Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

entropy

Article Modeling and Analysis of a Three-Terminal-Memristor-Based Conservative Chaotic System

Ze Wang and Guoyuan Qi *

Tianjin Key Laboratory of Advanced Technology of Electrical Engineering and Energy, Tiangong University, Tianjin 300387, China; [email protected] * Correspondence: [email protected]

Abstract: In this paper, a three-terminal memristor is constructed and studied through changing dual-port output instead of one-port. A new conservative memristor-based chaotic system is built by embedding this three-terminal memristor into a newly proposed four-dimensional (4D) Euler equation. The generalized Hamiltonian energy function has been given, and it is composed of conservative and non-conservative parts of the Hamiltonian. The Hamiltonian of the Euler equation remains constant, while the three-terminal memristor’s Hamiltonian is mutative, causing non- conservation in energy. Through proof, only centers or saddles equilibria exist, which meets the definition of the conservative system. A non-Hamiltonian conservative chaotic system is proposed. The Hamiltonian of the conservative part determines whether the system can produce chaos or not. The non-conservative part affects the dynamic of the system based on the conservative part. The chaotic and quasiperiodic orbits are generated when the system has different Hamiltonian levels. Lyapunov exponent (LE), Poincaré map, bifurcation and Hamiltonian diagrams are used to analyze the dynamical behavior of the non-Hamiltonian conservative chaotic system. The frequency and initial values of the system have an extensive variable range. Through the mechanism adjustment, instead of trial-and-error, the maximum LE of the system can even reach an incredible value of 963. An analog circuit is implemented to verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos. Citation: Wang, Z.; Qi, G. Modeling and Analysis of a Three-Terminal- Keywords: three-terminal memristor; non-Hamiltonian conservative chaotic system; conservative Memristor-Based Conservative chaos; analog circuit Chaotic System. Entropy 2021, 23, 71. https://doi.org/doi:10.3390/ e23010071 1. Introduction Received: 3 December 2020 Accepted: 28 December 2020 Since the HP laboratory [1] confirmed memristors’ physical existence in 2008, the Published: 4 January 2021 memristors [2] have received extensive attention from the academic community. Mem- ristor [3] is a kind of nonlinear resistor with memory function. The memristor has been Publisher’s Note: MDPI stays neu- investigated and applied in non-volatile memory [3], artificial neural network [4], confiden- tral with regard to jurisdictional clai- tial communication [5], analog circuit [6], an artificial intelligence computer [4,6], biological ms in published maps and institutio- behavior simulation [7], etc., showing great potential. nal affiliations. Although two-terminal memristors have proved the basic principle of neurons, the synapses of one neuron are far more than one, so it is necessary to study multi-terminal memristors. The three-terminal Widrow-Hoff memristor [8] has carried out this kind of attempt by adding a control terminal to realize a three-terminal chemical memristor. Copyright: © 2021 by the authors. Li- censee MDPI, Basel, Switzerland. A new floating gate silicon MOS (MOSFET) transistor [9] was similarly proposed, and This article is an open access article Lai proposed field-effect transistors with nano ionic gates [10]. Mouttet proposed the distributed under the terms and con- basic definition of the three-terminal memristor [11] based on the two-terminal memristor ditions of the Creative Commons At- passive nonlinear system [12] by Chua. Recently, using monolayer molybdenum disulfide, tribution (CC BY) license (https:// three-terminal [13], six-terminal, or even more synapses’ memristors were realized. creativecommons.org/licenses/by/ Chaos exists in mathematical models [14] and other aspects such as the macroeco- 4.0/). nomic model [15], the breaking of topological supersymmetry [16], etc. Chaotic systems

Entropy 2021, 23, 71. https://doi.org/10.3390/e23010071 https://www.mdpi.com/journal/entropy Entropy 2021, 23, 71 2 of 17

are divided into the conservative system and dissipative system [17]. The Lyapunov di- mensions of dissipative chaotic systems are fractional; for instance, if the system’s full dimension is three, the Lyapunov dimension is slightly higher than two. The divergence of the dissipative chaotic system is less than zero leading to the phase volume converging to zero with the exponential rate of the divergence. Hence, the passing trajectories of the dissipative system are not ergodic in the 3D space as the chaotic attractor looks like, but oc- cupies zero space of the fractional dimension. The system orbits cannot traverse the entire space given by the initial value, and even most of the spatial range cannot be experienced. The poor ergodicity caused by the fractional dimension is the disadvantage in chaos-based encryption. It is easy to obtain exhaustive attacks when used in encryption systems [17,18]. However, the conservative chaos has a full dimension in phase volume, and encryption based on conservative systems is better than dissipative systems in anti-attack [17–19]. Constructing a memristor-based conservative chaotic system is helpful to the encryption of information and provides better security about information-theory. A conservative chaotic system is a system in which the phase volume space remains unchanged [17–19], so the dimension is an integer [20], and the orbit can traverse and occupy the entire space given by the initial value. Compared with dissipative chaotic systems, conservative chaotic systems are scarce, and conservative chaos can be divided into two types: Hamiltonian conservative (energy conservative) and phase volume conservative [17]. Recently, Qi [17,18] first established the four-dimensional (4D) Euler equations. The 4D Euler equation modeling is essential in mathematics, rigid-body dynamics, and the structure of symplectic manifolds and fluid dynamics [17,18]. Based on this, a 4D conservative chaotic system was constructed, which is strictly conservative chaos. Nowadays, most of the research on memristors is based on dissipative chaos [7,14]. There are two research routes of two-terminal memristor chaotic systems based on dissipative system: (1) The memristor is used to replace nonlinear components such as the Chua circuits [14,21], oscillator circuits [22], etc. The hidden attractor, multistability [23], hyperchaotic and fractional-order form [24] were proposed. (2) The memristor is used as a feedback term to couple into a neuron model such as the Hindmarsh-Rose (HR) neuron model [7], etc. Some recent research methods on memristors, such as Chua’s periodic table [16,25,26] and the multidimensional scaling [27], have not explained the causes of chaos from the perspective of Hamiltonian energy. Only a few two-terminal memristor systems are driven by piecewise function [28] or sine function [29,30] to obtain conservative chaotic systems. However, the values of positive Lyapunov exponent (LE) of these systems in [28–30] are too small. The system [29] just satisfies that the sum of LEs is zero, but the system does not analytically meet zero divergence requirements. More importantly, to our best knowledge, no conservative chaos based on three-terminal memristor has been studied. A memristor-based conservative chaotic system is more complicated with more parameters, which increases the keyspace of the chaos generator. Besides, a memristor-based conservative chaotic system with a high positive Lyapunov exponent is necessary for chaos-based encryption to generate the pseudo-random number. We consider designing a conservative three-terminal memristor chaotic system based on the 4D Euler equation given in [18]. The 4D Euler equation is a very delicate ordinary differential equation based on mathematics, which is strictly conservative in both energy and divergence, but it just produces the periodic orbit instead of the chaos. By coupling the Euler equation with a three-terminal memristor, the energy conservation is broken, but the phase-volume conservation (divergence being zero) is still kept. Therefore, this paper constructs a three-terminal-memristor-based conservative chaotic system, and gives its energy function, pointing out the cause of its chaos. A large positive Lyapunov exponent is produced, which is advantageous over existing dissipative chaos and conservative chaos. An analog circuit is designed to prove the theory’s feasibility, and verify the existence of the non-Hamiltonian conservative chaotic system, which overcomes the challenge that a little bias will lead to the disappearance of conservative chaos. Entropy 2021, 23, 71 3 of 17

