Field coupling synchronization between chaotic circuits via a memristor

Journal Pre-proofs Regular paper Field coupling synchronization between chaotic circuits via a memristor Xiufang Zhang, Fuqiang Wu, Jun Ma, Aatef Hobiny, Faris Alzahrani, Guodong Ren PII: DOI: Reference:

S1434-8411(19)32913-9 https://doi.org/10.1016/j.aeue.2019.153050 AEUE 153050

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International Journal of Electronics and Communications

Received Date: Revised Date: Accepted Date:

23 November 2019 17 December 2019 22 December 2019

Please cite this article as: X. Zhang, F. Wu, J. Ma, A. Hobiny, F. Alzahrani, G. Ren, Field coupling synchronization between chaotic circuits via a memristor, International Journal of Electronics and Communications (2019), doi: https://doi.org/10.1016/j.aeue.2019.153050

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Field coupling synchronization between chaotic circuits via a memristor Xiufang Zhang1) Fuqiang Wu1) Jun Ma1)* Aatef Hobiny 2) 1)

2)

Faris Alzahrani2) Guodong Ren1)

Department of Physics, Lanzhou University of Technology, Lanzhou 730050, China

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Abstract: Continuous energy flow is pumped and propagated via the coupling components, which are activated to build coupling channel between two nonlinear circuits. When the coupling channels are bridged to different output ends, these coupling devices show much difference in pumping and balancing energy flow from the coupled circuits. In this paper, a memristor is used to bridge connection between different outputs ends of the Pikovsk-Rabinovich circuits, and coupling synchronization is investigated carefully. It is found that complete synchronization can be realized when the coupling channel is connected to the output ends for capacitors, while complete synchronization is blocked to stabilize phase synchronization when memristor is used to connect the output ends for tunnel diodes. Finally, the coupling mechanism is discussed by calculating the energy pumping in the coupling memristor. Indeed, the involvement of memristor in the coupling channel introduces a kind of nonlinear coupling as field coupling, and the synchronization transition and stability are much dependent on the intrinsic properties of the coupling channel. Key words: chaotic circuit; memristor; synchronization; magnetic field; energy flow

1 Introduction Nonlinear chaotic circuits can be tamed to show distinct periodicity by applying a variety of control schemes [1-4]. The involvement of nonlinear electric components is critical to build any chaotic and hyperchaotic systems. It is claimed that chaotic systems can be used for potential secure communication and even image encryption [5-8]. In generic way, nonlinear resistance such as tunnel diode, nonlinear capacitor such as artificial cell membrane, and nonlinear inductor such as Josephson junction [9-12], memristor [13-16] can be used to design a variety of nonlinear circuits and appropriate parameters can be selected to trigger chaotic series and periodic oscillation. Indeed, the involvement of memristor makes the dynamical systems become more sensitive to initial setting, which dynamics transition can be induced between periodical and chaotic states even all the parameters are fixed [17-20]. For two or more chaotic systems, the stability and realization of synchronization

