1. Introduction
The conceptional proposal of a memristor [1], the fabrication of a memristor [2] and its potential application have been widely explored in the control of a nonlinear circuit [3-5], neuromorphic computing [6-8], chaotic systems [9-11], and artificial neural networks [12-15]. In fact, a memristor can be embedded into a nonlinear circuit to develop a memristive circuit for presenting rich complex dynamical characteristics. For example, Yang et al [16] designed a simple chaotic circuit by coupling a magnetic flux-controlled memristor (MFCM) with two capacitors and an inductor. The memristive circuit has a complex hyperchaotic phenomenon and the parameter range for the chaotic region is large. Dou et al [17] proposed an RC bridge oscillatory circuit based on a memristor, and this memristive circuit can result in symmetric coexistence such as single-scroll, asymmetrical single-scroll, symmetric double-scroll and asymmetrical limit-cycle behaviors. Zhou et al [18] developed a new memristive chaotic system for inducing multistability. In addition, memristive functional neuron models [19-21] are obtained by connecting memristors into a simple neural circuit.
Recently, discrete memristors (DM) have become a research hotspot. DM's have some advantages such as low power consumption, programmability and anti-interference. Therefore, they have potential applications in the design of DM chaotic/hyperchaotic maps [22-24], neural networks [25-27] and embedded system development [28] fields. Furthermore, a variety of DM maps have been proposed for a dynamical approach and application in signal processing. For instance, Zhang et al [29, 30] proposed dual memristor hyperchaotic maps from mathematical assumption. Bao et al [31] designed a parallel bi-memristor hyperchaotic map. The discrete memristive Rulkov neuron map [32, 33] was developed by introducing a DM into a Rulkov neuron model. A fractional memristive map is proposed in [34]. In addition, DM chaotic maps are hardware implemented by applying DSP platform [35-37] and an analog circuit [38]. One important application is that DM chaotic maps are used in image encryption algorithms [39-41].
Most of the previous works about the approach of DM maps suggested that a DM map can be defined by introducing a DM into an existing discrete map. In fact, this kind of description is artificial and is a mathematical assumption without clarifying its inner physical property. The physical modeling of a DM map is not perfect, and the energy of a DM map is also an open problem. Inspired by the suggestion in [42], biophysical and memristive maps can be converted from memristive circuits, in which the energy function for its equivalent memristive oscillator is defined in a theoretical way. In this work, a dual memristive circuit is constructed by using two different memristor elements, a nonlinear resistor and a capacitor. The dual memristive oscillator is derived and its Hamilton energy function is obtained from physical approach. According to the transformation relationship between the oscillator and the map described in [43], the dual memristive map and the corresponding Hamilton energy function are obtained. The results provide further possible guidance for designing discrete maps and calculating their energy.
This study is organized as follows: in section 2 , the dual DM map is obtained. The numerical investigations are given in section 3 . In section 4 , the results are summarised.
2. Model and scheme
Due to the nonlinear properties of the memristor, the dynamics of a nonlinear circuit coupled by the memristor and the memristive circuit become more controllable. In this paper, a memristive circuit is constructed by including a MFCM and a CCM in two additive branch circuits, a nonlinear resistor and a capacitor are coupled in parallel in figure 1.
Figure 1. Schematic diagram for a dual memristive circuit. M(φ) denotes mem-conductance for a MFCM and M(q) represents mem-resistance for a CCM, respectively. C and NR means a capacitor and a nonlinear resistor, respectively. |
In figure 1, the current of the nonlinear resistance (NR) can be estimated by 2 ), the M(φ) and M(q) have simple forms with low order, and their internal state equations are only composed of linear terms. Compared with higher-order memristors, the memristors in this paper are easier to implement physically. Sinusoidal stimulus Asin(2πωt) is imposed on the memristor, where A means the amplitude of stimulus, and ω denotes the frequency of the stimulus. The v-i curve for the MFCM with different frequencies ω under A = 3 is shown in figure 2(a), the v-i curve for the MFCM with different amplitudes A at ω = 0.8 is shown in figure 2(b).
