 channel digital oscilloscope. By turning the precision potentiometer R, different periodic behaviors with different periodicities or periodic bursters are measured from the experimental prototype. When R is located in the region of smaller resistances, the memristive diode bridgebased oscillator operates in periodic oscillating states. For instance, when R = 21.5 k and 21.8 k, the time sequences of the voltage variables v1 (t) and v2 (t) for two types of the periodic behaviors are experimentally captured, as shown in Fig. 9. Whereas when R is fallen into the region of larger resistances, the memristive diode bridge-based oscillator works in symmetric periodic bursting states. Regarding R = 26.6 k and 41.5 k as examples, the phase plane plots in the v1 (t) − v2 (t) plane and time sequences of the voltage variables v1 (t) and v2 (t) for two types of the symmetric periodic bursting behaviors are experimentally captured, as shown in Fig. 10. There are some relatively larger differences between the values of R used in numerical simulations and hardware experiments, which are mainly caused by the equivalent series resistance (ESR) of the manually winding inductor. Ignoring these resistance differences, it can be observed that the experimentally captured results in Figs. 9 and 10 are well agreed with the numerically simulated results in Figs. 3, 4, and 7.
[1] Iu HHC, Yu D, Fitch AL, Sreeram V, Chen H. Controlling chaos in a memristor based circuit using a twin-T notch filter. IEEE Trans Circuits Syst I 2011;58(6):1337–44. [2] Xu Q, Lin Y, Bao B, Chen M. Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solit Fractals 2016;83:186–200. [3] Bao B, Wang N, Xu Q, Wu H, Hu Y. A simple third-order memristive band pass filter chaotic circuit. IEEE Trans Circuits Syst II 2017;64(8):977–81. [4] Wu R, Wang C. A new simple chaotic circuit based on memristor. Int J Bifurcation Chaos 2016;26(9):1650145. [5] Kengne J, Tabekoueng ZN, Tamba VK, Negou AN. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos 2015;25(10):103126. [6] Xu Q, Wang N, Bao B, Chen M, Li C. A feasible memristive Chua’s circuit via bridging a generalized memristor. J Appl Anal Comput 2016;6(4):1152–63. [7] Tabekoueng ZN, Kengne J, Fotsin HB, Negou AN, Tchiotsop D. Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit. Chaos Solit Fractals 2016;91:180–97. [8] Wu H, Bao B, Liu Z, Xu Q, Jiang P. Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn 2016;83(1):893–903. [9] Bao B, Wu P, Bao H, Chen M, Xu Q. Chaotic bursting in memristive diode bridge coupled Sallen-Key low-pass filter. Electron Lett 2017;53(16):1104–5. [10] Bao B, Wu P, Bao H, Xu Q, Chen M. Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator. Chaos Solit Fractals 2018;106:161–70. [11] Bi Q, Ma R, Zhang Z. Bifurcation mechanism of the bursting oscillations in periodically excited dynamical system with two time scales. Nonlinear Dyn 2015;79(1):101–10. [12] Bi Q, Zhang Z. Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales. Phys Lett A 2011;375(8):1183–90. [13] Babacan Y, Kaçar F, Gürkan K. A spiking and bursting neuron circuit based on memristor. Neurocomputing 2016;203:86–91. [14] Hu M, Chen Y, Yang JJ, Wang Y, Li H. A compact memristor-based dynamic synapse for spiking neural networks. IEEE Trans Comp Aid Des Int 2017;36(8):1353–66. [15] Feali MS, Ahmadi A. Transient response characteristic of memristor circuits and biological-like current spikes. Neural Comp Appl 2017;28(11):3295–305. [16] Izhikevich EM. Neural excitability, spiking and bursting. Int J Bifurcation Chaos 20 0 0;10(6):1171–266. [17] FröHlich F, Bazhenov M. Coexistence of tonic firing and bursting in cortical neurons. Phys Rev E 2006;74(1):031922.
