1 Introduction

Memristor, the fourth basic circuit element described by the relationship between magnetic flux and electric charge, exhibits special nonlinear memristance or memductance nonlinearity controlled by the inner electric charge or magnetic flux variable [1]. Owing to this inner state variable, the memristor has been used as a critical building block for the construction of novel chaotic oscillating circuits [2, 3]. Various memristive circuits and systems with complex dynamical behaviors, such as self-excited or hidden extreme multistability [4, 5], multi-scroll or multi-wing chaotic attractors [6, 7], hyperchaotic attractors [8, 9], and conservative chaotic motions [10, 11], have been proposed for theoretical investigations and physical applications. The natural plasticity of memristor gives it unique advantages in simulating biological neuron synapses [12, 13] and constructing neuromorphic circuits or artificial neural networks [14, 15].

Initial condition-dependent multistability with the coexistence of multiple disconnected attractors is relatively easy to achieve in memristor-based circuits and systems. These coexisting multiple-stable states can provide great flexibility for chaos-based applications [16, 17]. Hence, exploring the synchronous control strategies and detailed synchronicities of these coexisting behaviors is of great importance. Aiming at different application requirements, researchers have proposed a variety of synchronization control strategies, including sliding mode control, impulse control, adaptive control, and finite time control, and achieved important research achievements [18,19,20,21]. Memristor shows dynamic constraint relationships among the port voltage, port current, and inner electric charge or magnetic flux variable. It can provide a new coupling synchronization scheme for nonlinear dynamical circuits and systems [22,23,24]. When a memristor is connected between two dynamical circuits or systems as a nonlinear coupler, the synchronization error controls its inner state variable in real time to adjust the signal exchanged between the coupled circuits or systems and promote synchronization. Some examples of such memristor-coupled networks include memristor-coupled chaotic circuits or systems [22, 23], locally active memristor-coupled neural networks [25], and memristor synapse-coupled neural networks with electromagnetic induction [26, 27].

Memristive couplers have multiple effects on the formed networks. In addition to the coupling strengths, the initial conditions of their inner state variables greatly affect the synchronization performances of the coupled networks [28, 29]. For memristor-based subsystems, the dependence of synchronous behaviors on the initial conditions of the coupled network becomes complicated. The coupling strength and initial condition of the coupling memristor and the initial conditions of the coupled subsystems exert remarkable influence on the synchronous behaviors of the memristor-coupled network. Given that the attractors of memristive systems can be offset-boosted in the phase space by the initial conditions [30], the special parallel-offset synchronization behaviors could be obtained in the memristor-coupled memristive systems [31]. However, these phenomena have not been comprehensively investigated. Therefore, this work proposes a memristor-coupled homogeneous network consisting of two identical non-autonomous memristive Fitzhugh–Nagumo models and studies its initial condition-sensitive coexisting and synchronous behaviors. The proposed network exhibits hidden extreme multistability similar to that of the individual Fitzhugh–Nagumo model. For these hidden coexisting behaviors, the complete, parallel-offset, and phase synchronization behaviors related to the coupling strength and initial conditions of the memristor-coupled network are numerically revealed and experimentally verified.

The remainder of this paper is arranged as follows. Section 2 shows the mathematical model of the proposed memristor-coupled homogeneous network and studies its equilibrium state and stability. Section 3 describes the exploration of hidden extreme multistability. Section 4 presents the investigation of coexisting synchronous behaviors. Section 5 performs PSIM (power simulation) circuit simulations to confirm the numerical simulations. Finally, Sect. 6 concludes the whole paper.

2 Mathematical models and equilibrium points

The 3D non-autonomous memristive Fitzhugh–Nagumo circuit proposed in [32] is selected as the subsystem of the coupled network. Its dimensionless mathematical model is described as

$$ \left\{ \begin{aligned} &\dot{x} = y + (a - b\tanh \varphi )x + H\cos (Ft), \hfill \\ &\dot{y} = - cx - cy, \hfill \\ &\dot{\varphi } = - dx. \end{aligned} \right. $$
(1)

An ideal memristor with smooth hyperbolic tangent memductance nonlinearity W(φ) = a – btanhφ is adopted to implement the specific cubic nonlinearity of the Fitzhugh–Nagumo circuit. When Hcos(Ft) ≠ 0, system (1) has no equilibrium point. When Hcos(Ft) = 0, the system has a line equilibrium set. With the evolution of time, the equilibrium state of system (1) switches between no equilibrium points and the line equilibrium set. For the selected system parameters a = 0.5, b = 0.5, c = 1, d = 1, H = 1.8, and F = 1, this system is proven to generate hidden extreme multistability by employing the incremental flux-charge analysis method [4].