*

Corresponding author: [email protected], [email protected] (J. Ma) 1

become attractive and interesting [21-24]. For nonlinear oscillators, many variables can be used to realize synchronization via one or more channels. When more coupling channels are open, more variables are coupled to enable the realization of synchronization. In fact, from physical view, this kind of variable coupling can be explained as voltage coupling via linear resistor when these nonlinear oscillators are produced in nonlinear circuits. However, the coupling resistor has to consume a large Joule heat before reaching complete synchronization. Therefore, Ma et al. [25-30] suggested that capacitor connection can be used to activate electric field coupling [25-28], and also an induction coil connection can be applied to generate magnetic field coupling [29, 30] between chaotic circuits without consuming any Joule heat. For experimental implement, appropriate selection for capacitance with the coupling capacitor and suitable inductance with the coupling induction coil can synchronize the chaotic circuits and memristive circuits completely. Indeed, this field coupling seldom consumes Joule energy and the coupling devices just pump and propagate the energy between nonlinear circuits. In particular, Wang et al. [31] suggested that artificial hybrid synapse, which is made of capacitor, inductor and linear resistor, can be used to realize synchronization between neurons and they argued that the release of chemical synapse function is dependent on the formation of physical field. Indeed, time-varying electric field is triggered in the coupling capacitor which is effective in pumping energy from the output ends of the chaotic circuits and its dynamical mechanism is explained as differential coupling [28]. On the other hand, time-varying magnetic field is generated in the coupling induction coil which can pump energy from the output ends of the coupled circuits and its dynamical mechanism is explained as integral coupling [29]. As a result, continuous propagation and pumping of energy via the coupling capacitor or induction coil can balance the energy exchange in these chaotic circuits and thus synchronization can be stabilized. However, the synchronization approach can be much dependent on the initial setting for some variables of two identical memristive systems even all the parameters are fixed, and the potential mechanism is that memristor-based nonlinear function generates parameter mismatch when the memristive variable shows slight diversity in initial values [32-34]. For most of the previous works, the coupling intensity is kept beyond the threshold for stabilizing synchronization, which is dependent on the intrinsic parameters as well. In realistic systems, the coupling intensity could be time-varying [35] thus lower cost in energy can be supplied during the signal encoding and exchange. For biological neural networks, intrinsic response and propagation time delay [36] could also be changeable when pulse and/or wave are propagated. Therefore, the author of this paper suggested that saturation gain method [37] can be applied to optimize the synchronization between neurons by continuous self-learning. Considering the diversity in coupling channels and parameters setting, chimera states [38] can be 2

formed in the network due to self-organization, and coupling channels can be adjusted for triggering pattern formation. In fact, the involvement of memristor coupling between neurons and nonlinear circuits provides another kind of field coupling. According to the intrinsic dependence of outputs on inputs, memristor is estimated with two types, one is magnetic flux-controlled and another one is charge flux-controlled type. When the two output ends of a memristor are included into a branch of the circuit, a time-varying current will pass across it and magnetic field will be induced in the memristor. As a result, the charges are propagated and magnetic flux is changed to follow the flow of current along the memristor. For example, Xu et al. [39] investigated the synchronization between neurons with memristive synapses. Usha et al. [40] confirmed that the dynamics of neural activities become more multiple when memristor is coupled to develop a memristive synapse. Differs from the standard variable coupling with invariant coupling gain, memristor-based coupling can be explained as time-varying coupling [41] because the induction current across the memristor is variant with time. When Josephson junction is coupled with a resonator [42], the neural activities can be reproduced by generating spiking series [43]. For example, Zhang et al.[44] investigated the dynamics of neural circuit composed of Josephson junction and memristor. That is, the involvement of memristor enables the possibility of mode transition in neural activities by resetting the initial value for magnetic flux variable. The pure resistor will consume Joule heat when it is included into a branch of the circuit. Therefore, the voltage coupling via resistor can pump much energy from the coupled circuits by consuming Joule heat continuously thus the variable-dependent energy can be balanced, and then synchronization or partial synchronization can be reached. Both of capacitor coupling and induction coil coupling between chaotic circuits just pump energy via the coupling devices so that the energy propagation and exchange can be balanced without consuming any Joule heat. From physical view, the memristor holds the main properties of linear resistor and ideal induction coil because its memductance consists two parts, constant and magnetic-dependent term which shows field effect. When memristive synapse is included into the neural network, the firing patterns and mode selection will be controlled, for example, Xu et al. [45] studied the collective synchronization on chain network of neurons connected by memristor. On the other hand, the memristive synapse current can be thought of as induction current resulting from the electromagnetic induction in the media. Therefore, the field effect becomes more important and distinct when more neurons are included into network for estimating the collective neural activities. That is, magnetic flux can be used to estimate the changes of electromagnetic field and the induction current is suitable to estimate the effect of electromagnetic 3

induction on membrane potential for neurons. Therefore, Ma et al. [46-48] suggested that weight can be introduced for calculating exchange of magnetic flux of neurons, and thus the effect of electromagnetic induction and radiation on pattern selection, collective behaviors on neural network can be estimated. In this paper, a memristor is used to connect different output ends of two chaotic circuits, which can be described by the Pikovskii-Rabinovich (PR) oscillator [49], and the dynamical equations are mapped from the coupled circuits for nonlinear analysis and synchronization estimation by applying scale transformation on the circuit equations and intrinsic parameters.