A novel third-order autonomous memristive diode bridge-based oscillator with two time scales is designed, which has simple circuit topology and only consists of one op-amp, two capacitors, three resistors, four diodes, and one inductor. However, the presented memristive oscillator is always unstable and can exhibit an interesting symmetric periodic bursting behavior with a symmetric fold/Hopf cycle-cycle burster by MATLAB numerical simulations and hardware experimental confirmations. Through constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the evolution of the system parameter and slow-scale variable, the bifurcation mechanism of the symmetric periodic bursting behavior is theoretically and numerically explored, which demonstrates that the periodic bursting oscillation switching from a quiescent state to a spiking state and back via the fold and Hopf bifurcation routes are a symmetric fold/Hopf cycle-cycle bursting oscillation. The presented third-order autonomous memristive diode bridge-based oscillator is inexpensive and easy to practically realize, which can be taken as an example with a special dynamical phenomenon for the in-depth investigations of this kind of memristor-based dynamical circuits. Acknowledgements This research issue was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 51777016, 61601062, 61705021, 11602035, and 51607013, the Natural Science Foundations of Jiangsu Province, China under Grant No. BK20160282, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province, China under Grant No. KYCX17_2083.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153 [18] Ngouonkadi EBM, Fotsin HB, Fotso PL, Tamba VK, Cerdeira HA. Bifurcations and multistability in the extended Hindmarsh–Rose neuronal oscillator. Chaos Solit Fractals 2016;85:151–63. [19] Innocenti G, Morelli A, Genesio R, Torcini A. Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 2007;17(4):043128. [20] Malashchenko T, Shilnikov A, Cymbalyuk G. Six types of multistability in a neuronal model based on slow calcium current. Plos One 2011;6(7):e21782. [21] Ma J, Tang J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn 2017;89(3):1569–78. [22] Kingni ST, Nana B, Ngueuteu GSM, Woafo P, Danckaert J. Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect. Chaos Solit Fractals 2015;71:29–40. [23] Zhang Z, Liu B, Bi Q. Non-smooth bifurcations on the bursting oscillations in a dynamic system with two time scales. Nonlinear Dyn 2015;79(1):195–203. [24] Bi Q, Li S, Kurths J, Zhang Z. The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures. Nonlinear Dyn 2016;85(2):993–1005. [25] Zheng S, Han X, Bi Q. Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system. Commun Nonlinear Sci Numer Si 2011;16(4):1998–2005. [26] Zhou G, Bao B, Xu J. Complex dynamics and fast-slow scale instability in current-mode controlled buck converter with constant current load. Int J Bifurcation Chaos 2013;23(4):1350062.
153
[27] Talukdar A, Radwan AG, Salama KN. Generalized model for memristor-based Wien family oscillators. Microelectron J 2011;42(9):1032–8. [28] Yang X, Machado JAT, Cattani C, Gao F. On a fractal LC-electric circuit modeled by local fractional calculus. Commun Nonlinear Sci Numer Si 2017;47:200–6. [29] Yang X. A new integral transform with an application in heat-transfer problem. Therm Sci 2016;20:S677–81. [30] Yang X. A new integral transform operator for solving the heat-diffusion problem. Appl Math Lett 2017;64:193–7. [31] Yang X. New integral transforms for solving a steady heat transfer problem. Therm Sci 2017;21:S79–87. [32] Yang X, Yang Y, Cattani C, Zhu M. A new technique for solving the 1-D Burgers equation. Therm Sci 2017;21:S129–36. [33] González-Miranda JM. Block structured dynamics and neuronal coding. Phys Rev E 2005;72(1):051922. [34] Kim Y. A numerical study of the local K spectrum of strange nonchaotic attractors. J Korean Phys Soc 2014;64(1):6–10. [35] Savi MA, Pereira-Pinto FHI, Viola FM, Paula ASD, Bernardini D, Litak G, Rega G. Using 0-1 test to diagnose chaos on shape memory alloy dynamical systems. Chaos Solit Fractals 2017;103:307–24. [36] Gottwald GA, Melbourne I. On the validity of the 0-1 test for chaos. Nonlinearity 2009;22:1367–82.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Review
Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator B.C. Bao, P.Y. Wu, H. Bao, H.G. Wu, X. Zhang, M. Chen∗ School of Information Science and Engineering, Changzhou University, Changzhou 213164, China
a r t i c l e
i n f o
Article history: Received 22 January 2018 Revised 20 February 2018 Accepted 23 February 2018
Keywords: Symmetric periodic bursting Memristive oscillator Bifurcation sets Bifurcation mechanism
a b s t r a c t This paper presents a novel third-order autonomous memristive diode bridge-based oscillator with fastslow effect. Based on the modeling of the presented memristive oscillator, stability of the equilibrium point is analyzed by using the eigenvalues of the characteristic polynomial, and then symmetric periodic bursting behavior is revealed through bifurcation diagrams, phase plane plots, time sequences, and 0–1 test. Furthermore, bifurcation mechanism of the symmetric periodic bursting behavior is explored by constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the variations of the system parameter and slow-scale variable. Consequently, the presented memristive oscillator is always unstable and exhibits complex dynamical behavior of symmetric periodic bursting oscillations with a symmetric fold/Hopf cycle-cycle burster. In addition, experimental measurements are performed by hardware circuit to confirm the numerical simulations. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Based on multifarious memristor emulators using commercially available components, various kinds of memristor- based chaotic circuits have been physically implemented for breadboard experimental studies in recent years [1–10], which effectively promotes the development of the fundamentals of memristive circuits. Generally, the memristor emulators mainly have two types of the implementation circuits with the off-the-shelf discrete components, namely, the op-amps and analog multipliers-based equivalent circuits [1–4] and memristive diode bridges cascaded with L, RC, or LC components [5–10]. Like as the constructing method of memristor-based chaotic circuits reported in [1–10], this paper presents a novel third-order autonomous memristive oscillator, which is extended from the memristive Wien-bridge oscillator [8] or memristive diode bridge-coupled Sallen-Key low-pass filter [9,10]. However, in most of the previous reports except Ref. [8], all the state variables of the memristive circuits are specified on the same time scale. In this paper, the presented third-order memristive oscillator can be considered as two subsystems to have order gap between the two time scales related to the two subsystems [11,12]. Particularly, it should be clarified that the newly presented autonomous memristive oscillator is only third-order but
∗
Corresponding author. E-mail address: [email protected] (M. Chen).