This paper is organized as follows: Section2 proposes the three-terminal memristor and constructs the circuit to implement it. Section3 proposes the conservative chaotic system based on a three-terminal memristor from a strict conservative system [18]. Section4 gives the characteristics of equilibria of the three-terminal memristor conservative system. Section5 gives the dynamical analysis, and the cause of dynamical changing of the system in different levels and the impact of Hamiltonian on the system are investigated. Analog circuit implementation is provided in Section6. Section7 summarizes the paper.

2. Modeling of Three-Terminal Memristor The memristor predicted by Chua is a two-port device [31]. It also has three signiﬁcant features: the hysteresis loop passes the origin, the hysteresis loop is the shape of eight, and the area of the hysteresis curve decreases with increasing signal frequency. Here, we choose the cubic smooth memductance nonlinearity model [14,32]. The memristor model is described in the following form

. ϕ = v, W(ϕ) = α + βϕ2, (1) i = W(ϕ)v = (α + βϕ2)v.

According to the magnetic controlled three-terminal memristor model proposed in the memristive systems analysis [11], the following results are obtained

dw dt = f w, vg, vd , ig = g w, vg, vd , (2) id = h w, vg, vd .

Here vg, vd represent the input voltages, ig, id represent the output currents, w donates an n-dimensional state variable of the system, g, h are deﬁned as continuous function, and f is an n-dimensional continuous function. The memristor model in Equation (1) is appropriately deformed. By changing it to the model of single-port input and dual-port output, the model is transformed as

dϕ dt = f (w, v1, v2) = vin = v1v2, 2 i1 = g(w, v1, v2) = W(ϕ)v1 = (α + βϕ )v1, (3) 2 i2 = h(w, v1, v2) = W(ϕ)v2 = (α + βϕ )v2.

where vin = v1v2 is the input voltage of the model, ϕ is the magnetic ﬂux that controls the state of the model, W(ϕ) represents the memductance, and i1, i2 are the two current outputs. To verify the feature of the model Equation (3), we used Matlab for numerical simulation, as shown in Figure1, The product of v1 = A1 sin(ω1t) and v2 = A2 sin(ω2t) is the input of this device with parameters A1 = A2 = ω2 = 1, and ω1= 2 in Figure1a and ω1= 10 Figure1b. We found that the hysteresis curve does not converge to a single-valued function, but a multi-valued resistance with the frequency increasing, which means this model has a complex resistance value. The three-terminal memristor can be implemented by the circuit shown in Figure2. Entropy 2021, 23, x FOR PEER REVIEW 4 of 19 Entropy 2021, 23, x FOR PEER REVIEW 4 of 19

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(a) (b) (a) (b) Figure 1. I-V curves with different frequency inputs: (a) ωω=2， =1 , (b) ωω=10， =1. ωω12， ωω12， FigureFigure 1. I-V 1. I-V curves curves with with different different frequency frequency inputs: inputs: (a) (a)12ω=21= 2, =1ω2,= (b1), (b12)=10ω1= 10, =1ω.2 = 1.

Figure 2. Circuit implementation of the three-terminal memristor. Figure 2. Circuit implementation of the three-terminal memristor. Figure 2. Circuit implementation of the three-terminal memristor. In Figure2, v1, v2 are input voltages of the model, M1 to M4 multipliers, R1 to R5 In Figure 2, vv, are input voltages of the model, , M to M multipliers, R to resistors, C1 a capacitor,12 and i1, i2 output currents, respectively.1 The analytical4 mathematical1 In Figure 2, vv12, are input voltages of the model, , M1 to M4 multipliers, R1 to Rmodel5 resistors, is given C1 aas capacitor, and ii12, output currents, respectively. The analytical math- R5 resistors, C1 a capacitor, and ii12, output currents,(R )2 respectively. The analytical mathematical model is given as v1 v1v2dt v1 i1 = R + 2 2 , ematical model is given as 2 R1C1 R3 R 2 (4) v2 ( v1v2dt) v2 i2 = + 2 2 2 . R5 ()vvdt12R C R v 1 v1  1 1 4 i =+v ()vvdt12 v 1 , 1=+1  22 R1, C1 form the integrated circuit,i1 andRRCR2113 the remaining22 , four resistors match the output Entropy 2021, 23, x FOR PEER REVIEW RRCR 5 of(4) 19 coefﬁcients. Using Multisim to simulate2113 the model,2 input voltages are consistent with 2 (4) Matlab numerical simulation. The circuit simulation()vvdt12 results v 2 received are shown in Figure3. v2  i =+v ()vvdt12 v 2 . 2=+2  22 i2 RRCR511422 . RRCR5114

RC11, form the integrated circuit, and the remaining four resistors match the output co- RC, form the integrated circuit, and the remaining four resistors match the output co- efficients.11 Using Multisim to simulate the model, input voltages are consistent with efficients.Matlab numerical Using Multisim simulation. to simulateThe circuit the simula model,tion input results voltages received are are consistent shown in Figurewith Matlab3. numerical simulation. The circuit simulation results received are shown in Figure 3.