2 Model and scheme The Pikovskii-Rabinovich(PR) oscillator [49] is mapped from a simple three-variable nonlinear circuit, which is composed of two capacitors, one induction coil and one tunnel diode. The relation between voltage and current on this tunnel diode has distinct nonlinearity and the circuit can be presented in Fig.1.

Fig.1 Schematic diagram for PR circuit (a) and nonlinear relation for voltage-current on tunnel diode (b). The current across the tunnel diode is estimated at I1=I0+ I0[(U/U0−1)3−( U/U0−1)]. C, C1, L represents the capacitor and induction coil, respectively. The nonlinear resistor has negative conductance –g and r is linear resistor. According to the physical Kirchhoff law, the circuit equations for Fig.1 can be obtained as follows dI ì ïV - U = Ir + L dt ï dV ï - gV ; í- I = C dt ï dU ï ï I = I1 + C dt î

(1)

By further applying scale transformation on the physical variables and parameters as shown in Eq.(2), a dimensionless dynamical system can be described in Eq.(3).

4

ì U0 (V - Ir - U 0 ) I U 1 - gr - 1, t =tw , w = ,z= , d = , ï x = - 1, y = w LI 0 w I0 L I0 U0 LC ï (2) í ï2g = gL - rC , a = rU 0 , b = -1 + gU 0 , m = I 0 ï w LC w 2 L2 I 0 w 2 I 0 LC wC1U 0 î

ì dx ï dt = y - d z; ï ï dy = - x + 2g y + a z + b ; í ï dt ï dz 3 ï dt = m ( x + z - z ) î

(3)

As reported in Ref.[50], Gaussian white noise can be imposed to change the evolution of variable z, and two PR chaotic oscillators can reach complete synchronization even when no any variables are used to couple them. Here, a magnetic flux-controlled memristor is used to bridge connection between two PR circuits and the coupling channel can be open by connecting different output ends, the coupling circuits are presented in Fig.2.

Fig.2 Memristor connection between two PR circuits. M is the memristor, and 1, 2, 3, 4 are the outputs end of the PR circuits. The memdunctance for the memristor M is calculated by W(φ)= a+3bφ2, a, b are intrinsic parameters in the memristor and φ is magnetic flux. The memristive current I M = W (j )(V - Vˆ ) when the output ends 1, 2 are connected, while IˆM = W (j )(U - Uˆ ) when the output ends 3, 4 are connected.

As is well known, the magnetic flux φ across the memristor will be changed to induce the occurrence of induced electromotive force, which is balance against the outputs voltage from the output ends of the PR circuits. Here, the memristor can be estimated as an equivalent induction coil with N turns with constant conductance value a. For the memristor, the dependence of voltage on magnetic flux can be calculated by

5

ì ïï N í ïN ïî

dj = V - Vˆ , output ends 1, 2 are connected dt ; (4) dj ˆ = U - U , output ends 3, 4 are connected dt

where the equivalent parameter N can be replaced by a normalized parameter k=1/N, and the dynamics for electromagnetic induction can be rewritten by

ì dj ˆ ïï dt = k (V - V ), output ends 1, 2 are connected ; (5) í ï dj = k (U - Uˆ ), output ends 3, 4 are connected ïî dt Furthermore, the Eq.(5) can be replaced by an equivalent dimensionless dynamical equation when the physical magnetic flux variable is replaced by an dimensionless variable as follows

w=

jw U0

=

j

1 - gr ; (6) LC

U0

As a result, the Eq.(5) is rewritten by the following dimensionless dynamical equations

ì dw kI 0 = [w L( y - yˆ ) + r ( x - xˆ )], output ends 1, 2 are connected ï ï dt U 0 ; (7) í ï dw = k ( z - zˆ ), output ends 3, 4 are connected ï î dt By applying the same physical Kirchhoff law, the coupled PR circuits via memristor can be estimated by

ì dI ï L dt = V - U - Ir ï ï dV = gV - I - l12 I M ; (8a ) íC ï dt ï dU ˆ ïC1 dt = I - I1 - l34 I M î