https://doi.org/10.1016/j.chaos.2018.02.031 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
the memristive Wien-bridge oscillator reported in [8] is fourthorder, resulting in the briefness and easiness of the mathematical model and quantitative analyses. Recently, based on some special memristor emulators or pure mathematical memristor models, several memristor-based spiking and bursting neuron circuits have been developed [13–15], which denote that these memristor-based application circuits can exhibit the spiking and bursting firing behaviors with a biologically plausible spike shapes [13,16]. However, the hardware experimental level on a breadboard is hard to fabricate due to the complexity of these memristor emulators or pure mathematical memristor models. The spiking and bursting oscillations are often encountered in a large class of neuronal models [16–21] and nonlinear dynamical systems [11,12,22–26]. These neuronal models and nonlinear dynamical systems usually involve two time scales, leading to the occurrence of the bursting firing behaviors since the fast-scale variables are modulated by the slow-scale variables. Similarly, with the appropriate parameters, the presented third-order memristive oscillator referring to two time scales also can exhibit the novel phenomenon of symmetric periodic bursting oscillations as well, which, to the authors’ knowledge, is rarely appeared in the previously published literatures [8]. In particular, the symmetric periodic bursting phenomenon emerging in such a third-order memristive oscillator is very interesting, which can be taken as a paradigm in mathematical and experimental demonstrations of some special nonlinear dynamics.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
1 d i0 1 = ln(2IS cosh σ v1 ) − ln(i0 + 2IS ) dt σL σL dv1 kv1 (i0 + 2IS ) tanh σ v1 v2 = − − dt RC1 C1 RC1 dv2 kv1 v2 = − dt RC2 RC2
147
(2)
where k = R2 /R1 . By utilizing three dimensionless variables and four dimensionless parameters as
x=
σ Ri0 , y = σ v1 , z = σ v2 , t = RC1 τ ,
a = R2C1 /L, b = 2σ RIS , ε = C1 /C2 , k = R2 /R1 Fig. 1. Third-order autonomous memristive oscillator: (a) memristive diode bridgebased oscillator, (b) memristive diode bridge cascaded with an inductor.
The paper content is constructed as follows. In Sect. 2, based on a simplified first-order memristive diode bridge, a novel thirdorder autonomous memristive oscillator is presented and its corresponding three-dimensional system model is then built. Moreover, with the eigenvalues of the characteristic polynomial, stability of the equilibrium point is analyzed. In Sect. 3, by using several conventional dynamical methods, period oscillations and symmetric periodic bursting behaviors are revealed. In the following Sect. 4, based on the fast-scale and slow-scale subsystems divided from the three-dimensional system model, the evolutions of the equilibrium point and its stability of the fast-scale subsystem are discussed and its bifurcation mechanism is explored through constructing the fold and Hopf bifurcation sets of the system parameter and slow variable. In Sect. 5, hardware circuit experiments are performed to confirm the numerical simulations. The conclusions are summarized in the last section.
(3)
A normalized system model is thus built from (2) as
x˙ = a ln(b cosh y ) − a ln(x + b) y˙ = ky − (x + b) tanh y − z z˙ = ε (ky − z )
(4)
Therefore, the newly presented autonomous memristive oscillator is a three-dimensional nonlinear dynamical system, which has much simpler algebraic equations than the fourth-order memristive Wien-bridge oscillator reported in [8], more suitable for the quantitative analyses. The desired circuit parameters of linear elements in Fig. 1 are chosen as L = 20 mH, C1 = 1 nF, C2 = 33 nF, R = 30 k, R1 = 2 k, and R2 = 6 k, and the considered system parameters of (4) are then calculated by (3) as
a = 45, b = 0.0036, ε = 0.0303, and k = 3
(5)
Note that the inductor L and capacitors C1 and C2 considered in Fig. 1 are three differentiable elements, which are completely different from the non-differentiable elements reported in [28].
2. Third-order autonomous memristive oscillator
2.2. Stability for the equilibrium point
A novel third-order autonomous memristive oscillator is presented, as shown in Fig. 1(a), by introducing a first- order memristive diode bridge emulator only cascaded with an inductor, as shown in Fig. 1(b), to substitute the resistor of the parallel RC network in Wien-bridge oscillator [27]. Also, the proposed memristive oscillator can be derived from the fourth-order memristive Wien-bridge oscillator reported in [8] through simplifying the LC network in second-order memristive diode bridge emulator by an inductor or the third-order memristive diode bridge-coupled Sallen-Key low-pass filter reported in [9,10] through removing the grounded resistor. It is remarked that the third-order memristive diode bridge-based oscillator has the same topological structure as the traditional Wien-bridge oscillator.