When two identical systems as described in (1) are bidirectionally coupled by a memristor with the same smooth hyperbolic tangent memductance nonlinearity, a 7D homogenous network can be formulated as follows

$$ \left\{ \begin{aligned} &\dot{x}_{1} = y_{1} + (0.5 - 0.5\tanh \varphi_{1} )x_{1} + 1.8\cos (t) - k(1 - \tanh \varphi_{3} )(x_{1} - x_{2} ), \hfill \\ &\dot{y}_{1} = - x_{1} - y_{1} , \hfill \\ &\dot{\varphi }_{1} = - x_{1} , \hfill \\ &\dot{x}_{2} = y_{2} + (0.5 - 0.5\tanh \varphi_{2} )x_{2} + 1.8\cos (t) + k(1 - \tanh \varphi_{3} )(x_{1} - x_{2} ), \hfill \\ &\dot{y}_{2} = - x_{2} - y_{2} , \hfill \\ &\dot{\varphi }_{2} = - x_{2} , \hfill \\ &\dot{\varphi }_{3} = x_{1} - x_{2} . \end{aligned} \right. $$
(2)

The system control parameters of the two subsystems are kept unchanged, and the kinetic and synchronous effects of coupling strength k, initial condition φ30 of the coupling memristor, and six initial conditions x10, y10, φ10, x20, y20, and φ20 of the two subsystems are investigated.

Similar to the individual subsystem (1), the memristor-coupled network (2) possesses no equilibrium points for 1.8cos(t) ≠ 0 and has a space equilibrium set S = (0, 0, c1, 0, 0, c2, c3) for 1.8cos(t) = 0. The stability of this space equilibrium set is hard to determine due to the existence of zero eigenvalues. As a solution, network (2) is reconstituted in the integral domain via incremental integral transformation [33]. With this method, the parameterized conversion of initial conditions can be achieved and the equilibrium points of the original dynamical system can be reformed to facilitate the theoretical analyses. A 4D dimensionality reduction model is then formulated as follows

$$ \left\{ {\begin{array}{*{20}l} \begin{aligned} &\dot{X}_{1} = Y_{1} + 0.5X_{1} + 0.5\ln \cosh ( - X_{1} + \varphi_{10} ) - 0.5\ln \cosh \varphi_{10} + 1.8\sin (t) \hfill \\ &\, - k(X_{1} - X_{2} ) + k\ln \cosh (X_{1} - X_{2} + \varphi_{30} ) - k\ln \cosh \varphi_{30} + x_{10} , \end{aligned} \hfill \\ &{\dot{Y}_{1} = - X_{1} - Y_{1} + y_{10} ,} \hfill \\ &\begin{aligned} \dot{X}_{2} = Y_{2} + 0.5X_{2} + 0.5\ln \cosh ( - X_{2} + \varphi_{20} ) - 0.5\ln \cosh \varphi_{20} + 1.8\sin (t) \hfill \\ &\, + k(X_{1} - X_{2} ) - k\ln \cosh (X_{1} - X_{2} + \varphi_{30} ) + k\ln \cosh \varphi_{30} + x_{20} , \end{aligned} \hfill \\ &{\dot{Y}_{2} = - X_{2} - Y_{2} + y_{20} .} \end{array} } \right. $$
(3)

The state variables X1, Y1, X2, and Y2 are the incremental integrals of x1, y1, x2, and y2 in the time interval of [0, τ]. The constants x10, y10, φ10, x20, y20, φ20, and φ30 indicate the seven initial conditions of network (2). Moreover, the inner state variables of the three memristors, i.e., φ1, φ2, and φ3, are expressed as functions of the newly introduced state variables X1, X2, and their initial conditions φ10, φ20, and φ30. Therefore, the reconstituted system (3) is reduced in dimension.

In (3), the equilibrium points of the memristor-coupled network (2) are formulated as time-varying determined equilibrium points P = (ξ1, y10 − ξ1, ξ2, y20 − ξ2). The values of ξ1 and ξ2 are solutions of the following two equations

$$ \left\{ \begin{aligned} &- \xi_{1} + \ln \cosh ( - \xi_{1} + \varphi_{10} ) + 3.6\sin (t) - 2k(\xi_{1} - \xi_{2} ) \hfill \\ &\, + 2k\ln \cosh (\xi_{1} - \xi_{2} + \varphi_{30} ) - 2k\ln \cosh \varphi_{30} + 2(x_{10} + y_{10} ) - \ln \cosh \varphi_{10} = 0, \hfill \\ &- \xi_{2} + \ln \cosh ( - \xi_{2} + \varphi_{20} ) + 3.6\sin (t) + 2k(\xi_{1} - \xi_{2} ) \hfill \\ &\, - 2k\ln \cosh (\xi_{1} - \xi_{2} + \varphi_{30} ){ + }2k\ln \cosh \varphi_{30} + 2(x_{20} + y_{20} ) - \ln \cosh \varphi_{20} = 0. \end{aligned} \right. $$
(4)

Denoting

$$\begin{aligned} &f_{1} = - \xi_{1} + \ln \cosh ( - \xi_{1} + \varphi_{10} ) - 2k(\xi_{1} - \xi_{2} ) \hfill \\ &\, + 2k\ln \cosh (\xi_{1} - \xi_{2} + \varphi_{30} ) - 2k\ln \cosh \varphi_{30} + 2(x_{10} + y_{10} ) - \ln \cosh \varphi_{10} \end{aligned} $$
(5a)
$$ \begin{aligned} &f_{2} = - \xi_{2} + \ln \cosh ( - \xi_{2} + \varphi_{20} ) + 2k(\xi_{1} - \xi_{2} ) \hfill \\ &\, - 2k\ln \cosh (\xi_{1} - \xi_{2} + \varphi_{30} ){ + }2k\ln \cosh \varphi_{30} + 2(x_{20} + y_{20} ) - \ln \cosh \varphi_{20} , \end{aligned} $$
(5b)