ì dIˆ ˆ ˆ ˆ = V - U - Ir ïL ï dt ï dVˆ = gVˆ - Iˆ + l12 I M ; (8b) íC dt ï ï dUˆ = Iˆ - Iˆ1 + l34 IˆM ïC1 dt î

where λ12 , λ34 represents the on-off state, when the outputs end 1,2 (or 3,4) are connected via memristor, λ12 (or λ34)=1, otherwise, λ12 (or λ34) =0 in which means the two outputs are out of connection. By using the same scale transformation for the variables in Eq.(8), the dimensionless dynamical systems under memristive coupling can be obtained. At first, we consider the case when the outputs end 1, 2 are connected via a memristor, 6

ì dx = y -d z ï ï dt ï dy ar ï dt = - x + 2g y + a z + b - LCw 2 ( x - xˆ ) ï ï dz = m ( x + z - z 3 ) ï dt ï ï dxˆ = yˆ - d zˆ í ï dt ï dyˆ ar ( xˆ - x) = - xˆ + 2g yˆ + a zˆ + b ï LCw 2 ï dt ï dzˆ 3 ï dt = m ( xˆ + zˆ - zˆ ) ï ï dw = kI 0 [w L( y - yˆ ) + r ( x - xˆ )] ï î dt U 0

3bU 02 2 3bU 02 2 a w ( x - xˆ ) w ( y - yˆ ) ( y - yˆ ) 4 LCw Cw Cw 3

; (9) 3bU 02 2 3bU 02 2 a ˆ ˆ w x x y y w ( yˆ - y ) ( ) ( ) LCw 4 Cw Cw 3

For simplicity, the parameter is selected as β=0, and the normalized parameters are estimated according to the definition shown in Eq.(2). g=

I0 I U0 1 ; a = 0 d 2 r; g = (1 - a ); d = ; U 0 + rI 0 U0 I 0w L 2d

(10)

According to the setting criterion of parameters shown in Eq.(10), the isolated PR oscillator still can find chaotic parameter regions even when it sets U0/I0=1. In this way, the parameters can be selected with simplest form. By calculating the Lyapunov spectrum and the largest Lyapunov exponent, appropriate parameters can be selected to generate chaos in the isolated PR oscillator. For simplicity, we select δ=0.5, γ=0.2, α=0.8, and the parameter μ is changed to detect the chaotic region. For example, μ=1.4 (or 4.0), one positive Lyapunov exponent is found for obtaining chaos. On the other hand, when the output ends 3, 4 are connected; the dimensionless dynamical systems under memristor coupling can be obtained as follows ì dx ï dt = y - d z ï ï dy = - x + 2g y + a z + b ï dt ï 2 ï dz = m ( x + z - z 3 ) -( a + 3bU 0 w2 )( z - zˆ ) 3 ï dt wC1 w C1 ï ï dxˆ = yˆ - d zˆ í ï dt ï dyˆ ï dt = - xˆ + 2g yˆ + a zˆ + b ï 3bU 02 2 a ï dzˆ 3 ï dt = m ( xˆ + zˆ - zˆ ) -( wC + w 3C w )( zˆ - z ) 1 1 ï ï dw ï dt = k [ z - zˆ ] î 7

; (11)

From Eq.(9) and Eq.(11), the coefficients for the coupling terms are much complex. In fact, the memristor is reduced to a linear resistor at b=0 and the magnetic field effect is removed. For simplicity, together with the scale transformation in Eq.(2) and Eq.(10), here, we reduce the scale ratio U0=1, I0=1, ω=1, and the physical parameters for the electric devices are obtained as follows

r=

a 1 d3 1 d2 ; L ; C ; C ; g ; (12) = = = = 1 d2 d a +d 2 m a +d 2

As a result, the dynamical equations for the coupled PR oscillators with memristor coupling can be rewritten in a simpler form. For Eq.(9), they can be replaced by ì dx ï dt = y - d z ï 2 2 2 2 ï dy = - x + 2g y + a z - a a (a + d ) ( x - xˆ ) - 3b(a + d ) w2 ( x - xˆ ) - a (a + d ) ( y - yˆ ) - 3b(a + d ) w2 ( y - yˆ ) 4 2 3 3 ï dt d d d d ï dz 3 ï = m(x + z - z ) ï dt ï dxˆ ï ; (13) = yˆ - d zˆ í ï dt 2 2 2 2 ï dyˆ 3b(a + d ) 2 3b(a + d ) 2 a (a + d ) a (a + d ) w ( yˆ - y ) ( xˆ - x) ( yˆ - y ) w ( xˆ - x) = - xˆ + 2g yˆ + a zˆ - a ï 4 2 3 d d d d3 ï dt ï dzˆ 3 ï dt = m ( xˆ + zˆ - zˆ ) ï ï dw = k[ y - yˆ + a ( x - xˆ ) ] ï d d2 î dt