The system (4) has only one original equilibrium point (0, 0, 0). The characteristic polynomial of the system Jacobian at (0, 0, 0) is given by
2.1. State equations and normalized model For the emulator input voltage v1 and current i along with the inductor L current i0 , as shown in Fig. 1(b), the memristive diode bridge emulator is modeled by [9,10]
i = W (i0 , v1 )v = (i0 + 2IS ) tanh σ v1 1 d i0 1 = ln(2IS cosh ρv1 ) − ln(i0 + 2IS ) dt σL σL
(1)
where σ = 1/(2nVT ). Three considered model parameters of 1N4148 diode are assigned as IS = 5.84 nA, n = 1.94, and VT = 25 mV, respectively. With the two capacitor voltages v1 and v2 and one inductor current i0 in Fig. 1, the circuit equations of the third-order autonomous memristive diode bridge-based oscillator are established as
P (λ ) = (λ + a/b)[λ2 − (k − b − ε )λ + bε ] = 0
(6)
Thus, three eigenvalues are yielded as
λ1 = 0.5(k − b − ε ) + 0.5 (k − b − ε )2 − 4bε λ2 = 0.5(k − b − ε ) − 0.5 (k − b − ε )2 − 4bε λ3 = −a/b
(7)
As k is considerably larger than b and ɛ, the roots of λ1 and λ2 are positive real constants, whereas the root of λ3 is negative real constant, which indicate that the original equilibrium point (0, 0, 0) of (4) is an unstable saddle point. Considering the typical system parameters given in (5) as an example, three eigenvalues in (7) are calculated as λ1 = 2.9661, λ2 = 0.0 0 0 04, and λ3 = − 12,50 0. Consequently, the third-order autonomous memristive diode bridgebased oscillator is always unstable. 3. Symmetric periodic bursting behavior By utilizing MATLAB software tool, ODE23S algorithm is considered to draw bifurcation diagrams, phase plane plots, time sequences, and 0–1 test in our next works. Due to the existence of complex nonlinearities in system (4), pure mathematical analysis methods, for instance, integral transform methods proposed in [29–32], are inapplicable to the solutions of system (4), leading to that some computer-aided simulation tools should be utilized.
148
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
Fig. 2. Dynamics versus the resistance R: (a) bifurcation diagram, (b) partial enlarged view of (a), where ɛ = 0.0303 and k = 3 are fixed, while a and b are synchronously adjusted with the variation of the resistance R.
Fig. 3. Numerically periodic behaviors by using time sequences of y and z for different values of R: (a) R = 23 k, (b) R = 25 k, (c) R = 25.92 k, (d) R = 25.96 k.
Taking the resistance R as a bifurcation parameter, i.e., the system parameters a and b being synchronously adjustable, the bifurcation diagrams of the local maxima X of the state variable x of system (4) are plotted in Figs. 2(a) and 2(b), respectively, where Fig. 2(b) is the partial enlarged view of Fig. 2(a). It should be mentioned that the first Lyapunov exponent is always less than or equal to zero, indicating the inexistence of chaotic behavior in the third-order memristive diode bridge-based oscillator. As the adjusting resistance R is changed from 20 k to 50 k, the system orbit starting from the period 1 oscillation successively enters into multiple types of periodic oscillations with period 4, 6, and 8 at R = 24.06 k, 25.90 k, and 25.94 k, respectively. Beyond the region of the periodic oscillations, the system orbit suddenly shifts into symmetric periodic bursting oscillations with uncountable periodicities at R = 26.01 k, and then the periodicities and dynamic amplitudes of the system orbit will gradually shrink with the increase of the resistance R in a wider parameter region.
In the region of 20 k ≤ R < 26.01 k, only some simple periodically oscillating behaviors with period-adding routes are displayed in Fig. 2. Correspondingly, four types of time sequences of the variables y and z for four different values of R are depicted in Fig. 3, which demonstrate that the symmetric block structured dynamics exists in this parameter adjustable region [33]. Additionally, the sequence of the variable y in Fig. 3(a) is analogous to the bipolar spike firing, whereas the sequences of y in Figs. 3(b), 3(c), and 3(d) are similar to the bipolar burst firing [18]. However, in the region of 26.01 k ≤ R ≤ 50 k, complex symmetric periodic bursting behaviors with the fast-slow effects are exhibited in Fig. 2. When R = 30 k, an example of the symmetric periodic bursting dynamics is illustrated in Fig. 4, where Figs. 4(a) and 4(b) display two phase plane plots in the z − y and y − x planes, respectively, Fig. 4(c) shows the overlapped sequences of the fast variable y and slow variable z, and Fig. 4(d) given the p − q dynamics that behaves a grotesquely flywheel-shaped torus with
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
149
Fig. 4. Symmetric periodic bursting behavior by MATLAB numerical simulations at R = 30 k, where a = 45, b = 0.0036, ɛ = 0.0303, and k = 3: (a) phase plane plot in the z−y plane, (b) phase plane plot in the y−x plane, (c) time sequences of y and z, (d) p–q periodic bursting dynamics using 0–1 test.