and using the graphic analytic method, ξ1 and ξ2 are solved by inspecting the intersection points of two function curves of f1 + 3.6sin(t) = 0 and f2 + 3.6sin(t) = 0. Take k =  ± 1 as two examples. The curves of these two functions are depicted at sin(t) = 0 and ± 1 as shown in Fig. 1. During simulations, the seven initial condition constants are specified as x10 = 0.01 and y10 = φ10 = x20 = y20 = φ20 = φ30 = 0. Note that f1 + 3.6sin(t) = 0 has no solutions when k = 1 and 3.6sin(t) → 3.6, and f2 + 3.6sin(t) = 0 has no solutions when k =  − 1 and 3.6sin(t) → 3.6. For k > 0, the function curves of f1 + 3.6sin(t) = 0 and f2 + 3.6sin(t) = 0 are monotonically increased and could have one or no intersection point, as shown in Fig. 1a. For k < 0, the two function curves are non-monotonically increased and may have one, two, or no intersection points as depicted in Fig. 1b.

Fig. 1
figure 1

Graphical representations for the intersection points of two function curves f1 + 3.6sin(t) = 0 and f2 + 3.6sin(t) = 0, a k = 1; b k =  − 1

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When denoting

$$ \begin{aligned} &h_{1} = 0.5 - 0.5\tanh ( - \xi_{1} + \varphi_{10} ),\quad \hfill \\ &h_{2} = 0.5 - 0.5\tanh ( - \xi_{2} + \varphi_{20} ),\quad \hfill \\ &h_{3} = k - k\tanh (\xi_{1} - \xi_{2} + \varphi_{30} ), \end{aligned} $$
(6)

the characteristic polynomial at equilibrium state P is deduced as

$$ \lambda^{4} + a_{3} \lambda^{3} + a_{2} \lambda^{2} + a_{1} \lambda + a_{0} = 0, $$
(7)

with

$$ \begin{aligned} &a_{3} = - h_{1} - h_{2} + 2h_{3} + 2, \hfill \\ &a_{2} = h_{1} h_{2} - h_{1} h_{3} - h_{2} h_{3} - 2h_{1} - 2h_{2} + 4h_{3} + 3, \hfill \\ &a_{1} = 2h_{1} h_{2} - 2h_{1} h_{3} - 2h_{2} h_{3} - 2h_{1} - 2h_{2} + 4h_{3} + 2, \hfill \\ &a_{0} = h_{1} h_{2} - h_{1} h_{3} - h_{2} h_{3} - h_{1} - h_{2} + 2h_{3} + 1. \end{aligned} $$
(8)

If the following conditions

$$ \left\{ \begin{aligned} &a_{1} (a_{2} a_{3} - a_{1} ) - a_{0} a_{3}^{2} > 0 \hfill \\ &a_{i} > 0\quad (i = 0,\;1,\;2,\;3) \end{aligned} \right. $$
(9)

are all satisfied, the equilibrium points P should be stable and the generated dynamics is hidden. Given that the function tanh(•) is bounded within (− 1, 1), the three functions h1, h2, and h3 in (6) are bounded within (0, 1), (0, 1), and (0, 2 k), respectively, for the positive coupling strength k. The Routh–Hurwitz criteria described in (9) are numerically evaluated within these bounded regions. The results demonstrate that all the conditions given in (9) are satisfied, indicating that the memristor-coupled network (2) maintains the hidden property of the individual system when the positive coupling strength k is assigned. For clear illustrations, the characteristics of equilibrium points for k =  − 1, 0.5, and 1 are listed in Table 1, where USF and SNF represent unstable saddle focus and stable node focus, respectively. They verify the aforementioned theoretical deductions. If the negative coupling strength k is selected, self-excited dynamical behaviors could be obtained due to the existence of unstable equilibrium points. However, numerical simulations reveal that the memristor-coupled network easily tends to be unbounded when the negative coupling strength k is chosen.

Table 1 Characteristics of reformed equilibrium points for different values of k
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3 Hidden extreme multistability of the memristor-coupled homogenous network

This section discusses the hidden extreme multistability of the memristor-coupled homogenous network. The differential equations of network (2) are solved using the MATLAB ODE45 algorithm with a time step of 0.01, and the Lyapunov exponents (LEs) are calculated by the ODE45-based Wolf's Jacobian matrix method.

3.1 Coexisting behaviors induced by the coupling memristor

The coexisting behaviors induced by the coupling memristor are examined by taking coupling strength k and initial condition φ30 as two varying parameters. The initial conditions of the two subsystems are fixed as (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0). The varying parameters k and φ30 are adjusted in the regions of [− 0.05, 0.2] and [− 1, 6], respectively.