(a) (b) ωω== ωω== FigureFigure 3. 3. I-VI-V curves curves of of Multisim Multisim simulation simulation circuit circuit ( (aa)) ω112=2,2, ω2 1=(b1) (b)12ω110,= 10, 1ω.2 = 1.

It was found that the simulation results of the circuit model are consistent with the simulation of the mathematical model, so the model can be used to build the actual hardware circuit.

3. Modeling of Conservative Chaotic System Based on Three-Terminal Memristor Qi proposed a 4D Euler rigid body equation with a Hamiltonian vector field form [18]

x =∇Jx() Hx ()， (5)

where

− 00xx32 xxxx0 −− Jx()= 3143 , −+ − (6) xxx2140 x 2 − 00xx32

with

1 Hxxxxx()=+++ (ππππ2222 ). (7) 2 11 22 33 44

The system can be written as =−ππ xxx13223(), =−ππ +− ππ xxxxx213134334()(), =−ππ +− ππ (8) xxxxx321122424()(), =−ππ xxx43223().

The divergence of the 4D Euler equation is

4 ∂f ∇⋅x =xi =0.  ∂ (9) i=1 xi

Therefore, Equation (8) is a phase-volume conservative system. Because Jx() in Equation (6) is a skew-symmetric matrix, we have

HHxJxHx =∇()T () ∇ () = 0 . (10)

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It was found that the simulation results of the circuit model are consistent with the simulation of the mathematical model, so the model can be used to build the actual hardware circuit.

3. Modeling of Conservative Chaotic System Based on Three-Terminal Memristor Qi proposed a 4D Euler rigid body equation with a Hamiltonian vector ﬁeld form [18]

. x = J(x)∇H(x), (5)

where   0 −x3 x2 0  x 0 −x − x x  J(x) =  3 1 4 3 , (6)  −x2 x1 + x4 0 −x2  0 −x3 x2 0 with 1 H(x) = (π x2 + π x2 + π x2 + π x2). (7) 2 1 1 2 2 3 3 4 4 The system can be written as

. x1 = (π3 − π2)x2x3, . x2 = (π1 − π3)x1x3 + (π4 − π3)x3x4, . (8) x3 = (π2 − π1)x1x2 + (π2 − π4)x2x4, . x4 = (π3 − π2)x2x3. The divergence of the 4D Euler equation is

4 . ∂ fx ∇ · x = ∑ i = 0. (9) i=1 ∂xi Entropy 2021, 23, x FOR PEER REVIEW 6 of 19

Therefore, Equation (8) is a phase-volume conservative system. Because J(x) in Equation (6) is a skew-symmetric matrix, we have Thus, the system of Equation. (8) is a Hamiltonian conservative system. Therefore, it H = ∇H(x)T J(x)∇H(x) = 0. (10) preserves both the phase-volume and Hamiltonian. Using Matlab for numerical simulation, take parameters [ππππ , , , ]TT= [2, 3, 4, 5] , initial conditions Thus, the system of Equation (8)1234 is a Hamiltonian conservative system. Therefore, it TT=−− = preserves[xxxx10 , 20 , 30 both , 40 the ] phase-volume [5, 5, 5, 5] and, and Hamiltonian. sampling time Using T Matlab0.001 for s. numerical Since the simulation, conserva- T T T tiontake of parameters both phase-volume[π1, π2, π and3, π Hamiltonian,4] = [2, 3, 4, th 5is] ,system initial conditionsonly produces[x10 ,periodicx20, x30 orbit, x40 ][17–= 19],[5, 5,as− shown5, −5 ]inT, Figure and sampling 4. time T = 0.001 s. Since the conservation of both phase- volume and Hamiltonian, this system only produces periodic orbit [17–19], as shown in Figure4.

Figure 4. Periodic orbit of the sy systemstem of Equation (8).

The 4-D Hamiltonian Hamiltonian conservative conservative system system on onlyly produces produces a aperiod periodicic orbit orbit because because of theof the conservation conservation of both of both phase phase volume volume and andHamiltonian Hamiltonian energy. energy. Does Does it generate it generate conservativeconservative chaos chaos by breaking by breaking one of one the of conser the conservations,vations, like the like Hamiltonian the Hamiltonian energy? energy? Qi [17] proposed a Hamiltonian conservative chaotic system by changing the Casimir conservation and keeping the Hamiltonian constant. Only one parameter was changed. So far, to our best knowledge, no memristor has been applied in all the conservative chaotic systems generation. To generate chaos, we should break the conservation of Hamiltonian by adding a ()ππ pair of constants c and 41c in the symplectic matrix Jx() in Equation (6), and then Equation (5) can be written as =∇ x Jxc () Hx (), (11)

where − 0 x32xc xxxx0 −− 3143 = −+ − Jxc ()x214xx0 x 2 , (12) π −−4 cx x 0 π 32 1

For simplification, the system parameters are fixed as

[][]ππππTT= 1234,,, 2,3,4,5. (13)

Then Equation (11) becomes =+ xxxcx12345, =− + x213342,xx xx =− (14) x31224xx2, xx =− xxxcx42315.

with

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Qi [17] proposed a Hamiltonian conservative chaotic system by changing the Casimir conservation and keeping the Hamiltonian constant. Only one parameter was changed. So far, to our best knowledge, no memristor has been applied in all the conservative chaotic systems generation. To generate chaos, we should break the conservation of Hamiltonian by adding a pair of constants c and (π4/π1)c in the symplectic matrix J(x) in Equation (6), and then Equation (5) can be written as . x = Jc(x)∇H(x), (11) where   0 −x3 x2 c  x3 0 −x1 − x4 x3  Jc(x) =  , (12)  −x2 x1 + x4 0 −x2  − π4 c −x x 0 π1 3 2 For simpliﬁcation, the system parameters are ﬁxed as

T T [π1, π2, π3, π4] = [2, 3, 4, 5] . (13)