Since ɛ = 0.0303, the system (4) can be considered as two subsystems with the order gap between the two time scales. The fastscale subsystem is described by the first two equations of (4) as
x˙ = a ln(b cosh y ) − a ln(x + b) (8a)
and the slow-scale subsystem is expressed by the third equation of (4) as
z˙ = ε (ky − z )
(8b)
4.1. Evolutions of the equilibrium point and its stability For the fast-scale subsystem in (8a), the equilibrium points are determined as EFS = (b cosh y¯ − b, y¯ ), in which y¯ can be numerically solved by
G1 = ky¯ − b sinh y¯ − z = 0
*
h2 = z = 19.2455 h2 = z = 0 * *
h1 *
h2 = z = –19.2455 *
* y
Fig. 5. Evolution of the equilibrium points of the fast-scale subsystem illustrated by the crossing points of two curves.
Differentiating G1 with respect to y¯ , there yields
4. Bifurcation mechanism of the fast-scale subsystem
y˙ = ky − (x + b) tanh y − z
*
h1, h2
a 0–1 test median value close zero [34–36]. Additionally, three Lyapunov exponents are calculated as L1 = 0.0, L2 = − 0.5164, and L3 = − 13.6290, respectively, which further declare that the moving trajectory of the symmetric periodic bursting behavior shown in Fig. 4 is in a periodic pattern with large number of periodicities. The numerical results shown in Fig. 4 denote that the symmetric periodic bursting behavior appears in such a third-order autonomous memristive oscillator. Furthermore, it is noted that the variables y and z in system (4) have the order difference between the two non-dimensional time scales due to the appearance of the very small controlling strength ɛ in third equation, leading to that the three-dimensional system of (4) may be considered as the coupling of two subsystems of the two-dimensional fast-scale subsystem and one-dimensional slow-scale subsystem.
(9)
G2 = k − b cosh y¯ = 0
(10)
which has two roots, obtained as
y¯ 1 = arccosh kb , y¯ 2 = −arccosh kb
(11)
At these roots y¯ 1 and y¯ 2 , G1 has two extreme values, which can be given by
G1 (y¯ 1 ) = k arccosh
k b
− b sinh(arccosh kb ) − z
G1 (y¯ 2 ) = b sinh(arccosh kb ) − k arccosh
k b
−z
(12)
respectively. Take the specified system parameters listed in (5) as an example. Considering that
h1 = ky¯ − b sinh y¯ , h2 = z
(13)
the values of y¯ can be then determined by the crossing points of two function curves of (13), as shown in Fig. 5. When z > 19.2455 or z < − 19.2455, i.e., G1 (y¯ 1 ) < 0orG1 (y¯ 2 ) > 0, only one crossing point exists; when − 19.2455 < z < 19.2455, i.e., G1 (y¯ 2 ) < 0 <
150
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
R = 45 kΩ *H H*4 F4* 3 HB2 HB1 FB1 FB2
R, kΩ
*F 3
*F
1
H*2
*H 1
GND
R
F2*
U
R2
R = 30 kΩ
R1
C2
z
C1 Second-order autonomous oscillator
Fig. 6. Fold and Hopf bifurcation sets of the fast-scale subsystem distributed in the z – R phase plane.
G1 (y¯ 1 ), three crossing points appear; while when z = 19.2455 or z = −19.2455, i.e., G1 (y¯ 1 ) = 0orG1 (y¯ 2 ) = 0, one tangent point along with one crossing point occur, leading to the generation of fold bifurcation [11]. To estimate the stability of the equilibrium point EFS , the Jacobian matrix at EFS is derived and simplified as
JFS
− ab sech y¯ = − tanh y¯
a tanh y¯ k − bsech y¯
(14)
The corresponding characteristic polynomial is deduces as
P (λ ) = λ2 + αλ + β = 0
(15)
y¯ − k] and β = a − y¯ . With the where α = [(b + characteristic polynomial (15), bifurcation analyses of the fast-scale subsystem can be performed. It should be remarkable that the coefficients α and β of the characteristic polynomial (15) are only determined by the solutions of the equilibrium point Eq. (9), whose values are closely associated with the slow variable z. Therefore, the stability of the fastscale subsystem is dependent to the slow-scale subsystem. a )sech b
ka sech b
4.2. Fold and Hopf bifurcation sets
D4
D1
D3
D2
L Memristive diode bridge Fig. 8. Screen capture of the experimental prototype for the presented third-order autonomous memristive diode bridge-based oscillator, the left is a global graph of the oscilloscope linking to the circuit breadboard and the right is the partial enlarged drawing of the circuit breadboard.