For Eq.(11), the dynamical systems can be rewritten by an equivalent form as follows ì dx ï dt = y - d z ï ï dy = - x + 2g y + a z ï dt ï ï dz = m ( x + z - z 3 ) - ( a m + 3bm w2 )( z - zˆ ) ï dt ï ï dxˆ = yˆ - d zˆ í ï dt ï dyˆ ï dt = - xˆ + 2g yˆ + a zˆ ï ï dzˆ = m ( xˆ + zˆ - zˆ 3 ) - ( a m + 3bm w2 )( zˆ - z ) ï dt ï ï dw = k [ z - zˆ ] ï dt î

; (14)

When the intrinsic parameters for PR circuits are fixed, for example, δ=0.5, γ=0.2, α=0.8, the normalized induction coefficient k and intrinsic parameters a, b for memristor are selected to realize synchronization between two chaotic PR oscillators when the memristor connection is switched on. The error function is 8

defined as follows

q (ex , ey , ez ) = ex2 + ey2 + ez2 = ( x - xˆ )2 + ( y - yˆ )2 + ( z - zˆ)2 ; (15) When the error function decreases to zero within finite period, complete synchronization is reached between two chaotic PR oscillators. Also, the flow of magnetic flux φ becomes invariant when complete synchronization is reached no matter whether the coupling channel is activated between the output ends 1, 2 or 3, 4. In fact, a single variable or channel coupling can realize complete synchronization between low-dimensional chaotic and hyperchaotic systems. However, when another variable is used to couple the chaotic systems, complete synchronization becomes difficult while partial synchronization, while phase synchronization can be reached. The sampled time series for observable variables are often calculated to obtain the phase series via Hilbert transformation and other feasible algorithm, and then the phase synchronization can be explored [25, 51-54].

3 Numerical results and discussion In this section, the fourth order Runge-Kutta algorithm

is used to find solutions for an

isolated PR system and the coupled PR systems. The parameters are fixed at δ=0.5, γ=0.2, α=0.8, time step h=0.01, and the parameter μ is changed to find chaotic region by calculating the Lyapunov exponent spectrum in Fig.3.

 Fig. 3 (a) Distribution for Lyapunov exponent spectrum is estimated vs. μ and (b) formation of chaotic attractor at parameters δ=0.5, γ=0.2, α=0.8, μ=1.4.

From Fig.3, it shows that the PR oscillator can present chaotic state by setting appropriate value for μ, and multi-scroll attractors can be formed at μ=1.4. In the following studies, the intrinsic parameters of PR oscillator are fixed at δ=0.5, γ=0.2, α=0.8, μ=1.4, and the synchronization between two identical chaotic PR oscillators 9

are investigated by selecting different parameters for the coupling memristor. The initial values are selected as (x0, y0, z0, xˆ0 , yˆ0 , zˆ0 , φ)=(0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.0).When all the known parameters are fixed in the coupled PR oscillators and the coupling channel is open to connect the outputs end 1, 2, and then the coupled dynamical systems for Eq.(13) are obtained by

