Interpreting initial offset boosting via reconstitution in integral domain

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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Interpreting initial offset boosting via reconstitution in integral domain Mo Chen, Xue Ren, Huagan Wu, Quan Xu, Bocheng Bao∗ 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 15 August 2019 Revised 2 November 2019 Accepted 19 November 2019 Available online xxx Keywords: Memristive system Initial offset boosting Reconstitution Integral domain

a b s t r a c t Initial offset boosting behaviors with homogenous, heterogeneous or extreme multistability have been reported in several nonlinear systems, but the forming mechanisms were rarely discussed. To figure out this problem, a four-dimensional (4-D) memristive system with cosine memductance is presented, which can exhibit initial offset boosting related to extreme multistability. Taking this 4-D memristive system as paradigm, a three-dimensional (3-D) system with standalone initials-related parameters is reconstructed in an integral domain. Thus, the original line equilibrium set is mapped as some periodically varied equilibrium points, which allows that the initial offset boosting is modeled as variable offset boosting with infinite topologically different attractors. Besides, the reconstituted 3-D model exhibits bi-stability or quadri-stability for fixed parameters, but it maintains the dynamics of the 4-D memristive system when initiated from the neighborhood of the origin point. Finally, circuit synthesis, PSIM simulations, and experimental measurements are carried out to validate the reconstituted variable offset boosting behaviors. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Dynamical behaviors of nonlinear systems are highly related to their equilibrium points. For nonlinear systems with two or more equilibrium points, the system trajectories may drop into attraction basins with differentiated dynamical behaviors, leading to the generation of multistability [1-7]. If the number of attraction basins tends to infinite, the system is said to be of extreme multistability [8-11], which can be easily achieved through introducing a certain number of ideal memristors into pre-existing oscillating circuits [9,12-15], nonlinear systems [16-20], or even neural models [21,22]. The memristor-based dynamical systems commonly have infinite equilibrium points located in the line [9], plane [12,17], or even space [13] defined by the inner variables of the memristors, whose stabilities have rarely been rigorously discussed in the literature [23]. The system trajectory, initiated from different memristor initials, will locate at a nearby point within the equilibrium set. Consequently, the attractor positions are naturally tuned by the memristor initials, resulting to the striking phenomenon defined as memristor-initial offset boosting [24-26]. Moreover, due to the detailed stability property of the equilibrium set, attractors with

∗

Corresponding author. E-mail address: [email protected] (B. Bao).

the same shape but different amplitudes or frequencies, i.e. the homogenous multistability [26], could be found along the offset boosting route; otherwise several or even infinitely many topologically different coexisting attractors, i.e. the heterogeneous multistability [24] or extreme multistability [25], could be revealed in the initial space. More specially, besides the memristor initials, the other initials can also induce attractor position offsets or attractor type changes [17,25]. However, all these initials are not explicitly expressed in the mathematical models, thus the initial offset boosting behaviors, especially their forming mechanisms, were rarely discussed in the previous works. The recently proposed incremental integral transformation method [27,28] can provide a new viewpoint for interpreting the initial offset boosting behaviors since it can formulate the initials as standalone parameters. This method has been applied for mechanism analyses and physical controls of the sensitive extreme multistability phenomena in memristive circuits and systems [17,21,29,30]. It describes a nonlinear system using state variables with different measuring scales, thus some latent dynamical characteristics could be revealed, which is helpful for seeking deeper understanding for the intrinsic dynamical behaviors. Inspired by these scientific considerations, we try to interpret the initial offset boosting behaviors of memristive systems in an integral domain. A four-dimensional (4-D) memristive system with cosine memductance is selected as paradigm. The trigonometric functions, such as sine/cosine function, hyperbolic sine/cosine

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Fig. 1. Memristor initial dependent behaviors with x(0) = 10−6 , y(0) = 0, z(0) = 0, and ϕ (0) = δ , the top is stability distributions of line equilibrium set E illustrated by the nonzero eigenvalues and the bottom is the first three Lyapunov exponent spectra.