i.e., β = 0. Therefore, with the Eq. (9), the fold bifurcation set can be deduced as
ky¯ − b sinh y¯ − z = 0 a − ka sech y¯ = 0 b
FB :
and simplified as
FB : ± k arccosh
k b
(17)
− b sinh arccosh
k b
−z=0
(18)
The Hopf bifurcation is associated with the appearance of a pair of pure imaginary eigenvalues. Then, (15) can be expressed as
PHB (λ ) = λ2 + β = 0
Observed from Fig. 5, the equilibrium points of the fast-scale subsystem have a transition from three to one with the variation of the slow variable z. At the critical condition of G1 (y¯ 1 ) = 0orG1 (y¯ 2 ) = 0, a small perturbation of z may cause the tangent point to disappear or to split into two crossing points, resulting in the occurrence of fold bifurcation. Based on the critical condition, the fold bifurcation set can be easily derived. Alternatively, in consideration of that the fold bifurcation is referred to one zero eigenvalue, (14) can be thereby rewritten as
PFB (λ ) = λ(λ + α ) = 0
+15 V
−15 V
(16)
(19)
i.e., α = 0. Therefore, with the Eq. (9), the Hopf bifurcation set can be gotten as
HB :
k y¯ − b sinh y¯ − z = 0 (b + ab )sech y¯ − k = 0
and simplified as
HB : ± k arccosh
a/b+b k
(20)
− b sinh arccosh
a/b+b k
−z=0
(21)
Consider that the resistance R is adjusted in the region [20 k, 50 k]. By MATLAB numerical simulations, the fold bifurcation
Fig. 7. Bifurcation mechanism of periodic bursters at (a) R = 30 k and (b) R = 45 k, where F1 , F2 , F3 , and F4 stand for the fold bifurcation points, H1 , H2 , H3 , and H4 represent the Hopf bifurcation points, and EP1 and EP2 are the equilibrium point curves of the fast-scale subsystem with the z variation.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
151
Fig. 9. Experimentally periodic behaviors by using time sequences of v1 (t) and v2 (t) for different R: (a) R = 21.5 k, (b) R = 21.8 k.
Fig. 10. Symmetric periodic bursting behaviors by hardware circuit experiments: (a1) phase plane plot in the v1 (t) − v2 (t) plane and (a2) time sequences of v1 (t) and v2 (t) at R = 26.6 k, (b1) phase plane plot in the v1 (t) − v2 (t) plane and (b2) time sequences of v1 (t) and v2 (t) at R = 41.5 k.
set of (18) (marked as FB1 and FB2 ) and Hopf bifurcation set of (21) (marked as HB1 and HB2 ) in the z – R phase plane can be together drawn, as shown in Fig. 6. The bifurcation sets in Fig. 6 are the possible bifurcations of the fast-scale subsystem, which may be correspondence to the turning points between the quiescent state and spiking state when periodic bursting oscillation takes place in the system (4).
4.3. Bifurcation mechanism of periodic bursters Let R = 30 k and R = 45 k be two examples using for explaining the mechanism of the periodic bursters. Referring to Fig. 6, when R = 30 k, two fold bifurcation points locate at z = ± 19.2455 V, marked as F2 and F1 , whereas two Hopf bifurcation points are at z = ± 12.0738 V, marked as H2 and H1 . The phase plane plot of system (4) and the equilibrium point curve EP1
152
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153
of the fast-scale subsystem dotted by the four bifurcation points are depicted in Fig. 7(a). The moving trajectory of the system (4) beginning at H1 gradually enters into the quiescent state with the decrease of the oscillating amplitude. The quiescent state ends once the trajectory moves at the fold bifurcation point F1 , leading to that the trajectory rises to the upper and the system (4) goes into the spiking stat The spiking oscillation around the equilibrium point curve EP1 is maintained with the large oscillating amplitude. After the Hopf bifurcation occurs at H2 , the moving trajectory re-enters into the quiescent state and dives to the lower via the fold bifurcation route at F2 . The spiking state appears again and ends once the trajectory moves at the Hopf bifurcation point H1 , which initiates next period of the periodic burster. Due to the small oscillation of the quiescent state, the periodic burster can be regarded as a symmetric fold/Hopf cycle-cycle burster [16]. Similarly, when R = 45 k, two fold bifurcation points locate at z = ± 18.0291, marked as F4 and F 3 , whereas two Hopf bifurcation points are at z = ± 5.4598, marked as H3 and H4 . The phase plane plot of the system (4) and the equilibrium point curve EP2 of the fast-scale subsystem dotted by the four bifurcation points are demonstrated in Fig. 7(b). By utilizing the bifurcation analysis in the above section, analogous bifurcation details of EP2 are encountered for this case, i.e., the system (4) has a transition from the quiescent state to spiking state and back via the fold and Hopf bifurcation routes. It should be addressed that the spanning time length of the quiescent oscillation increases with the resistance R.