The 2D bifurcation diagram of network (2) is plotted in Fig. 2a by checking the periodicities of state variable φ1. The chaotic (CH), periodic, and unbounded (UB) behaviors are indicated by different colors. The labels P2, P3, P4, P6, and P8 represent periodic behaviors with different periodicities. The label MP represents the multi-period behaviors, whose periodicities are between 9 and 15. If the periodicity goes beyond 15, the corresponding dynamical behaviors are identified as chaotic ones and labeled as CH in Fig. 2a. Meanwhile, the dynamical regions in the k-φ30 plane are further distinguished by the signs of LEs and illustrated in Fig. 2b for intuitive comparisons. In Fig. 2b, the periodic (PE) region covers the P2 to MP regions of Fig. 2a. The hyperchaotic (HCH) behaviors, which have two positive, three zero, and two negative LEs, are distinguished from the chaotic ones. The dynamical patterns in Fig. 2a, b match well with each other. In the 2D bifurcation diagram of Fig. 2a, the hyperchaotic and chaotic behaviors are not distinguished from each other and the periodic phenomena that only occur in extremely narrow parameter regions are not displayed. The 2D bifurcation plots given in later sections follow the same principle.

Fig. 2
figure 2

Bifurcation plots in the k-φ30 plane with fixed subsystems’ initial conditions (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0), a 2D bifurcation diagram, b distributions of dynamical behaviors distinguished by the sign of LEs

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As shown in Fig. 2, the homogeneous network easily goes infinite when the negative k is assigned. If the two subsystems are weakly coupled, hyperchaotic behaviors tend to occur. The chaotic or periodic behaviors emerge as the positive coupling strength k increases gradually. k = 0.05 and 0.2 are taken as the representative cases to explore the detailed bifurcation behaviors induced by the coupling memristor coefficients. The 1D bifurcation diagrams (top) and the corresponding first four LE spectra (bottom) are presented in Fig. 3. The hyperchaotic, chaotic, and periodic behaviors and the period-doubling bifurcation, tangential bifurcation, and crisis scenarios, are illustrated. They further verify the dynamical evolution characteristics revealed in Fig. 2.

Fig. 3
figure 3

1D bifurcation plots induced by φ30 with fixed subsystems’ initial conditions (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0), a k = 0.05, b k = 0.2

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As φ30 approaches 0 or becomes negative, the bifurcation plots of φ1 and φ2 tend to overlap. Meanwhile, the bifurcation plot of φ3 tends to be constant. According to the seventh equation of (2), the time derivation of φ3 equals the difference between x1 and x2, indicating that φ3 tends to be constant when the two subsystems reach complete synchronization, i.e., x1 − x2 = 0. These phenomena indicate that the two subsystems can achieve complete synchronization for small or negative φ30 when a positive k is selected. This synchronous region becomes enlarged with the increase of the coupling strength k.

Take k = 0.2 as an example. Four different sets of phase diagrams and time-domain waveforms are plotted in Fig. 4 for φ30 =  − 1, 2.5, 3.5, and 4.8. Figure 4a shows the chaotic motions at φ30 =  − 1 with one positive, three zero, and three negative LEs. In this case, the phase trajectories and time-domain waveforms of the two subsystems fit together, indicating the generation of completely synchronized chaotic motions. Figure 4b exhibits asynchronous periodic behavior at φ30 = 2.5 with three zero and four negative LEs; Fig. 4c displays asynchronous hyperchaotic behavior at φ30 = 3.5 with two positive, three zero, and two negative LEs; and Fig. 4d shows asynchronous quasi-periodic behavior at φ30 = 4.8 with four zero and three negative LEs. For φ30 = 3.5, the memristor-coupled network reaches complete synchronization in partial periods, as illustrated by the time-domain waveforms in Fig. 4c.

Fig. 4
figure 4

Phase portraits in the x1-φ1 and x2-φ2 planes for the representative dynamical behaviors with fixed subsystems’ initial conditions (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0) and coupling strength k = 0.2, a φ30 =  − 1, b φ30 = 2.5, c φ30 = 3.5, d φ30 = 4.8

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3.2 Coexisting behaviors induced by the subsystems’ initial conditions

The coexisting behaviors induced by the initial conditions of the two subsystems are examined by setting the coupling memristor-related coefficients as k = 0.2 and φ30 = 0. The subsystems’ initial conditions are set as (x10, 0, φ10, 0, 0, 0), and the examined initial conditions x10 and φ10 are varied in the range of [− 1, 1].

The 2D bifurcation plot in the x10-φ10 plane is depicted in Fig. 5a. The chaotic (red) and periodic (magenta, yellow, blue, light blue, and green) behaviors with different topological structures or locations coexist in the memristor-based homogenous network. In addition, Fig. 5b displays the 2D bifurcation plot in the φ10-φ20 plane for the subsystems’ initial conditions (0.01, 0, φ10, 0, 0, φ20). It also reveals coexisting chaotic and periodic behaviors with different topological structures or locations. Similar to Fig. 2a, the coexisting phenomena that occur only in extremely narrow initial condition regions are ignored in Fig. 5.