Then Equation (11) becomes

. Entropy 2021, 23, x FOR PEER REVIEW x1 = x2x3 + 5cx4, 7 of 19 . x2 = −2x1x3 + x3x4, . (14) x3 = x1x2 − 2x2x4, . x41= x2x3 − 5cx1. H ()xxxxx=+++ (22222 3 4 5 ). (15) 2 1234 with Because the main diagonal of Equation1 2 (12) is2 zero2 as Equation2 (6), the phase volume H(x) = (2x1 + 3x2 + 4x3 + 5x4). (15) is still conservative. 2 However,Because thefor the main Hamiltonian diagonal of energy, Equation we (12) have is zero as Equation (6), the phase volume is still conservative. However, for the =∇ HamiltonianT ∇ energy, = weππ have() − π =− HHxJxHxc()c () ()41 414 xxcxx 15 14 . (16) . T H = ∇H(x) Jc(x)∇H(x) = cπ4(π1 − π4)x1x4 = −15cx1x4. (16) Thus, the system is non-conservative in Hamiltonian energy but conservative in TT=−− phase Thus,volume. the systemSetting isinitial non-conservative conditions in[xxxx Hamiltonian10 , 20 , 30 , 40 energy ] [5,5, but conservative 5, 5] , constant in phase = = T T c volume.1 and Settingsampling initial time conditions T 0.001[xs,10 this, x20 system, x30, x 40produces] = [5 , 5chaotic , −5 , orbits−5] , constant[Figure 5a];c = 1 therefore,and sampling the Hamiltonian time T = 0.001 of Equations, this system (12) is produces non-conservative chaotic orbits because [Figure the 5positivea]; therefore, and the Hamiltonian of Equation (12) is non-conservative because the positive and negative negative changes of x14x [Figure 5b]. changes of x1x4 [Figure5b].

(a) (b) Figure 5. Chaotic orbits of system Equation (11), (a) time series of x , (b) time series of x x and Figure 5. Chaotic orbits of system Equation (11), (a) time series of1 x1,(b) time series of14x1x4 and Hamiltonian.Hamiltonian.

When the three-terminal memristor is coupled with the 4D Euler equation, can chaos be generated using a similar way? The memristor model is regarded as a device inserted in the 4D rigid body to replace 5c and −5c . To get the input of the memristor, we added αβ+ 2 another variable x5 as the input of the memristor. Thus, the memristor is x5 , and the new system is described as

=++γα β 2 x123xx(), x 54 x =− + xxxxx213342, =− xxxxx312242, (17) =−+γα β 2 x423xx(), x 51 x = xxx514.

Here γ is the three-terminal memristor weight parameter. Therefore, the memristor is added as a feedback term to Equation (12). The divergence of Equation (17) is

5 ∂f ∇⋅x =xi =0.  ∂ (18) i=1 xi

which means the phase volume of the system (17) is still conservative.

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When the three-terminal memristor is coupled with the 4D Euler equation, can chaos be generated using a similar way? The memristor model is regarded as a device inserted in the 4D rigid body to replace 5c and −5c. To get the input of the memristor, we added 2 another variable x5 as the input of the memristor. Thus, the memristor is α + βx5, and the new system is described as

. 2 x1 = x2x3 + γ(α + βx5)x4, . x2 = −2x1x3 + x3x4, . x3 = x1x2 − 2x2x4, (17) . 2 x4 = x2x3 − γ(α + βx )x1, . 5 x5 = x1x4.

Here γ is the three-terminal memristor weight parameter. Therefore, the memristor is added as a feedback term to Equation (12). The divergence of Equation (17) is

5 . ∂ fx ∇ · x = ∑ i = 0. (18) i=1 ∂xi

which means the phase volume of the system (17) is still conservative. Now, we test whether the Hamiltonian energy is still conservative. The generalized Hamiltonian form was used [33]. We consider the input of the ﬁfth term as a non- conservative force, and get . x = M(x)∇H(x), (19) where M(x)∇H(x) = Jm(x)∇H(x) + R(x)∇H(x) = fc(x) + fd(x), (20) with new Hamiltonian 1 H = (x2 + x2 + x2 + x2 + x2), (21) 2 1 2 3 4 5 and  2  0 0.5x3 0.5x2 γ(α + βx5) 0  −0.5x3 0 1.5(x4 − x1) −0.5x3 0    Jm(x) =  −0.5x2 1.5(x1 − x4) 0 −0.5x2 0 ,  2   −γ(α + βx5) 0.5x3 0.5x2 0 0  0 0 0 0 0    2    0 0 0 0 0 x2x3 + γ(α + βx5)x4 0  0 0 0 0 0   −2x1x3 + x3x4   0        R(x) =  0 0 0 0 0 , fc(x) =  x1x2 − 2x2x4 , fd(x) =  0 .    2     0 0 0 0 0   x2x3 − γ(α + βx5)x1   0  0 0 0 0 x1x4 0 x x x5 1 4 M(x) is no longer skew-symmetric but is decomposed into the sum of a skew- symmetric matrix J(x) and a symmetric matrix R(x). The total force exerted on the system is non-conservative. Differentiating the Hamilton function, we get

...... T H = (x1x1 + x2x2 + x3x3 + x4x4 + x5x5) = x1x4x5 = ∇H fd(x) = x1x4x5, (22)

which indicates Hamiltonian energy function is no longer conservative. We divide H of Equation (21) into H = Hc + Hn, (23) with 1 1 H = (x2 + x2 + x2 + x2), H = x2. c 2 1 2 3 4 n 2 5 Entropy 2021, 23, 71 8 of 17

. . It can be proved that Hc = 0, but Hn = x1x4x5. Therefore, Hc represents the conservative energy of the system, and Hn is the non-conservative part. According to [17], there are two categories of conservative systems: the Hamiltonian conservative chaotic system, in which both the volume and Hamiltonian of the system are constant, and the non-Hamiltonian conservative chaotic system, in which only the phase volume is conservative. Thus, the proposed three-terminal-memristor-based system (i.e., Equation (17)) is a typical non-Hamiltonian conservative chaotic system.

4. Equilibria and Their Stability of Three-Terminal Memristor Conservative System The equilibria point plays an essential role in analyzing the system’s properties and we examined whether it meets the requirements of the conservative chaotic system. For a conservative system, only saddles and centers exist. There are no stable or unstable nodes and foci to exist in a conservative system. Equation (17) can be rewritten as

. 2 x1 = x2x3 + αγx4 + βγx4x5, . x2 = −2x1x3 + x3x4, . x3 = x1x2 − 2x2x4, (24) . 2 x4 = x2x3 − αγx1 − βγx1x , . 5 x5 = x1x4.