ì dx ï dt = y - 0.5 z ï ï dy = - x + 0.4 y + 0.8 z - 13.44a( x - xˆ ) - 12.6bw2 ( x - xˆ ) - 8.4a( y - yˆ ) - 25.2bw2 ( y - yˆ ) ï dt ï ï dz = 1.4( x + z - z 3 ) ï dt ïï dxˆ ;(16) í = yˆ - 0.5 zˆ ï dt ï dyˆ 2 2 ï dt = - xˆ + 0.4 yˆ + 0.8 zˆ - 13.44a( xˆ - x) - 12.6bw ( xˆ - x) - 8.4a( yˆ - y ) - 25.2bw ( yˆ - y ) ï ï dzˆ = 1.4( xˆ + zˆ - zˆ 3 ) ï dt ï ï dw = k[0.5( y - yˆ ) + 3.2( x - xˆ )] îï dt From dynamical viewpoint, parameters a, b, k can be extensively selected to find the synchronization region when the largest Lyapunov exponent becomes negative. To discern the main contribution from magnetic flux, parameter is selected with very small value as b=0.0001, k=0.01, and then the parameter a is selected with a=0.0001, 0.0011, 0.01, 0.1. Then the error function in Eq. (15) and evolution of magnetic flux are estimated for checking synchronization approach, and the results are shown in Fig. 4.

q

q

4 2 0 0

200

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t

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(c)

5

3 2 1 0 -1 0

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2

-3

0

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0

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(h)

0.020

0.005

-4

0.0

0

400

0.010

-2

0.2

1

200

w

w

0.4

0

0.015

-1

q

3

(f)

7 6 5 4 3 2 1 0

1000

(g)

1

(d)

0.8

4

q

4

w

6

(e)

5

(b)

7 6 5 4 3 2 1 0

w

(a)

8

0.000 0

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t

Fig. 4 Evolution of error function and magnetic flux w is calculated at b=0.0001, k=0.01 when outputs end 1, 2 are connected by a memristor. For (a, e) a=0.0001; (b, f) a=0.0011; (c, g) a=0.01; (d, f) a=0.1. It is confirmed that appropriate setting for parameters a, b in the coupling memristor can realize complete 10

synchronization between two chaotic PR circuits. Also, the magnetic flux is stabilized to constant when complete synchronization is reached. When the normalized induction coefficient k and parameters a, b are further increased, for example, b=0.0003, k=0.3, the transient period for reaching complete synchronization is shortened greatly by setting the same value a=0.0001, 0.0011, 0.01, 0.1. It is interesting to discuss the synchronization dependence on the induction coefficient k and the initial setting for magnetic flux φ, which is mapped into the variable w. Therefore, the evolution of maximal error function and Largest Lyapunov exponents are calculated by changing the normalized induction coefficient k, respectively. Then the parameters a, b, and k are fixed, the same Lyapunov exponents are calculated vs. initial value w0, and then the synchronization dependence on initial value is estimated in Fig. 5.

 Fig. 5 Evolution of maximal error function (a) and largest Lyapunov exponent spectrum (b) are calculated under different induction coefficients k, and w0=0.0. Dependence of error function (c), and the largest Lyapunov exponent (d) on initial value w0 is calculated. The parameters are selected as a=0.01, b=0.0001, k=0.01. When the intrinsic parameters (a, b) in the coupling memristor are fixed, appropriate selection for the normalized parameter k can cause distinct transition for the error function and Lyapunov exponent, and appropriate value for parameter k can enhance the coupling of memristor between two chaotic circuits and then complete synchronization is reached. That is, the synchronization is much dependent on the 11

normalized induction coefficient k when the same initial value and parameters are used. As a result, changing the value for k can cause transition from synchronization to non-synchronization. On the other hand, when parameters are selected in the synchronization region, slight switch on initial value will induce non-synchronization. Also, non-synchronization can be modulated to find synchronization by changing the initial value for the magnetic flux variable. In fact, the involvement of memristor makes the nonlinear circuits and coupled PR oscillators present multistability and the synchronization will be dependent on the initial value. Therefore, resetting or feedback on the magnetic flux can induce intermittent synchronization between chaotic oscillators. In addition, the same parameters are endowed with Eq. (14), which the outputs end 3, 4 are connected by a memristor, and the coupled dynamical systems are expressed by ì dx ï dt = y - 0.5 z ï ï dy = - x + 0.4 y + 0.8 z ï dt ï ï dz = 1.4( x + z - z 3 ) - (1.4a + 4.2bw2 )( z - zˆ ) ï dt ï ï dxˆ = yˆ - 0.5 zˆ í ï dt ï dyˆ ï dt = - xˆ + 0.4 yˆ + 0.8 zˆ ï ï dzˆ = 1.4( xˆ + zˆ - zˆ 3 ) - (1.4a + 4.2bw2 )( zˆ - z ) ï dt ï ï dw = k[ z - zˆ ] ï dt î

; (17)

To compare with the case that outputs end 1, 2 are connected via the same memristor, the same parameters value are selected to find the synchronization region by finding numerical solution for Eq. (17), which memristor is connected to the outputs 3,4 from two chaotic PR oscillators. Then the error function and evolution of magnetic flux are calculated in Fig. 6.