Fig. 5
figure 5

2D bifurcation plots induced by the subsystems’ initial conditions with fixed k = 0.2 and φ30 = 0, a 2D bifurcation plot in the x10-φ10 plane for (x10, 0, φ10, 0, 0, 0), b 2D bifurcation plot in the φ10-φ20 plane for (0.01, 0, φ10, 0, 0, φ20)

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Corresponding to Fig. 5a, the representative phase portraits for coexisting attractors with varied x10 and fixed φ10 = 0 are plotted in Fig. 6a. Meanwhile, the representative phase portraits for coexisting attractors of Fig. 5b are depicted in Fig. 6b with varied φ10 and fixed φ20 = 0. With the variation of initial condition x10 or φ10, the dynamical behaviors of two subsystems are changed. Moreover, the attractor position of the first subsystem shifts along the φ1-coordinate, as reflected by the relative positions between the red (subsystem 1) and blue (subsystem 2) trajectories. This phenomenon may be related to the space equilibrium set S = (0, 0, c1, 0, 0, c2, c3) that exists at Hcos(Ft) = 0. For dynamical systems with the line or space equilibrium sets, the generated attractors are located around one nearby equilibrium point around the assigned initial conditions. Correspondingly, the generated attractors move along the axis or plane where the equilibrium points are located. However, in network (2), the equilibrium state switches between a space equilibrium set and no equilibrium points because the stimulus Hcos(Ft) is not always equal to zero. Thus, the cause of attractors’ displacement cannot be explicitly identified and still needs further exploration.

Fig. 6
figure 6

Phase portraits in the x1(x2) -φ1(φ2) planes for the coexisting attractors induced by the subsystems’ initial conditions with fixed k = 0.2 and φ30 = 0, a x10 =  − 1, − 0.2, 0, and 0.4 for (x10, 0, 0, 0, 0, 0), b φ10 =  − 1, − 0.5, 0.4, and 1 for (0.01, 0, φ10, 0, 0, 0)

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The dynamics of the memristor-coupled homogenous network is flexibly controlled by the initial conditions of the coupling memristor and two subsystems. In particular, the complete period-doubling bifurcation routes are found with the variations of x10, φ10, φ20, and φ30, leading to the generation of hidden extreme multistability [34] with infinite hidden coexisting attractors. The detailed synchronous behaviors of these coexisting attractors are discussed in Sect. 4.

4 Synchronous behaviors of the memristor-coupled homogenous network

This section depicts the synchronicities of the memristor-coupled homogenous network (2) using several numerical measures.

4.1 Complete synchronization

As revealed in Figs. 3 and 4, the memristor-coupled network (2) can achieve complete synchronization states when only a minor initial condition mismatch occurs between the two subsystems. In this case, the synchronization error of two subsystems is quantitatively evaluated using the normalized mean synchronization error [28, 35] defined as

$$ E = \frac{1}{N}\mathop {\mathop \sum \nolimits }\limits_{n = 1}^{N} \frac{{\sqrt {\left[ {x_{1} (n) - x_{2} (n)} \right]^{2} + \left[ {y_{1} (n) - y_{2} (n)} \right]^{2} + \left[ {\varphi_{1} (n) - \varphi_{2} (n)} \right]^{2} } }}{{\sqrt {x_{1} (n)^{2} + y_{1} (n)^{2} + \varphi_{1} (n)^{2} + x_{2} (n)^{2} + y_{2} (n)^{2} + \varphi_{2} (n)^{2} } }} $$
(10)

where xj(n), yj(n), and φj(n) (j = 1, 2) are the sampling values of state variables within a certain time interval. N samples are used to calculate the normalized mean synchronization error. If E approaches 0, complete synchronization is realized.

For the fixed subsystems’ initial conditions (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0), the normalized mean synchronization errors in the k-φ30 plane are calculated using the data samples within the time interval of [3900, 4000]. The obtained results are presented in Fig. 7a. The red area indicates that the normalized mean synchronization error E is close to 0 and the complete synchronization state is achieved. The other colored areas indicate that E is positive and the two coupled subsystems are out of synchronization. We can find that the memristor-coupled network (2) realizes complete synchronization with a larger positive coupling strength k and a more negative memristor initial condition φ30.

Fig. 7
figure 7

The complete synchronization areas depicted by the normalized mean synchronization error E, a the distribution of E in the k-φ30 plane for (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0, 0, 0, 0), b the distribution of E in the x10-φ10 plane for (x10, 0, φ10, 0, 0, 0), k = 0.2 and φ30 = 0, c the distribution of E in the x10-φ10 plane for (x10, 0, φ10, 0, 0, 0), k = 5 and φ30 =  − 5

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As revealed in Figs. 5 and 6, infinite coexisting attractors controlled by the subsystems’ initial conditions are observed in the memristor-coupled homogenous network. Their synchronization states are of particular interest for understanding the synchronicities of memristor-coupled network (2). For coexisting attractors revealed in Fig. 5a, the normalized mean synchronization errors in the x10-φ10 plane are presented in Fig. 7b. Network (2) can reach complete synchronization when the initial conditions x10 and φ10 are located in the red region nearby the diagonal line. The non-memristor initial condition x10 and memristor initial condition φ10 show different shifting effects on the phase space positions of the generated attractors [36]. Their shifting effects cancel each other out when x10 and φ10 are properly tuned. For example, if x10 is specified as − 0.8, − 0.5, and − 0.2, zero E can be achieved by optimizing φ10 as − 0.93, − 0.57, and − 0.226, respectively. Therefore, completely synchronized periodic and chaotic motions can be obtained.

However, when large positive coupling strengths and negative coupling memristor initial conditions are selected, i.e., k = 5 and φ30 =  − 5, the synchronization error significantly decreases as illustrated in Fig. 7c. The red complete synchronization region in the x10-φ10 plane migrates to the vicinity of φ10 = 0 and its scope is slightly expanded. This finding further verifies the synchronous effects of the coupling memristor coefficients.