Setting the left of Equation (24) equal to 0, we can get three cases: Case 1: x1 = 0, x4 = 0, we can get x2x3 = 0, x5 ∈ R. This case has three sub-cases as follows: Case 1.1: x1 = 0, x2 = 0, x3 = 0, x4 = 0, x5 = x5, Case 1.2: x1 = 0, x2 = 0, x3 6= 0, x4 = 0, x5 = x5, Case 1.3: x1 = 0, x2 = 0, x3 6= 0, x4 = 0, x5 = x5, Case 2: x1 6= 0, x4 = 0. Case 3: x1 = 0, x4 6= 0.

In Case 2, we derive x2 = x3 = 0, but parameters α, β > 0 and the weight parameter 2 2 γ 6= 0, γ(α + βx5)x1 = 0 holds. Since x1 6= 0, we have γ α + βx5 = 0 which contradicts the premise. Therefore, Case 2 does not hold. Case 3 has the same problem as Case 2. T Therefore, from Case 1, the system has line equilibria E5 = [0, 0, 0, 0, x5] , plane equilibria T T E3,5 = [0, 0, x3, 0, x5] , and E2,5 = [0, x2, 0, 0, x5] . The Jacobi matrix of the system is  ∗ ∗ ∗2 ∗ ∗  0 x3 x2 γ(α + βx5 ) 2γβx4 x5 ∗ ∗ ∗ ∗  −2x3 0 x4 − 2x1 x3 0   ∗ ∗ ∗ ∗  J =  x2 x1 − 2x4 0 −2x2 0 . (25)  ∗2 ∗ ∗ ∗ ∗   −γ(α + βx5 ) x3 x2 0 −2γβx1 x5  ∗ ∗ x4 0 0 x1 0

By substituting E5 into the characteristic equation, we ﬁnd the eigenvalues of E5 as

2 2 E5 = (0, 0, 0, 0, x5) → λ = (0, 0, 0, −γj(βx5 + α), γj(βx5 + α)). (26)

Hence, line equilibria E5 are centers. For the plane equilibrium E3,5, we ﬁnd the eigenvalues as √ √ 1 3 1 3 E = (0, 0, x , 0, x ) → λ = (0, 0, a , − a − j a , − a + j a ), (27) 3,5 3 5 4 2 4 2 5 2 4 2 5 Entropy 2021, 23, 71 9 of 17

where 2 2 2 ∗2 2 2 ∗4 ∗2 α γ 2αβγ x5 β γ x5 x3 a1 = 3 + 3 + 3 + 3 , ∗2 ∗2 ∗2 3βγx3 x5 3αγx3 a2 = 2 + 2 , rq 3 3 2 a3 = a1 + a2 − a2, a = a − a1 , 4 3 a3 a5 = 2a3 − a4.

where j is the imaginary unit. If a4 = 0 holds, E3,5 must be centers from Equation (24). If a4 6= 0, we can ﬁnd out if a4 > 0, λ3 is a positive real number and the real parts of λ4, λ5 must be negative. If a4 > 0, λ3 is a negative real number and the real parts of λ4, λ5 must be positive. Therefore, if a4 6= 0, E3,5 must be saddles. Likewise, plane equilibrium E2,5 has the same properties as E3,5. In sum, both E2,5 and E3,5 are either natural elliptic (centers) or saddles. Thus, from the perspective of these equilibria, the system fully meets the characteristics that the conservative system in phase space only has either saddle or center [17–19].

5. Dynamical Analysis of Three-Terminal Memristor Conservative Chaotic System 5.1. Memristor Effect in Chaos Generation T T For system (19), take initial values [x10, x20, x30, x40, x50] = [1, 1, −1, −1, 0] , pa- Entropy 2021, 23, x FOR PEER REVIEW T T 11 of 19 rameters [α, β, γ] = [1, 1, 0] , and sampling time T = 0.001 s. Since γ = 0, the outputs of the three-terminal memristor do not affect the system. According to the analysis of Equation (10), it produces periodic orbit, as shown in Figure6a.

(a) (b)

Figure 6. Cont.

(e) (f)

(g) (h)

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(a) (b)

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(e) (f)

(g) (h)

Figure 6. Numerical characteristics of the system of Equation (19): (a) 3D view of x1 − x2 − x3 with γ = 0,(b) Lyapunov exponents with γ = 1,(c) 3D view of x1 − x2 − x3 with γ = 1,(d) phase portrait of x1 − x2 with γ = 1,(e) Poincaré map of x2 − x3 with x1 = 0,(f) Hamiltonian energy, (g) Lyapunov exponent (LEs) with large coefﬁcients, and (h) Poincaré map with large coefﬁcients.

Now fixing α = β = γ = 1 and initial values as above, the orbits of different phase spaces are shown in Figure6c,d. Therefore, the three-terminal-memristor-based conservative system produces chaos, which is called the three-terminal-memristor-based conservative chaotic system. The Poincaré map from Figure6e with x1 = 0 shows the orbits are chaotic and do not form a chaotic attractor. The chaotic attractor of a dissipative system has several little branches of hair-like Poincaré map because of the fractal dimension, but this memristor-based conservative chaos has a wide-banding Poincaré map that almost evenly fills the space initially occupied. This is because it has an integer dimension. The ergodicity of the memristor-based conservative chaotic system is much better than general dissipative chaotic systems, which is beneficial in chaos-based encryption. T The LEs L1,2,3,4,5 = [0.4446, 0.0003, 0.0000, −0.0003, −0.4446] in Figure6b, indicating the sum of the LEs is zero. From Ref. [20], the Kaplan-Yorke Lyapunov dimension is LKY = 4 + (L1 + L2 + L3 + L4)/|L5| = 5. (28)