6. Conclusion
5. Hardware experimental confirmations
References
According to the circuit designed in Fig. 1, an experimental prototype of the presented third-order autonomous memristive diode bridge-based oscillator is photographed and its screen capture is shown in Fig. 8. Three precision potentiometers, two monolithic ceramic capacitors, four 1N4148 diodes, a manually winding inductor, and an AD711KN op-amp with ± 15 V power supply are chosen. The linear element parameters used during numerical simulations are selected in hardware experiments and the experimental phase plane orbits and time sequences are measured by a 4 channel digital oscilloscope. By turning the precision potentiometer R, different periodic behaviors with different periodicities or periodic bursters are measured from the experimental prototype. When R is located in the region of smaller resistances, the memristive diode bridgebased oscillator operates in periodic oscillating states. For instance, when R = 21.5 k and 21.8 k, the time sequences of the voltage variables v1 (t) and v2 (t) for two types of the periodic behaviors are experimentally captured, as shown in Fig. 9. Whereas when R is fallen into the region of larger resistances, the memristive diode bridge-based oscillator works in symmetric periodic bursting states. Regarding R = 26.6 k and 41.5 k as examples, the phase plane plots in the v1 (t) − v2 (t) plane and time sequences of the voltage variables v1 (t) and v2 (t) for two types of the symmetric periodic bursting behaviors are experimentally captured, as shown in Fig. 10. There are some relatively larger differences between the values of R used in numerical simulations and hardware experiments, which are mainly caused by the equivalent series resistance (ESR) of the manually winding inductor. Ignoring these resistance differences, it can be observed that the experimentally captured results in Figs. 9 and 10 are well agreed with the numerically simulated results in Figs. 3, 4, and 7.
[1] Iu HHC, Yu D, Fitch AL, Sreeram V, Chen H. Controlling chaos in a memristor based circuit using a twin-T notch filter. IEEE Trans Circuits Syst I 2011;58(6):1337–44. [2] Xu Q, Lin Y, Bao B, Chen M. Multiple attractors in a non-ideal active voltage-controlled memristor based Chua’s circuit. Chaos Solit Fractals 2016;83:186–200. [3] Bao B, Wang N, Xu Q, Wu H, Hu Y. A simple third-order memristive band pass filter chaotic circuit. IEEE Trans Circuits Syst II 2017;64(8):977–81. [4] Wu R, Wang C. A new simple chaotic circuit based on memristor. Int J Bifurcation Chaos 2016;26(9):1650145. [5] Kengne J, Tabekoueng ZN, Tamba VK, Negou AN. Periodicity, chaos, and multiple attractors in a memristor-based Shinriki’s circuit. Chaos 2015;25(10):103126. [6] Xu Q, Wang N, Bao B, Chen M, Li C. A feasible memristive Chua’s circuit via bridging a generalized memristor. J Appl Anal Comput 2016;6(4):1152–63. [7] Tabekoueng ZN, Kengne J, Fotsin HB, Negou AN, Tchiotsop D. Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bridge-based Jerk circuit. Chaos Solit Fractals 2016;91:180–97. [8] Wu H, Bao B, Liu Z, Xu Q, Jiang P. Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dyn 2016;83(1):893–903. [9] Bao B, Wu P, Bao H, Chen M, Xu Q. Chaotic bursting in memristive diode bridge coupled Sallen-Key low-pass filter. Electron Lett 2017;53(16):1104–5. [10] Bao B, Wu P, Bao H, Xu Q, Chen M. Numerical and experimental confirmations of quasi-periodic behavior and chaotic bursting in third-order autonomous memristive oscillator. Chaos Solit Fractals 2018;106:161–70. [11] Bi Q, Ma R, Zhang Z. Bifurcation mechanism of the bursting oscillations in periodically excited dynamical system with two time scales. Nonlinear Dyn 2015;79(1):101–10. [12] Bi Q, Zhang Z. Bursting phenomena as well as the bifurcation mechanism in controlled Lorenz oscillator with two time scales. Phys Lett A 2011;375(8):1183–90. [13] Babacan Y, Kaçar F, Gürkan K. A spiking and bursting neuron circuit based on memristor. Neurocomputing 2016;203:86–91. [14] Hu M, Chen Y, Yang JJ, Wang Y, Li H. A compact memristor-based dynamic synapse for spiking neural networks. IEEE Trans Comp Aid Des Int 2017;36(8):1353–66. [15] Feali MS, Ahmadi A. Transient response characteristic of memristor circuits and biological-like current spikes. Neural Comp Appl 2017;28(11):3295–305. [16] Izhikevich EM. Neural excitability, spiking and bursting. Int J Bifurcation Chaos 20 0 0;10(6):1171–266. [17] FröHlich F, Bazhenov M. Coexistence of tonic firing and bursting in cortical neurons. Phys Rev E 2006;74(1):031922.