function, and hyperbolic tangent function, haven been introduced into some dynamical systems as nonlinear terms to obtain N-scroll or N-wing attractors [31-33], hidden attractors [34,35], and coexisting attractors [36-39]. When the cosine function is used as the memductance nonlinearity, periodic initial offset boosting behaviors with extreme multistability can readily be observed in an infinite region of the memristor initial space [40], which are quite different from previously reported extreme multistability in the memristor-based circuits and systems with aperiodic memductance nonlinearities [17,21,29,30]. The exhibited dynamical behaviors are highly sensitive to the initials but can provide more freedom for chaos-based engineering applications [41-43]. To further dig out the latent dynamical characteristics of the periodic initial offset boosting behaviors, a three-dimensional (3-D) model is formulated based on the 4-D memristive system through incremental integral transformation. Thus, the original line equilibrium set is mapped as some periodically varied equilibrium points, which allows that the periodic initial offset boosting behaviors are modeled as variable offset boosting behaviors [44,45] with infinite topologically different attractors. As a result, the characteristics of the original initial offset boosting behaviors can be evaluated by investigating the evolutions of the mapped determined equilibrium points and the dynamics of the transformed variable offset boosting behaviors. The rest paper is organized as follows. In Sect. 2, the 4D memristive system with cosine memductance is reconstituted in the integral domain. The transformed equilibrium points and dynamical behaviors are simply discussed. In Sect. 3, the reconstituted initial offset boosting behaviors are investigated. In Sect. 4, multistability characteristics of the reconstituted system are revealed. In Sect. 5, the circuit synthesis, PSIM simulations, and experiment measurements are performed to verify the numerical simulations. The conclusion is summarized in Sect. 6.

2. Mathematical model in integral domain 2.1. Memristive system with cosine memductance Through coupling a memristor with cosine memductance W(ϕ ) = cos(ϕ ) into a linear system, a 4-D memristive chaotic system is established as follows

⎧ ⎪ ⎨x˙ = y + z − k cos(ϕ )y y˙ = z

(1)

⎪ ⎩z˙ = −x − z ϕ˙ = y

In (1), k is the unique control parameters. Since the initialsdependent behaviors are mainly concerned in this paper, this control parameter is fixed as k = 3. This memristive system has line equilibrium set E = (0, 0, 0, δ ) and the eigenpolynomial is formulated as

 3  λ + λ2 + λ + 1 − k cos(δ ) = 0

det(Iλ − J ) = λ

(2)

Based on (2), the stability distribution of E is illustrated by the real parts of the calculated eigenvalues, as plotted in the top of Fig. 1. Setting the initials as x(0) = 10−6 , y(0) = z(0) = 0, and ϕ (0) = δ , the first three Lyapunov exponent spectra are depicted in the bottom to illustrate the memristor-initial-dependent dynamical behaviors. Due to the periodic memductance nonlinearity, the stability distributions and induced dynamical behaviors exhibit distinct periodicity in a wide range of δ = [−∞, ∞] in theory. In addition, within the yellow shadow regions of Fig. 1, the equilibrium points are unstable saddle-foci, but point attractors are generated in system (1). This mismatch is mainly caused by the critically stable zero eigenvalues [12,17]. Phase portraits of typical coexisting attractors are depicted in the ϕ -z plane and presented in Fig. 2. With the change of ϕ (0), disconnected attractors with different attractor types, different at-

Fig. 2. Memristor initial offset boosting behaviors in the ϕ -z plane with x(0) = 10−6 , y(0) = 0, and z(0) = 0.