The integer result proves the conservativeness of the system. From Figure6f, Hc is constant, indicating this part is conservative; however, the total Hamiltonian of the system H changes, which is caused by the non-conservative part Hn. Thus, we confirmed the system is a typical non-Hamiltonian conservative chaotic system. In this case, the maximum LE value, L1 = 0.4446, is small, and the Poincaré map area is not large enough, although it fills evenly. Normally, in chaos-based encryption, the larger maximum LE value, the higher-order in pseudo-randomness of the chaos generator, the better in security, the more difficult to break. Besides, the larger area of Poincaré map, the more choices in the selection of pseudo-randomness. To enhance the maximum LE, many references took a trial-and-error method through bifurcation; we took a mechanism Entropy 2021, 23, 71 11 of 17

way in this paper. Provided that the frequency and energy of the system are increased, the positive Lyapunov exponent can be increased. Therefore, we adjusted the ﬁve initial values and four parameters according to their functions in frequency and energy. Accord- ing to the 4D Euler equation, increasing πi can speed up the operating frequency of the system, and increasing the initial values can store the kinetic energy. The initial values also determine the spatial magnitude of orbits and the area of the Poincaré map. Initial val- T T T T ues [x] = [1000, 1000, −1000, −1000, 0] and [π1, π2, π3, π4] = [100, 150, 200, 250] , and sampling time T = 10−7 s, were chosen to calculate Lyapunov exponents using the 3 Wolf method [34]. As shown in Figure6g,h, the maximum of the LEs LE1 = 0.9631 × 10 . The maximum of the LEs is far greater than those of the most chaotic systems; it can be observed from the Poincaré map with x1 = 0 that the space of the orbits is quite large, and the frequency is very high, which can be tested by frequency spectrum (the ﬁgure is omitted). Usually, the maximum LE cannot be adjusted too large through mechanical analysis. To increase the LE, the scaling time can be adjusted; however, this proposed memristor-based chaotic system can produce exceptionally large maximum LE through Entropy 2021, 23, x FOR PEER REVIEW the adjustments of parameter and initial, which is based on the physical13 of mechanism.19 The

large LE and magnitude of Poincaré map are greatly helpful in chaos-based encryption.

5.2. Dynamical Analysis with Different Initial Conditions of Hc 5.2. Dynamical Analysis with Different Initial Conditions of Hc We have analyzed the three-terminal memristor excitation of chaos, so how does We have Hamiltoniananalyzed the affectthree-terminal system dynamics? memristor Fixing excitation parameter of chaos,[α, β ,soγ ]howT = [does1, 1, 1]T, the initial [][]αβγTT= Hamiltonian affectvalue system determines dynamics?H0 from Fixi Equationng parameter (29). ,, 1,1,1, the initial value determines H from Equation (29). 0 1 H ≥ H = (x2 + x2 + x2 + x2 ) (29) 0 c0 2 10 20 30 40 ≥=1 2222 +++ HH0c 0() xxxx 10 20 30 40 (29) 2 T Choose the initial value x0 = [0.1, x20, 0.1, 0.1, 0] with x20 ∈ [−2, 2], T x = [x , 0.1, 0.1, 0.1, 0] with x ∈ [−2, 2]. TheT bifurcation diagrams are plotted by re- Choose the0 10initial value xx= []0.1,10 ,0.1,0.1,0 with x ∈−[ 2, 2] , moving 80% of transients. Both020 initials correspond to the Hamiltonian20 H ∈ [0.015, 2.015]. T 0 = []∈−[ ] xx010, 0.1, 0.1,Figure 0.1, 07a,b with are thex10 bifurcation2, 2 . The diagrams bifurcation based diagrams on the change are plotted of initials. by There- stripe colors H moving 80% ofrepresent transients. Hamiltonian Both initials correspond0 marked byto the the Hamiltonian color bar. The H low∈[0.015, Hamiltonian 2.015] energy (dark blue and blue) represents quasiperiodic orbits, and the high0 Hamiltonian energy (light . Figure 7a,b areblue, the bifurcation yellow, and diagrams red) represents based chaos.on the change of initials. The stripe colors represent Hamiltonian H marked by the color bar. The low Hamiltonian energy (dark Figure70 c is the zoom of Figure7b within x10 ∈ [1, 1.2]. There are some bifurca- blue and blue)tions, represents for instance, quasiperiodic when xorbi10 =ts,1 andand thex10 high= 1.2 Hamiltonian, the system energy produces (light chaotic orbits, as blue, yellow, andshown red) inrepresents Figure7d,f. chaos. However, it also generates the periodic orbits, period-2 orbits, and quasiperiodic orbits with x10 = 1.1, as shown in Figure7e.

(a) (b)

Figure 7. Cont.

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5.2. Dynamical Analysis with Different Initial Conditions of Hc We have analyzed the three-terminal memristor excitation of chaos, so how does Hamiltonian affect system dynamics? Fixing parameter [][]αβγ,,TT= 1,1,1, the initial

value determines H0 from Equation (29).

1 HH≥=() xxxx2222 +++ (29) 0c 02 10 20 30 40

= []T ∈−[ ] Choose the initial value xx0200.1, ,0.1,0.1,0 with x20 2, 2 , T = []∈−[ ] xx010, 0.1, 0.1, 0.1, 0 with x10 2, 2 . The bifurcation diagrams are plotted by re- ∈[ ] moving 80% of transients. Both initials correspond to the Hamiltonian H0 0.015, 2.015 . Figure 7a,b are the bifurcation diagrams based on the change of initials. The stripe colors

represent Hamiltonian H0 marked by the color bar. The low Hamiltonian energy (dark blue and blue) represents quasiperiodic orbits, and the high Hamiltonian energy (light blue, yellow, and red) represents chaos.

Entropy 2021, 23, 71 12 of 17

(a) (b)

Entropy 2021, 23, x FOR PEER REVIEW 14 of 19

(e) (f)

Figure 7. DynamicalFigure 7. analysisDynamical with analysis different with initial different conditions: initial conditions: (a) Bifurcation (a) Bifurcation within initial withinx20 initial∈ [−2, 2],(b) bifurcation ∈−[ ] =−[ ] ∈[ ] within initialxx2010 = 2,[− 22,, 2 (]b,()c bifurcation) bifurcation within within initial initialx10x10 ∈ 2,[1, 2 1.2, (]c,() dbifurcation) chaotic orbits within with initialx10 =x101,(e1,) quasiperiodic 1.2 , (d) orbits

with x10 = 1.1, (f) chaotic orbits with= x10 = 1.2. = = chaotic orbits with x10 1 , (e) quasiperiodic orbits with x10 1.1 , (f) chaotic orbits with x10 1.2 .