12

2

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0 0

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(c)

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0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 -0.1

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0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

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Fig. 6 Evolution of error function and magnetic flux w are calculated at b=0.0001, k=0.01. When outputs end 3, 4 are connected by a memristor. For (a, e) a=0.001; (b, f) a= 0.002; (c, g) a=0.004; (d, f) a=0.005. The results in Fig.6 show that complete synchronization can’t be stabilized when memristor is used to bridge connection between the outputs end 3, 4 from the chaotic PR circuits. Then the phase series are calculated from the sampled time series for the first variable of the PR circuits and the evolution of phases are estimated in Fig.7.

 Fig. 7 Evolution of phase difference is calculated at b=0.0001, k=0.01. When the outputs end 3, 4 are connected by a memristor. For (a, e) a=0.001; (b, f) a=0.002; (c, g) a=0.004; (d,f) a=0.005. The phase series f1 , f2 for the two PR oscillators can be calculated from the sampled time series for outputs x, xˆ , and phase error is estimated with Df = f1 - f2 . From Fig.7, it is demonstrated that phase synchronization can be reached within finite transient time under memristor coupling, which just modulates the channel current across the tunnel diode when output ends 3, 4 are connected. However, memristor connection between the outputs end 1, 2 can regulate the outputs voltage of 13

capacitors C directly and then the variables y and yˆ are controlled completely. In fact, the dynamics of the PR oscillator is much dependent on the second variable y shown in Eq. (3), as a result, complete synchronization becomes available when the second variable of PR oscillators are driven to keep pace with each other. It is also important to estimate energy pumping of the coupling memristor. When a linear resistor R is used to couple two chaotic circuits with outputs voltage V1, V2 , the Joule heat consumption of the coupling resistor WE=0.5∫(V1–V2)2dt/R is dependent on the transient period. More Joule heat costs before realization of complete synchronization. When a capacitor C is used to couple two chaotic circuits, the energy pumping for the coupling capacitor is estimated by WE=0.5C(V1–V2)2 and the energy flow is time-varying without accumulation, which indicates energy pumping can be reversing and bidirectional vs. time. For connecting two nonlinear circuits with an induction coil L, the energy pumping for the coupling inductor is estimated by WE=0.5LI2, which is dependent on the current across this induction coil, and the energy flow is also bidirectional for two PR oscillators. With respect to the coupling memristor, the energy pumping is calculated by

dq(j ) dj = 0.5DV ò dq(j ) dt ìï0.5(aj + bj 3 )(V - Vˆ ) 2 , output ends 1, 2 are connected =í ; (18) 3 2 ïî0.5(aj + bj )(U - Uˆ ) , output ends 3, 4 are connected

WM = 0.5ò I M (j )dj = 0.5ò

According to Eq. (18), the energy pumping of the coupling memristor is balanced when the two PR circuits are coupled to reach synchronization completely. For simplicity, we calculate the energy pumping and propagation across the coupling memristor by using the same parameters in Fig.4, and the results are shown in Fig.8. 50

(a)

2

30

WM

WM

(b)

4

40

20

0 -2 -4

10

-6

0 0

200

400

t

600

800

-8

1000

(c)

        

0

200

400

t

600

800

1000

(d)

0.00006

WM

WM

0.00005 0.00004 0.00003 0.00002 0.00001 0.00000 





t







0

200

400

t

600

800

1000

Fig. 8 Energy pumping WM in the coupling memristor when the output ends 1, 2 are connected. The parameters are selected as b=0.0001, k=0.01. For (a) a=0.0001; (b) a=0.0011; (c) a=0.01; (d) a=0.1. 14