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Fig. 3. The transformed equilibrium points and dynamical behaviors for η1 = 10−6 , η2 = η3 = 0, and X(0) = Y(0) = Z(0) = 0, the top is the periodically varied equilibrium points and the bottom is the first two Lyapunov exponent spectra.

tractor structures, and different positions are observed along the ϕ -axis, leading to memristor initial offset boosting behaviors with extreme multistability. Specially, other initials, such as x(0) and y(0), also can boost the generated attractors along the ϕ -axis, but the mechanisms are unpredictable from the mathematical model. 2.2. Reconstructed model in integral domain To interpret the forming mechanisms of the initial offset boosting related to extreme multistability, a 3-D dimensionality reduction model is necessary to be formulated in an integral domain by the incremental integral transformation [27,28]. Towards this end, three new state variables and four initial-related parameters are introduced as



X=

t 0

xdτ , Y =



t 0

ydτ , Z =



t 0

zdτ

(3)

and

η0 = ϕ ( 0 ) , η1 = x ( 0 ) , η2 = y ( 0 ) , η3 = z ( 0 )

(4)

Integrating the four equations of the system (1) from 0 to t, there yields

⎧ ⎨X˙ = Y + Z − k sin(Y + η0 ) + k sin(η0 ) + η1 Y˙ = Z + η2 ⎩Z˙ = −X − Z + η 3

(5)

The newly constructed 3-D model possesses four parameters

η0 , η1 , η2 , and η3 standing for the initials of system (1). These explicitly expressed initial parameters are benefit for interpreting the dynamical mechanism of the revealed initial offset boosting behaviors. According to (3), the initials of system (5) all equal to 0, i.e. X(0) = 0, Y(0) = 0, and Z(0) = 0. In this case, the reconstituted system can maintain the same dynamics of system (1). However, as a nonlinear dynamical system, system (5) itself also exhibits multistability phenomena, which will be discussed in later sections. 2.3. Transformed equilibrium points and dynamical behaviors The equilibrium point of system (5) is characterized as

P = (η2 + η3 , Yˆ , −η2 )

(6)

in which, Yˆ is the root of the following equation

k sin(Yˆ + η0 ) = Yˆ − η2 + k sin(η0 ) + η1

3. Reconstitution of initial offset boosting behaviors The initials x(0), y(0), z(0), and ϕ (0) of system (1) are transformed as the standalone parameters η1 , η2 , η3 , and η0 in the newly reconstituted system (5). To illustrate the offset boosting effect of each initial-related parameter, system (5) is rewritten as given in Table 1. It can be seen that η0 and η1 appear in two terms associated with Y; η2 simultaneously appears in several terms associated with all three state variables; and η3 linearly appears in one term related to X. They can exert different influences on the average values of the state variables, leading to diverse kinds of variable offset boosting behaviors. In addition, these transformed variable offset boosting behaviors are different from those revealed in [46-48], which were achieved through adding a constant controller to one desired state variable and the attractors were boosted with a fixed topological structure along the axis of the boostable-variable. Taking η1 = 10−6 , η0 = η2 = η3 = 0, and X(0) = Y(0) = Z(0) = 0 as the typical parameter setting, the transformed variable offset boosting behaviors are revealed and discussed through tuning one of these four initials-related parameters. Correspondingly, the dynamical effects of each initial in system (1) can readily be deduced.

(7)

To evaluate the stability of P, the characteristic equation is deduced as

F (λ ) = λ3 + λ2 + λ + 1 − k cos(Yˆ + η0 ) = 0

Obviously, the stabilities of P mainly relate to Yˆ and η0 , while Yˆ is determined by η0 and –η2 +η1 . In a word, the initials-related parameters η0 , η1 , η2 , and η3 have a significant influence on the characteristics of the transformed equilibrium point P, which further affects the exhibited dynamical behaviors. When η0 = [−10, 10], η1 = 10−6 , and η2 = η3 = 0 are considered, the line equilibrium set E of system (1) is transformed as some periodically varied equilibrium points, as depicted in the top of Fig. 3. The red points indicate the index-2 saddle-foci, the blue ones stand for the index-1 saddle-foci, and the green ones represent the stable node-foci. Fixing X(0), Y(0), and Z(0) to 0, the first two Lyapunov exponent spectra are given in the bottom of Fig. 3, which confirms that system (5) exhibits the same dynamical behaviors of system (1). The mismatches between the stability distributions of the line equilibrium set E and the observed dynamical behaviors indicated in Fig. 1 are explained by the asymmetrically distributed equilibrium points of system (5), which are not expounded in this paper and the details can refer to [17] and [29].