∈ [ ] Figure 7c5.3. is the Dynamical zoom of Analysis Figure 7b with within Fixed Hx10c and1, Varied1.2 . There Hn are some bifurcations, = = for instance, when Althoughx10 1 and the non-conservativex10 1.2 , the system part producesHn changes chaotic with orbits, time at as the shown rate ofin Equation (22), Figure 7d,f. However,in Sections it also 5.1 generates and 5.2, the pe initialriodic value orbits, of period-2x50 6= 0 orbits,is crucial. and quasiperi- When it is set to zero, = odic orbits withthe x inﬂuence10 1.1, as of shown three-terminal in Figure 7e. memristor variable is reduced as much as possible from T Equation (23). In this section, the four initial values [x10, x20, x30, x40] corresponding

5.3. Dynamical toAnalysis the conservative with Fixed H partc andH cVaried0 being H ﬁxed,n we analyze how the initial value x50 corre- T sponding to the Hn0 effects of the system. Fix the initial values [x10, x20, x30, x40] = Although the non-conservative part H n changes with time at the rate of Equation [0.1, 0.1, 0.1, 0.1]T with H = 0.02 and [x , x , x , x ]T = [1, 1, 1, 1]T with H = 2, (22), in Section 5.1 and 5.2, the initial valuec0 of x ≠ 0 is10 crucial20 30. When40 it is set to zero, c0 and 50 the influence of three-terminal memristor variable is reduced1 as much as possible from H = x2 . (30) []n0 2 50 T Equation (23). In this section, the four initial values xxxx10,,, 20 30 40 corresponding to The low Hc0 energy determines that the system generates quasiperiodic orbits the conservative part H c 0 being fixed, we analyze how the initial value x50 corre- [Figure8a] and the high Hc energy makes chaos [Figure8c], which corresponds to the H 0 sponding toresults the givenn0 in Sections effects 5.1of and the 5.2 . Regardlesssystem. Fix of the the value initial of conservative values Hamiltonian [][]TT= = [][]TT= xxxx10, 20 , 30 , 40Hc0, x 0.1,50 promotes 0.1, 0.1, 0.1 the chaoticwith H degreec0 0.02 of the and system xxxx10, within 20 , 30 , a 40 range. 1,1,1,1 Beyond this range, it = suppresses the system’s dynamics and transits from chaotic motion to quasiperiodic motion. with Hc0 2 , and As the Hamiltonian level increases, the range becomes wider, as shown in Figure8b,d. 1 Hx= 2 . (30) n0502

The low Hc0 energy determines that the system generates quasiperiodic orbits [Fig-

ure 8a] and the high H c 0 energy makes chaos [Figure 8c], which corresponds to the results given in Section 5.1 and 5.2. Regardless of the value of conservative Hamiltonian

H c 0 , x50 promotes the chaotic degree of the system within a range. Beyond this range, it suppresses the system’s dynamics and transits from chaotic motion to quasiperiodic motion. As the Hamiltonian level increases, the range becomes wider, as shown in Figure 8b,d.

Entropy 2021, 23, x FOR PEER REVIEW 15 of 19 Entropy 2021, 23, 71 13 of 17

(a) (b)

(c) (d) = Figure 8. DynamicalFigure 8. Dynamical analysis with analysis ﬁxed withHc0 ,(fixeda) Bifurcation Hc0 , (a) Bifurcation of x50 with of Hxc500 =with0.02 H,(cb0 ) LE0.02s of, (bx)50 LEswith of Hc0 = 0.02, = = = = = (c) bifurcationx50 of withx50 with Hc0Hc0.020 2,, (c (d) )bifurcationLEs of x50 ofwith x50H cwith0 2. Hc0 2 , (d) LEs of x50 with Hc0 2 .

6. Circuit Implementation6. Circuit Implementation There are threeThere methods are for three the methods simulation for of the nonlinear simulation systems: of nonlinear chaotic system systems: sim- chaotic system ulation circuit,simulation FPGA (Field circuit, Programmable FPGA (Field Gate Programmable Array), and computer Gate Array), numerical and computer simu- numerical lation. For thesimulation. most dissipative For the systems, most dissipative the required systems, accuracy the required is not high, accuracy so all is notthree high, so all three simulation methodssimulation can methodsbe implemented. can be implemented. However, for However, conservative for conservative systems, systems,the re- the required quired accuracyaccuracy is very ishigh very, especially high, especially FPGA FPGAand numerical and numerical simulation. simulation. Even if Even the ac- if the accuracy is curacy is high,high, there there still stillare problems are problems such such as algorithms, as algorithms, sampling, sampling, discretization, discretization, and and the number the number of ofbits bits of ofcomputer computer operations operations causing causing errors. errors. The The conservative conservative system system diver- divergence is zero, gence is zero, soso a a little little bit bit of of error error will will make make the the divergence divergence either either larger larger than than zero zero leading leading to instability, to instability, oror less less than than zero zero leading leading to shrinking. to shrinking. Both Both situations situations cause cause the conservative the conservative chaotic chaotic phase diagramphase diagram to disappear. to disappear. The hardware The hardware implementation implementation is challenging. is challenging. There- Therefore, for fore, for conservativeconservative systems, systems, analog analog circuit circuit simulation simulation is indispensable is indispensable because because it is it a is a real analog real analog simulation.simulation. As shown in FigureAs shown 9, the in system Figure is9 ,implem the systemented is by implemented an analog electronic by an analog circuit electronic with- circuit with- out scale transformation.out scale transformation. A total of 10 A analog total of multipliers 10 analog multipliers accomplish accomplish seven quadratic seven quadratic terms terms and twoand cubic two terms cubic in terms the system. in the system.Besides, Besides, there are there five areintegrators ﬁve integrators and four and in- four inverters, verters, which whichare composed are composed of some of capacito some capacitorsrs and resistors. and resistors. The three-terminal The three-terminal memris- memristor is tor is constructedconstructed at the bottom at the of bottom Figure of 9. Figure 9. []TT= [ ] Selecting the initial values x0 0.2, 0.2, 0.2, 0.2, 0 and parameters [][]αβγ, ,TT= 1, 0.1, 1 , the phase portraits of numerical simulation are shown in Figure 10a,b. In the analog circuit, the chips have a slight voltage deviation that corresponds to a small initial value, so there is no need to add the initial voltage between the capacitor’s two pins.