A novel third-order autonomous memristive diode bridge-based oscillator with two time scales is designed, which has simple circuit topology and only consists of one op-amp, two capacitors, three resistors, four diodes, and one inductor. However, the presented memristive oscillator is always unstable and can exhibit an interesting symmetric periodic bursting behavior with a symmetric fold/Hopf cycle-cycle burster by MATLAB numerical simulations and hardware experimental confirmations. Through constructing the fold and Hopf bifurcation sets of the fast-scale subsystem with the evolution of the system parameter and slow-scale variable, the bifurcation mechanism of the symmetric periodic bursting behavior is theoretically and numerically explored, which demonstrates that the periodic bursting oscillation switching from a quiescent state to a spiking state and back via the fold and Hopf bifurcation routes are a symmetric fold/Hopf cycle-cycle bursting oscillation. The presented third-order autonomous memristive diode bridge-based oscillator is inexpensive and easy to practically realize, which can be taken as an example with a special dynamical phenomenon for the in-depth investigations of this kind of memristor-based dynamical circuits. Acknowledgements This research issue was supported by the grants from the National Natural Science Foundations of China under Grant Nos. 51777016, 61601062, 61705021, 11602035, and 51607013, the Natural Science Foundations of Jiangsu Province, China under Grant No. BK20160282, and the Postgraduate Research & Practice Innovation Program of Jiangsu Province, China under Grant No. KYCX17_2083.
B.C. Bao et al. / Chaos, Solitons and Fractals 109 (2018) 146–153 [18] Ngouonkadi EBM, Fotsin HB, Fotso PL, Tamba VK, Cerdeira HA. Bifurcations and multistability in the extended Hindmarsh–Rose neuronal oscillator. Chaos Solit Fractals 2016;85:151–63. [19] Innocenti G, Morelli A, Genesio R, Torcini A. Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos. Chaos 2007;17(4):043128. [20] Malashchenko T, Shilnikov A, Cymbalyuk G. Six types of multistability in a neuronal model based on slow calcium current. Plos One 2011;6(7):e21782. [21] Ma J, Tang J. A review for dynamics in neuron and neuronal network. Nonlinear Dyn 2017;89(3):1569–78. [22] Kingni ST, Nana B, Ngueuteu GSM, Woafo P, Danckaert J. Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect. Chaos Solit Fractals 2015;71:29–40. [23] Zhang Z, Liu B, Bi Q. Non-smooth bifurcations on the bursting oscillations in a dynamic system with two time scales. Nonlinear Dyn 2015;79(1):195–203. [24] Bi Q, Li S, Kurths J, Zhang Z. The mechanism of bursting oscillations with different codimensional bifurcations and nonlinear structures. Nonlinear Dyn 2016;85(2):993–1005. [25] Zheng S, Han X, Bi Q. Bifurcations and fast-slow behaviors in a hyperchaotic dynamical system. Commun Nonlinear Sci Numer Si 2011;16(4):1998–2005. [26] Zhou G, Bao B, Xu J. Complex dynamics and fast-slow scale instability in current-mode controlled buck converter with constant current load. Int J Bifurcation Chaos 2013;23(4):1350062.
153
[27] Talukdar A, Radwan AG, Salama KN. Generalized model for memristor-based Wien family oscillators. Microelectron J 2011;42(9):1032–8. [28] Yang X, Machado JAT, Cattani C, Gao F. On a fractal LC-electric circuit modeled by local fractional calculus. Commun Nonlinear Sci Numer Si 2017;47:200–6. [29] Yang X. A new integral transform with an application in heat-transfer problem. Therm Sci 2016;20:S677–81. [30] Yang X. A new integral transform operator for solving the heat-diffusion problem. Appl Math Lett 2017;64:193–7. [31] Yang X. New integral transforms for solving a steady heat transfer problem. Therm Sci 2017;21:S79–87. [32] Yang X, Yang Y, Cattani C, Zhu M. A new technique for solving the 1-D Burgers equation. Therm Sci 2017;21:S129–36. [33] González-Miranda JM. Block structured dynamics and neuronal coding. Phys Rev E 2005;72(1):051922. [34] Kim Y. A numerical study of the local K spectrum of strange nonchaotic attractors. J Korean Phys Soc 2014;64(1):6–10. [35] Savi MA, Pereira-Pinto FHI, Viola FM, Paula ASD, Bernardini D, Litak G, Rega G. Using 0-1 test to diagnose chaos on shape memory alloy dynamical systems. Chaos Solit Fractals 2017;103:307–24. [36] Gottwald GA, Melbourne I. On the validity of the 0-1 test for chaos. Nonlinearity 2009;22:1367–82.