(8)

3.1. Offset boosting behaviors induced by memristor-initial In system (5), the memristor-initial ϕ (0) of system (1) is represented by the parameter η0 . When η0 is chosen as a single

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M. Chen, X. Ren and H. Wu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx Table 1 The transformed variable offset boosting behaviors. Boosting controller

η0

η1

η2

η3

Re–expression of system model

⎧ ⎨X˙ = (Y + k sin η0 ) + Z − k sin(Y + η0 ) + η1 Y˙ = Z + η2 ⎩ Z˙ = −X − Z + η3 ⎧ X˙ = (Y + η1 ) + Z ⎪ ⎨ −k sin(Y + η1 + η0 − η1 ) + k sin(η0 ) Y˙ = Z + η2 ⎪ ⎩ Z˙ = −X − Z + η3 ⎧ X˙ = (Y − η2 ) + (Z + η2 ) ⎪ ⎨ −k sin(Y − η2 + η0 + η2 ) + k sin(η0 ) + η1 Y˙ = Z + η2 ⎪ ⎩ Z˙ = −(X − η2 ) − (Z + η2 ) + η3 ⎧ ⎨X˙ = Y + Z − k sin(Y + η0 ) + k sin(η0 ) + η1 Y˙ = Z + η2 ⎩ Z˙ = −(X − η3 ) − Z

Property of variable offset boosting Nonlinear, periodic, within Y = [−5, 5]

Nonlinear, periodic, negative direction of Y

Nonlinear, periodic, positive direction of Y; Linear, positive direction of X; Linear, negative direction of Z Linear, positive direction of X

Fig. 4. The transformed initial offset boosting behaviors with η1 = 10−6 , η2 = η3 = 0, and X(0) = Y(0) = Z(0) = 0. (a) Periodic behaviors with η0 = −11/−9.15/1.0/2.9/14, (b) periodic behaviors with η0 = −3.3/3.7/9.0/9.6, (c) chaotic behaviors with η0 = −9.0/0/8.75, (d) mean values of the state variables in system (5) (top) and system (1) (bottom).

changing variable and the specific memristor-initials of Fig. 2 are adopted, attractors with different attractor types, different attractor structures, and different positions are observed in the Y-Z plane, as depicted in Fig. 4. Since the other parameters are fixed as η1 = 10−6 ≈ 0, η2 = η3 = 0, the equilibrium points are expressed as P = (0, Yˆ, 0). With the variation of η0 , the trajectory of Yˆ is confined in the range of [−5, 5] as illustrated by the top figure of Fig. 3. Thus, the generated phase portraits, which evolve around one of these equilibrium points, are restricted in a bounded region around the origin point. The mean values of state variables X, Y, and Z are depicted in the top of Fig. 4(d). Correspondingly, those of x, y, z, and ϕ in systems (1) with x(0) = 10−6 , y(0) = 0, and z(0) = 0 are depicted in the bottom of Fig. 4(d). It can be found that in system (1) the attractors nonlinearly move along the positive direction of ϕ -axis, while in system (5) the attractors periodically sweep around the Y-axis within a bounded region.