4.2 Parallel-offset synchronization

In addition to the complete synchronization behaviors, parallel-offset synchronization behaviors could be expected, that is, the state variables x1(x2) and y1(y2) fit each other perfectly but the state variables φ1 and φ2 oscillate synchronously with a certain position offset determined by the initial condition mismatches between the two subsystems. The synchronization errors of the non-memristor state variables x1(x2) and y1(y2) can be evaluated by the normalized mean synchronization error E′ defined as

$$ E^{\prime} = \frac{1}{N}\mathop {\mathop \sum \nolimits }\limits_{n = 1}^{N} \frac{{\sqrt {\left[ {x_{1} (n) - x_{2} (n)} \right]^{2} + \left[ {y_{1} (n) - y_{2} (n)} \right]^{2} } }}{{\sqrt {x_{1} (n)^{2} + y_{1} (n)^{2} + x_{2} (n)^{2} + y_{2} (n)^{2} } }} $$
(11)

The parallel-offset synchronization behaviors are examined by comparing the normalized mean synchronization errors E and E′.

When the subsystems’ initial conditions are fixed as (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0.3, 0, 0, 0), initial condition mismatches are observed between the two subsystems. Figure 8a displays the 1D plots of E and E′ depicted within k = [0, 5] for φ30 = 0 and − 5, and Fig. 8b presents the 1D plots of E and E′ depicted within φ30 = [− 5, 4] for k = 0.2 and 5. For fixed φ30 = 0 and − 5, E tends to be a nonzero constant, and E′ gradually reaches 0 with the increase of k. When the positive coupling strength k is large enough, the error between state variables x1(x2) and y1(y2) asymptotically approaches zero and that between φ1 and φ2 oscillates around a constant. This finding indicates the generation of parallel-offset synchronization. The synchronization error between x1(x2) and y1(y2) continuously decreases with the increase of k. For fixed k = 0.2 and 5 in Fig. 8b, the memristor-coupled network enters a parallel-offset synchronization state when φ30 decreases below a certain threshold. Afterward, φ30 no longer has a significant influence on the synchronization errors.

Fig.8
figure 8

Normalized mean synchronization errors E and E′ depicted concerning k or φ30 for (x10, y10, φ10, x20, y20, φ20) = (0.01, 0, 0.3, 0, 0, 0), a plots about variable k with fixed φ30 = 0 and φ30 =  − 5, b plots about variable φ30 with fixed k = 0.2 and k = 5

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In Fig. 9a, the distribution of E′ in the x10-φ10 plane is depicted with fixed k = 5 and φ30 =  − 5. It demonstrates the parallel-offset synchronization regions in the x10-φ10 plane when initial condition mismatches occur between the two subsystems. Note that the complete synchronization is a special case of the parallel-offset synchronization. Excluding the red complete synchronization regions depicted in Fig. 7c, the remaining blue regions can be identified as parallel-offset synchronization regions. The corresponding 2D bifurcation plot is depicted in Fig. 9b to reveal the dynamical behaviors within the parallel-offset synchronization regions. From these two figures, parallel-offset synchronized chaotic and periodic behaviors are discovered.

Fig. 9
figure 9

Coexisting and synchronous dynamical behaviors in the x10-φ10 plane with k = 5, φ30 =  − 5, and (x10, y10, φ10, x20, y20, φ20) = (x10, 0, φ10, 0, 0, 0), a normalized mean synchronization error E′ in the x10-φ10 plane, b 2D bifurcation plot in the x10 – φ10 plane

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The phase portraits and time-domain waveforms of three representative synchronous behaviors of Figs. 7 and 9 are illustrated in Fig. 10a, b, respectively. In detail, Fig. 10a1, b1 shows the phase portraits and time waveforms for the completely synchronized periodic behavior occurring at k = 0.2 and (x10, y10, φ10, x20, y20, φ20, φ30) = (− 0.5, 0, − 0.57, 0, 0, 0, 0). When k = 5, φ30 =  − 5 are assigned and the other initial conditions remain unchanged, the dynamics turns into parallel-offset synchronized periodic behavior as depicted in Fig. 10a2, b2. Furthermore, when k = 5 and φ30 =  − 5 remain unchanged and x10 = 0.01 and φ10 = 0.3 are specified, a parallel-offset synchronized chaotic behavior is obtained as shown in Fig. 10a3, b3. These results demonstrate that the synchronous behaviors of the memristor-coupled homogenous network are flexibly controlled by the coefficients of coupling memristor and the initial conditions of the two subsystems.

Fig.10
figure 10

Phase portraits and time-domain waveforms of typical synchronous behaviors with varied k and initial conditions (x10, y10, φ10, x20, y20, φ20, φ30) = (x10, 0, φ10, 0, 0, 0, φ30), a1 and b1 complete synchronization for k = 0.2, x10 =  − 0.5, φ10 =  − 0.57, φ30 = 0, a2 and b2 parallel-offset synchronization for k = 5, x10 =  − 0.5, φ10 =  − 0.57, and φ30 =  − 5, a3 and b3 parallel-offset synchronization for k = 5, x10 = 0.01, φ10 = 0.3, and φ30 =  − 5

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4.3 Phase synchronization

Phase synchronization can easily be achieved beyond the complete and parallel-offset synchronization regions. In this case, the phases of two subsystems become locked and their amplitudes remain highly uncorrelated [37]. Using the method described in [25], the phase of a periodic or chaotic motion can be defined as

$$ \theta (t) = 2{\uppi }n + 2{\uppi }\frac{{t - t_{n} }}{{t_{n + 1} - t_{n} }},\left( {t_{n} < t < t_{n + 1} } \right) $$
(12)

where tn is the time of the n-th crossing of the motion on an appropriate Poincare section. Successive crossing with the Poincare section can be associated with a phase increase of 2π and the phases in between are computed with a linear interpolation as described in (12). Then, the phase synchronization can be identified by detecting the phase difference between two coupled subsystems.