Entropy 2021, 23, 71 14 of 17 Entropy 2021, 23, x FOR PEER REVIEW 16 of 19

Entropy 2021, 23, x FOR PEER REVIEW 16 of 19

Figure Figure9. Circuit 9. implementationCircuit implementation of system Equati ofon system (15), with Equation electronic (15), parameters: with electronic parameters: R1,R2,R3,R5,R6,R7,R8,R10,R11,R12,R13,R14,R15,R17,R19=10KΩ;R1, R2, R3, R5, R6, R7, R8, R10, R11, R12, R13, R14, R15, R17, R19 R4,R9= 10=5KΩ;KΩ; R R16,R18=100KΩ;4, R9 = 5 KΩ; R16, R18 = C1,C2,C3,C4,C5=10nF.100 KΩ; C1, C2,. C3, C4, C5 = 10 nF.

T T T Figure 9. Circuit implementationSelecting the of initialsystem values Equation[x0 (15),] = with[0.2, electronic 0.2, 0.2, parameters: 0.2, 0] and parameters [α, β, γ] = R1,R2,R3,R5,R6,R7,R8,R10,R11,R12,R13,R14,R15,R17,R19=10KΩ;[1, 0.1, 1]T, the phase portraits of numerical simulation R4,R9=5KΩ; are R16,R18=100KΩ; shown in Figure 10a,b. In the C1,C2,C3,C4,C5=10nF.analog. circuit, the chips have a slight voltage deviation that corresponds to a small initial value, so there is no need to add the initial voltage between the capacitor’s two pins.

(a) (b) − Figure 10. Phase portrait of numerical simulation: (a) Phase portrait of xx23, (b) phase portrait − of xx14. (a) (b) Figure 11a,b showing different phase diagrams are consistent with the results of a xx− FigureFigure 10. Phase 10.numerical Phase portrait portrait ofsimulation numerical of numerical showing simulation: simulation: in (Figurea) Phase (a) Phase10a,b. portrait portrait It is of verifiedx2 of− x323,( thatb) phase, ( theb) phase proposed portrait portrait of xsystem1 − x4. pro- − of xx14duces. conservative chaos.

Figure 11a,b showing different phase diagrams are consistent with the results of a numerical simulation showing in Figure 10a,b. It is verified that the proposed system produces conservative chaos.

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Entropy 2021, 23, x FOR PEER REVIEW Figure 11a,b showing different phase diagrams are consistent with the results17 of of 19 a numerical simulation showing in Figure 10a,b. It is veriﬁed that the proposed system produces conservative chaos.

(a) (b)

(c) (d) − − FigureFigure 11. 11. ImplementationImplementation of ofcircuit: circuit: (a) (Phasea) Phase portrait portrait of ofx23x2x−, (xb3),( phaseb) phase portrait portrait of x14 of x1, −(c)x 4, implementation(c) implementation of the of thecircuit, circuit, (d) implementation (d) implementation of the of thecircuit. circuit. 7. Conclusions 7. Conclusion This paper proposed a non-Hamiltonian conservative chaotic system by integrating This paper proposed a non-Hamiltonian conservative chaotic system by integrating three-terminal memristor and 4D Euler equations. The dual-output pins of the three- three-terminal memristor and 4D Euler equations. The dual-output pins of the three-ter- terminal memristor, satisfying the nature of the skew-symmetric matrix. The system has minal memristor, satisfying the nature of the skew-symmetric matrix. The system has gen- generalized Hamiltonian; the conservation of the 4D Euler equation has been preserved. eralized Hamiltonian; the conservation of the 4D Euler equation has been preserved. The The characteristics of either centers or saddles of the equilibrias of the system proved characteristics of either centers or saddles of the equilibrias of the system proved the con- the conservation property. Chaotic dynamics have been revealed by varying the weight servation property. Chaotic dynamics have been revealed by varying the weight parame- parameter of the three-terminal memristor. Changing the initial Hamiltonian of the system ter of the three-terminal memristor. Changing the initial Hamiltonian of the system will will produce rich dynamics, which provides the way of producing quasiperiodic orbit and produce rich dynamics, which provides the way of producing quasiperiodic orbit and chaos. The routes and mechanisms from quasiperiodic orbit to chaos have been provided chaos. The routes and mechanisms from quasiperiodic orbit to chaos have been provided through energy bifurcation. With different initial Hamiltonian levels, the system will have throughdifferent energy dynamic bifurcation. ranges. UsingWith different energy and initial frequency Hamiltonian adjustment, levels, the instead system of will trial-and- have differenterror, the dynamic system produced ranges. Using the huge energyLEs, and which frequency is more suitableadjustment, for encryption instead of thantrial-and- other error,chaotic the systems. system produced The analog the circuit huge LEs, of the which system is more was builtsuitable physically, for encryption which than conﬁrmed other chaoticthe chaotic systems. existence The analog of the system,circuit of and the combined system was the built three-terminal physically, memristor which confirmed and 4D theEuler chaotic equation existence successfully. of the system, By changing and combined the different the types three-terminal of memristors, memristor modifying and 4D the Eulerdual outputs,equation andsuccessfully. embedding By it changing into the 4Dthe Eulerdifferent equation, types of keeping memristors, the conservative modifying partthe dualskew-symmetric outputs, and nature,embedding chaos it can into be the generated 4D Euler under equation, reasonable keeping parameters. the conservative part skew-symmetric nature, chaos can be generated under reasonable parameters. Author Contributions: Conceptualization, G.Q. and Z.W.; methodology, G.Q.; software, Z.W.; val- Authoridation, Contributions: Z.W.; data curation, Conceptualization, Z.W.; writing—original G.Q. and Z.W.; draft methodology, preparation, G.Q. Z.W.;; software, writing—review Z.W.; & validation, Z.W.; data curation, Z.W.; writing—original draft preparation, Z.W.; writing—review & editing, G.Q.; supervision, G.Q.; funding acquisition, G.Q. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the National Natural Science Foundation of China (61873186).

Entropy 2021, 23, 71 16 of 17

editing, G.Q.; supervision, G.Q.; funding acquisition, G.Q. All authors have read and agreed to the published version of the manuscript. Funding: This work was funded by the National Natural Science Foundation of China (61873186). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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