3.2. Offset boosting behaviors induced by other initials In system (1), the initials x(0) and y(0) can also nonlinearly boost the attractors along the ϕ -axis, while z(0) can tune the attractor from chaotic to periodic with nearly fixed attractor position. But the reasons why x(0), y(0), and z(0) can induce these kinds of dynamical behaviors are hard to be explored from the mathematical model of system (1). With the explicitly expressed initialsrelated parameters, system (5) provides a novel viewpoint for seeking the inner mechanism. When η2 is chosen as the boosting controller, the equilibrium points are expressed as P = (η2 , Yˆ, −η2 ). The X and Z coordinates of P are linearly changed with the variation of η2 , and the Y coordinate is periodically evolved as depicted in Fig. 5(a). Referring to the mean values of the state variables given in Fig. 5(b), it can be seen that the offset boosting route of the generated attractors is nonlinear and multi-dimensional. Moreover, with the variation of η2 ,

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Fig. 5. Variable offset boosting behaviors with η2 as control parameter, η1 = 10−6 , η0 = η3 = 0, and X(0) = Y(0) = Z(0) = 0. (a) Trajectory of the equilibrium points, (b) mean values of the state variables, (c) bifurcation diagram of Y in a narrow region of η2 = [−5, 5], (d) attractors with position offset boosting within η2 = [−5, 5].

Fig. 6. Variable offset boosting behaviors induced by η1 and η3 , in which the tops are bifurcation diagrams and the bottoms are mean values of the state variables. (a) η1 = [−10, 10], η2 = 10−6 , η0 = η3 = 0, and X(0) = Y(0) = Z(0) = 0, (b) η3 = [−10, 10], η1 = 10−6 , η0 = η2 = 0, and X(0) = Y(0) = Z(0) = 0.

countless attractors with different topological structures are found along the boosting route. The bifurcation diagram of Y and phase portraits of typical coexisting attractors in the region of η2 = [–5, 5] are presented in Figs. 5(c) and 5(d). According to (7), −η1 influence Yˆ in the same way as η2 does. When η1 is considered as a boosting controller, the equilibrium points, denoted as P = (0, Yˆ, 0), and the generated attractors are periodically moved along the negative direction of Y-axis, as illustrated by the bifurcation diagram and mean values of X/Y/Z given in Fig. 6(a). While for η3 , the equilibrium points become P = (η3 , Yˆ, 0), in which the X coordinate is linearly changed with the variation of η3 and the Y coordinate is kept unchanged. Thus, the linear and onedimensional offset boosting is observed along the positive direction of X-axis, as shown by Fig. 6(b). In addition, since the number and

stabilities of the equilibrium points are invariant, only two kinds of attractors, i.e. chaotic attractor and period-1 limit cycle, are found along the boosting route. Interestingly, the initial offset boosting behaviors induced by x(0) and y(0), as well as the non-boosted behaviors caused by z(0) in system (1), are all formulated as the variable offset boosting behaviors in system (5) as summarized in Table. 1. It is revealed from (7) and (8) that the locations and stabilities of the equilibrium P are all tuned by the control parameters η0 , η1 , η2 , and η3 , and thus the offset boosting of the attractor positions and variations of the attractor types are naturally observed. The mysterious effects of x(0), y(0), z(0), and ϕ (0) in system (1) are readily reflected by the mathematical models listed in Table 1 and the evolutions of the equilibrium point P plotted in Figs. 3 and 5.

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4. Multistability of the reconstituted system Multistability can easily be found in system with more than one equilibrium points [49,50]. According to Figs. 3 and 5(a), system (5) could have one, or two, or three equilibrium points for a given set of control parameters. Consequently, there is a great possibility for system (5) itself to exhibit multistability. The multistability

phenomena revealed within Y(0) = [−9, 9] are listed in Table 2, in which η0 is specified as 0.5/1.0/1.5/2.8 and the other parameters are fixed as η1 = 10−6 and η2 = η3 = 0. Considering that the equilibrium points are distributed along the Y-axis, the influences of Y(0) are mainly investigated and X(0), Z(0) are all settled down to 0. The bifurcation diagrams of Y and the Lyapunov exponent spectra are plotted versus Y(0) and presented in Figs. 7(a1)-(d1), from

Fig. 7. Dynamics with the variation of Y(0), where (a1)-(d1) are bifurcation diagram and Lyapunov exponent spectra, and (a2)-(d2) are coexisting attractors in Y-Z plane, in which the equilibrium points are marked using red (index-2 saddle foci), blue (index-1 saddle foci), and green (stable node foci) stars. (a) η0 = 0.5, (b) η0 = 1.0, (c) η0 = 1.5, (d) η0 = 2.8.