The phase difference Δθ1(t) = θx1(t) − θx2(t) calculated from the x1 and x2 components of network (2) is mainly examined. Its time-domain waveforms are plotted under different sets of control parameters as shown in Fig. 11. Corresponding to the asynchronous motion depicted in Fig. 4c for k = 0.2 and (x10, y10, φ10, x20, y20, φ20, φ30) = (0.01, 0, 0, 0, 0, 0, 3.5), the time evolution of phase difference Δ1θ(t) is depicted by a red line in Fig. 11a. The phase difference between x1(t) and x2(t) fluctuates around zero and the amplitudes of their trajectories are uncorrelated in most time regions, indicating the generation of phase synchronization. When k = 0.2 is assigned and the initial conditions are tuned as (0.01, 0, 1, 0, 0, 0, 0) and (− 1, 0, 1, 0, 0, 0, 0), the phase errors are illustrated by the green and black lines in Fig. 11a. The fluctuation amplitudes of phase errors are bounded within relatively small values and directed by the initial conditions of network (2). However, when y1, y2 or φ1, φ2 are taken as the analytic targets, the calculated phase differences Δθ2(t) = θy1(t) − θy2(t) and Δθ3(t) = θφ1(t) − θφ2(t) are slightly different from Δθ1(t) as depicted in Fig. 11a. Their time-domain waveforms are illustrated in Fig. 11b, c, respectively. In summary, the natural frequencies of two subsystems are easily correlated due to the same stimulus of 1.8 cos(t) and their phase difference can be limited within an arbitrary constant.

Fig. 11
figure 11

Phase synchronization with the same stimulus frequency F = 1, a time evolutions of phase difference Δθ1(t), b time evolutions of phase difference Δθ2(t), c time evolutions of phase difference Δθ3(t)

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When the stimulus frequency of the first subsystem is maintained as F1 = 1 and that of the second subsystem is slightly increased to F2 = 1.01, the frequency correlation breaks and the phases of the two subsystems become unlocked. The blue trajectory in Fig. 12a depicts the linearly decreased phase difference Δθ1(t) for k = 0.2 and (x10, y10, φ10, x20, y20, φ20, φ30) = (0.01, 0, 0, 0, 0, 0, 3.5). In this case, the two subsystems are asynchronous. If the coupling memristor coefficients are changed to φ30 =  − 5 and k = 5, the memristor-coupled network (2) enters an imperfect phase synchronization state with perfect synchronization epochs interrupted by regular phase slips. The corresponding time evolution of Δθ1(t) is given at the bottom of Fig. 12a. In these two cases, the evolution patterns of Δθ1(t), Δθ2(t), and Δθ3(t) are consistent.

Fig. 12
figure 12

Phase synchronization with different stimulus frequencies F1 = 1 and F2 = 1.01, a asynchronous motion and imperfect phase synchronization with regular phase slips, b imperfectly synchronized x1 and x2 state variables with 22π phase jumps, and asynchronous y1, y2, and φ1, φ2 state variables, c imperfect phase synchronization with − 2π phase jumps

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In another case, we set k = 0.2 and (x10, y10, φ10, x20, y20, φ20, φ30) = (− 1, 0, 1, 0, 0, 0, 0). The state variables x1 and x2 are imperfectly synchronized with 22π phase jumps, as shown by the blue line of Fig. 12b. However, the other two pairs of state variables are out of synchronization because their phase differences linearly decrease as depicted by the red and black lines of Fig. 12b. If φ30 and k are adjusted as − 5 and 5, respectively, the whole network can reach imperfect phase synchronization and the amplitude of the phase jumps is reduced to 2π, as shown in Fig. 12c.

5 Circuit realization and PSIM simulations

In this section, the synchronous behaviors of the memristor-coupled network are equivalently verified in an analog circuit through PSIM circuit simulations. The PSIM software is commonly used to verify initial condition-sensitive dynamical behaviors [38, 39].