From Fig.8, it is found that the energy flow of across the coupling memristor is suppressed and then stabilized with the increase of the intrinsic parameter value a, as a result, the pumping of energy flow is terminated when complete synchronization is reached by connecting the outputs end 1, 2. In case of phase synchronization, the energy flow across the coupling memristor is also estimated when the outputs end 3, 4 are connected, and the results are shown in Fig.9. (a)

0.10

0.04

WM

WM

0.00

0.02 0.00

-0.02

-0.05

-0.04

-0.10

-0.06 0

200

400

0.03

t

600

800

1000

(c)

0.02

0           

200

400

t

600

800

1000

(d)

WM

0.01

WM

(b)

0.06

0.05

0.00 -0.01 -0.02 -0.03 0

200

400

t

600

800

1000

0

200

400

t

600

800

1000

Fig. 9 Energy pumping WM in the coupling memristor when the output ends 3, 4 are connected. The parameters are selected as b=0.0001, k=0.01. For (a) a=0.001; (b) a= 0.002; (c) a=0.004; (d) a=0.005. By applying the same parameters as the case for Fig.6 and Fig.7, it is confirmed that energy flow is pumped across the coupling memristor from one output end to another when phase synchronization is reached by connecting the output ends 3,4. That is, energy flow will be propagated to support continuous exchange of energy from the two PR circuits kept in phase synchronization. Extensive numerical results confirmed that the field energy in the memristor can present distinct fluctuation, which is dependent on the selection values for intrinsic parameters (a, b) in the coupling memristor. When the intrinsic parameter a is selected with larger value while parameter b is selected with very slight value, the energy pumping of the coupling memristor is decreased monotonously. Otherwise, the energy pumping function in Eq.(18) will show a distinct peak due to the nonlinearity in the memductance. That is, the ratio for the two intrinsic parameters can control the period for reaching complete synchronization and phase synchronization. In a summary, the physical property of coupling device and coupling channel are important for reaching synchronization between chaotic circuits. In particular, the involvement of memristor makes the occurrence of initial value dependence, which the dynamics of nonlinear circuits and oscillators is dependent on the initial setting. Therefore, the synchronization approach depends on the coupling intensity and also the initial values. As a result, resetting or feedback modulation on the memristive variable will 15

change the stability of synchronization. Similar to the application of coupling capacitor and induction coil, both resistor coupling and magnetic field coupling are generated when memristor is used to bridge connection to two or more nonlinear circuits. In the recent works, memristive synapse [55, 56] is introduced into the well-known neuron models and neural activities are measured. In fact, it is important to explore the physical expression for the memristive synapse current so that appropriate experiments can be finished in practice [57]. Readers can extend this study on neural network, which neurons are connected with memristive synapses, and then the network synchronization and pattern selection can be further evaluated to understand the cooperation of collective behaviors [58-62].

4 Conclusions Generic voltage coupling is switched on when linear resistor connects the nonlinear circuits via appropriate channels (outputs ends), and the consumption of Joule heat is effective in pumping energy from the coupled chaotic circuits for reaching synchronization. Capacitor and induction coil can connect the linear circuits by generating time-varying fields, and the field energy is generated and propagated for balancing the energy outputs for finding phase synchronization and complete synchronization between chaotic circuits. In this paper, a memristor is used to couple two PR circuits and the coupling mechanism is discussed. In particular, scale transformation is considered on the magnetic flux, which is used to estimate the effect of electromagnetic induction and radiation. When time-varying current passed across the memristor, an equivalent induced electromotive force is generated to balance the outputs voltage from the two PR circuits and then field energy is pumped from one to another. The dimensionless dynamical systems under coupling are obtained to find complete synchronization by setting appropriate values for the intrinsic parameters (a, b) for memristor and normalized induction coefficient k. Numerical results have confirmed its effectiveness with our scheme. Acknowledgement: This project is supported by National Natural Science Foundation of China under Grants 11765011, and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.

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Highlights: 1, Synchronization is dependent on the coupling channel and coupling device; 2, Memristor coupling introduces a kind of field coupling and voltage coupling; 3, Energy pumping in the coupling memristor is estimated; 4, Resetting and modulation of initial value can induces synchronization transition.

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Interest conflict: The authors declare there are no any conflicts of interests with this publication.

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