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Fig. 8. Equivalent realization circuit and hardware experiment setup of system (5). (a) Circuit schematic structure, (b) snapshot of the hardware experiment setup.

Table 2 Multistability of system (5) for different η0 within Y(0) = [−9, 9].

η0

Equilibrium points

Coexisting states

Figures

0.5

P+ (0, 1.40, 0) Index-2 saddle-focus P0 (0, 0, 0) Index-1 saddle-focus P− (0, −3.067, 0) Index-2 saddle-focus P+ (0, 0.456, 0) Stable node-focus P0 (0, 0, 0) Index-1 saddle-focus P− (0, −3.729, 0) Index-2 saddle-focus P0 (0, 0, 0) Stable node-focus P−1 (0, −0.547, 0) Index-1 saddle-focus P−2 (0, −4.220, 0) Index-2 saddle-focus P0 (0, 0, 0) Index-2 saddle-focus

Bi-stability: Chaotic attractor and period-1 limit cycle

Fig. 7a

Quadri-stability: Point, large-amplitude period-1 limit cycle, and two topological different period-2 limit cycles

Fig. 7b

Quadri-stability: Point, chaotic attractor, period-2 limit cycle, and large-amplitude period-1 limit cycle

Fig. 7c

Bi-stability: Period-2 and large-amplitude period-1 limit cycles

Fig. 7d

1.0

1.5

2.8

focus located at the origin point and two coexisting stable states of period-2 and large-amplitude period-1 limit cycles are found in Fig. 7(d). Compared with system (1), the reconstituted system (5) possesses relatively simple attraction basins. Moreover, it exhibits the same kinds of dynamical behaviors as those observed in system (1) when initiated from the neighborhood of the origin point, which is important for physical realization of the initialsdependent behaviors of system (1). 5. Circuit realization and PSIM simulations By employing operation amplifier, multiplier, capacitor, resistor, and trigonometric function converter, an analog circuit for implementing the reconstituted system described by (5) is designed as given in Fig. 8(a). The circuit equations are modeled as

⎧ R dvX R ⎪ ⎪ ⎪RC dt = vY + vZ − R sin(vY + V0 ) + R sin(V0 ) + V1 ⎪ k k ⎨ dvY RC = vZ + V2 ⎪ dt ⎪ ⎪ ⎪ ⎩RC dvZ = −vX − vZ + V3

(9)

dt

which bi-stability and quadri-stability are revealed. The phase portraits of these coexisting stable states are illustrated in the Y-Z plane, as shown in Figs. 7(a2)-(d2), in which the equilibrium points are indicated using red (index-2 saddle-foci), blue (index-1 saddlefoci) and green (stable node-foci) stars. In detail, for η0 = 0.5, there are two index-2 saddle-foci and one index-1 saddle-focus (located at the origin point). Two coexisting stable states including a chaotic attractor and a large-amplitude period-1 limit cycle are observed in Fig. 7(a). While for η0 = 1.0, there are one stable node-focus, one index-1 saddle-focus (located at the origin point), and one index-2 saddle-focus. Four coexisting stable states including a point and three limit cycles with different periodicities and topologies are revealed from Fig. 7(b). When η0 = 1.5 is considered, there exist one stable node-focus (located at the origin point), one index-1 saddle-focus, and one index-2 saddlefocus. Four coexisting stable states of a point, a chaotic attractor, and two limit cycles with different periodicities are discovered in Fig. 7(c). Finally, with η0 = 2.8, there is only one index-2 saddle