The implementation circuit of network (2) is designed as depicted in Fig. 13. It consists of seven integrators, an inverse addition circuit, four inverters, three hyperbolic tangent function converters, and four multipliers. The circuit equations for the seven capacitor voltages vx1, vy1, vφ1, vx2, vy2, vφ2, and vφ3 are established as

$$ \left\{ \begin{aligned} &{\rm RC}\frac{{{\text{d}}v_{x1} }}{{{\text{d}}t}} = v_{y1} + \frac{R}{{R_{a} }}(1 - \tanh v_{\varphi 1} )v_{x1} + 1.8\cos (2{\uppi }ft) - \frac{R}{{R_{k} }}(1 - \tanh v_{\varphi 3} )(v_{x1} - v_{x2} ), \hfill \\ &RC\frac{{{\text{d}}v_{y1} }}{{{\text{d}}t}} = - v_{x1} - v_{y1} , \hfill \\ &RC\frac{{{\text{d}}v_{\varphi 1} }}{{{\text{d}}t}} = - v_{x1} , \hfill \\ &RC\frac{{{\text{d}}v_{x2} }}{{{\text{d}}t}} = v_{y2} + \frac{R}{{R_{a} }}(1 - \tanh v_{\varphi 2} )v_{x2} + 1.8\cos (2{\uppi }ft) + \frac{R}{{R_{k} }}(1 - \tanh v_{\varphi 3} )(v_{x1} - v_{x2} ), \hfill \\ &RC\frac{{{\text{d}}v_{y2} }}{{{\text{d}}t}} = - v_{x2} - v_{y2} , \hfill \\ &RC\frac{{{\text{d}}v_{\varphi 2} }}{{{\text{d}}t}} = - v_{x2} , \hfill \\ &RC\frac{{{\text{d}}v_{\varphi 3} }}{{{\text{d}}t}} = v_{x1} - v_{x2} . \end{aligned} \right. $$
(13)
Fig. 13
figure 13

Circuit schematics for the equivalent realization circuits of network (2)

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In Fig. 13, the gains of four multipliers M1, M2, M3, and M4 are set as 1. The time-constant-related circuit elements are optimized as R = 10 kΩ and C = 100 nF. Other circuit elements are calculated as Ra = R/0.5 = 20 kΩ, Rk = 10/k kΩ, f = F/(2πRC) = 159 Hz. The memristor Rk and initial conditions of seven capacitor voltages vx1(0), vy1(0), vφ1(0), vx2(0), vy2(0), vφ2(0), and vφ3(0) are the main control parameters of this equivalent realization circuit.

Referring to Fig. 13, a circuit simulation model is established using PSIM software, in which the hyperbolic tangent function is realized by the editable math function modules. The simulation control parameters “time step”, “total time”, and “print time” are adjusted to 10 μs, 600 ms, and 400 ms, respectively. The three representative synchronous behaviors demonstrated in Fig. 10 are measured by adjusting Rk and seven initial capacitor voltages as Rk = 50 kΩ, (− 0.5 V, 0 V, − 0.57 V, 0 V, 0 V, 0 V, 0 V), then as Rk = 2 kΩ, (− 0.5 V, 0 V, − 0.57 V, 0 V, 0 V, 0 V, − 5 V), and finally as Rk = 2 kΩ, (0.01 V, 0 V, 0.3 V, 0 V, 0 V, 0 V, − 5 V). The captured phase portraits and time-domain waveforms are presented in Fig. 14(a) and (b), respectively. The circuit simulation results in Fig. 14 agree well with the numerical simulation results in Fig. 10, implying the feasibility of the implementation circuit and verifying the correctness of theoretical and numerical analysis results.

Fig. 14
figure 14

PSIM circuit simulation results with fixed initial conditions vy1(0) = vx2(0) = vy2(0) = vφ2(0) = 0 V, (a1) and (b1) complete synchronization for Rk = 50 kΩ, vx1(0) = –0.5 V, vφ1(0) = –0.57 V, and vφ3(0) = 0 V, a2 and b2 Parallel-offset synchronization for Rk = 2 kΩ, vx1(0) = –0.5 V, vφ1(0) = –0.57 V, and vφ3(0) = –5 V, a3 and b3 Parallel-offset synchronization for Rk = 2 kΩ, vx1(0) = 0.01 V, vφ1(0) = 0.3 V, and vφ3(0) = –5 V

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6 Conclusion

This work investigated the initial condition-sensitive coexisting and synchronous behaviors of a memristor-coupled homogeneous network consisting of two identical non-autonomous memristive Fitzhugh–Nagumo models. Affected by the external cosine stimuli, this homogenous network possesses a space equilibrium set S = (0, 0, c1, 0, 0, c2, c3) at discrete time points satisfying 1.8cos(t) = 0; otherwise, it has no equilibrium points. Similar to memristive systems with the line, plane, or space equilibrium sets, the memristor-coupled homogeneous network exhibits extreme multistability with coexisting hyperchaotic, chaotic, periodic, and quasi-periodic attractors. Moreover, a 4D dimensionality reduction model was built through incremental integral transformation, based on which the revealed extreme multistability phenomenon was proved to be hidden by stability analyses of the reformed equilibrium points. Numerical simulations demonstrated that the synchronicities of these coexisting hidden behaviors depend not only on the initial condition and coupling strength of the coupling memristor but also on the subsystems’ initial conditions. When large positive coupling strengths and negative coupling memristor initial conditions are chosen, the two coupled Fitzhugh–Nagumo models can enter complete or parallel-offset synchronization states with properly selected subsystems’ initial conditions. In addition to these two synchronous behaviors, phase synchronization can easily be achieved due to the existence of external stimuli. This initial condition-sensitive synchronization property can benefit the flexible control of the coupled homogeneous network.

In this study, only a pair of state variables is concerned in the coupling channel. The constructed network exhibits various coexisting and synchronous behaviors flexibly controlled by the coupling strength and initial conditions of the network. Future research will focus on introducing more than a pair of state variables in the coupling channel.