where, vX , vY , and vZ , are three circuit variables corresponding to the state variables of system (5), and V0 ~V3 are four DC bias voltages corresponding to the four parameters η0 ~η3 in (5). The time constant of the three integrators is specified as τ 0 = RC = 10 k × 33 nF = 330 μs. Consequently, the circuit parameter Rk = R/k is determined as 3.33 k. According to the realization circuit in Fig. 8(a), a breadboard experiment setup is welded and connected to a digital oscilloscope, as presented in Fig. 8(b), in which two AD639AD trigonometric function converters are adopted to realize the two nonlinear functions of sin(•) in (9). However, in AD639AD, 1 V input voltage corresponds to 50°; while for nonlinear functions of sin(•) in (9), 1 V input voltage corresponds to 1 rad. Thus, in the hardware circuit, the input voltage of AD639AD should be magnified 1.146 times using an op-amp-based proportional circuit. When tuning the four DC bias voltages V0 , V1 , V2 , and V3 , the transformed variable offset boosting behaviors illustrated in Figs. 4 and 5(d) are measured in the breadboard experimental cir-

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Fig. 9. Experimentally measured variable boosted attractors in the vY – vZ plane. (a) Measured chaotic and periodic behaviors with η0 = −3.3/0/1.0/2.9, (b) measured chaotic and periodic behaviors with η2 = ±0.6/±2.0/±4.0/±5.0.

Fig. 10. PSIM simulated variable boosted attractors in the vY -vZ plane. (a) Periodic behaviors with η0 = −11/−9.15/1.0/2.9/ 14, (b) periodic behaviors with η0 = −3.3/3.7/9.0/9.6, (c) chaotic behaviors with η0 = −9.0/0/8.75, (d) chaotic and periodic behaviors with η2 = ±0.6/ ±2.0/ ±4.0/ ±5.0.

cuit and the captured results are illustrated in Fig. 9. Note, the three capacitors in the integrators U2 , U3 , and U5 are fully discharged to make the circuit initiate from the origins. Due to the multistability of the reconstituted system (5) and physical restriction of the AD639AD output voltage, only partial phase portraits of the transformed variable offset boosting behaviors are successfully observed in the hardware circuit. To fully verify the numerical simulation results, PSIM simulations are further performed. During simulations, the three initial capacitor voltages are fixed as 0. The desired phase portraits in the vY – vZ plane are acquired and superimposed in the four figures of Fig. 10. It confirms that the variable offset boosted behaviors can be exhibited in the physical circuit of the reconstituted system. 6. Conclusion By introducing an ideal memristors with cosine memductance into a linear system, a 4-D memristive system was presented, from which the initial offset boosting behaviors with periodic offset-

boosted coexisting attractors were observed. The mysterious boosting effects of memristor initial and other three initials, in an implicit form in the original system, were interpreted based on a 3-D dimensionality reduction model reconstituted in the integral domain. In this dimensionality reduction model, the original initial offset boosting behaviors were reconstituted as variable offset boosting behaviors, which were easily comprehensible through inspecting the evolution of the mapped determined equilibrium points. Thus, the latent characteristics of the initial offset boosting behaviors were explicitly revealed and thoroughly investigated. Moreover, bi-stability and quadri-stability were revealed from the reconstituted model, which confirmed that the dynamical behaviors of the original system can be reconstituted in the 3-D model when initiated from the neighborhood of the origin point. At last, with a designed circuit model, PSIM simulations and experimental measurements were carried out to validate the transformed variable offset boosting behaviors. It is demonstrated that through choosing the proper state variables, the mathematical model of nonlinear systems could be simplified and some latent parameters

Please cite this article as: M. Chen, X. Ren and H. Wu et al., Interpreting initial offset boosting via reconstitution in integral domain, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109544

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or characteristics could be uncovered, which are benefit for comprehensive investigations of the intrinsic dynamics. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Please cite this article as: M. Chen, X. Ren and H. Wu et al., Interpreting initial offset boosting via reconstitution in integral domain, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109544