$\begin{eqnarray}{i}_{{\rm{N}}{\rm{R}}}=-\displaystyle \frac{r}{\rho }\left(V-\displaystyle \frac{{V}^{2}}{{V}_{0}}\right).\end{eqnarray}$
where ρ is the resistance within the linear region, V0 denotes a cut-off voltage and r means a dimensionless gain. V is the across voltage of the nonlinear resistor, and the dependence of voltage on the memristive channels current for the MFCM and CCM is defined by $\begin{eqnarray}\left\{\begin{array}{l}{i}_{M1}=M(\varphi ){V}_{M1}=\varphi V;\,\displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}t}=a\varphi +bV;\\ {V}_{M2}=M(q){i}_{M2}=\csc q\cdot {i}_{M2};\,\displaystyle \frac{{\rm{d}}q}{{\rm{d}}t}={\rm{d}}q+{i}_{M2}.\end{array}\right.\end{eqnarray}$
As shown in equation (
Figure 2. The v-i curve for the MFCM with different stimuli. For (a) A = 3; (b) ω = 0.8. |
Figure 2 shows that the v-i curve for the MFCM is 8-shaped. The 8-shaped area decreases when the frequency of the stimulus is increased. While the 8-shaped area increases when the amplitude of the stimulus is increased. By applying the same sinusoidal stimulus Asin(2πωt), the i-v curve for the CCM with different stimuli are displayed in figure 3.
Figure 3. The i-v curve for the MFCM with different stimuli. For (a) A = 3; (b) ω = 0.8. |
The results in figure 3 confirm that the i-v curve for the CCM exhibits a similar 8-shaped characteristic. The 8-shaped area becomes larger with the increase of amplitude and frequency of the stimulus. According to Kirchhoff's laws, the circuit equations for the memristive circuit in figure 1 are described by 3 ) are defined as follows 3 ) can be rewritten by 6 ). 6 ) for the memristive oscillator in equation (5 ) can be proven by applying the Helmholtz's theorem, when the memristive oscillator in equation (5 ) is rewritten in a vector form 6 ) satisfies the criterion in equations (8 ) and (9 ) completely. Furthermore, linear transformation is imposed on the sampled time series for variables from equation (5 ) with time step Δτ during a numerical approach. 6 ), the Hamilton energy for the map in equation (11 ) is described by
$\begin{eqnarray}\left\{\begin{array}{l}C\displaystyle \frac{{\rm{d}}V}{{\rm{d}}t}=\displaystyle \frac{r}{\rho }\left(V-\displaystyle \frac{{V}^{2}}{{V}_{0}}\right)-\varphi V-\,\sin \left(q\right)V;\\ \displaystyle \frac{{\rm{d}}\varphi }{{\rm{d}}t}=a\varphi +bV;\\ \displaystyle \frac{{\rm{d}}q}{{\rm{d}}t}={\rm{d}}q+\,\sin \left(q\right)V.\end{array}\right.\end{eqnarray}$
To facilitate the analysis, dimensionless parameters and variables in equation ( $\begin{eqnarray}\left\{\begin{array}{c}x=\displaystyle \frac{V}{{V}_{0}},w=\displaystyle \frac{\varphi }{\rho C{V}_{0}},u=\displaystyle \frac{q}{C{V}_{0}},\tau =\displaystyle \frac{t}{\rho C},a={\rho }^{2}C{V}_{0},\\ c=a\rho C,\alpha =\rho ,k=C{V}_{0},{\rm{d}}^{\prime} ={\rm{d}}C\rho .\end{array}\right.\end{eqnarray}$
The dynamic equations in equation ( $\begin{eqnarray}\left\{\begin{array}{c}\displaystyle \frac{{\rm{d}}x}{{\rm{d}}t}=r(x-{x}^{2})-awx-\alpha \,\sin (ku)x;\\ \displaystyle \frac{{\rm{d}}w}{{\rm{d}}t}=cw+bx\,;\\ \displaystyle \frac{{\rm{d}}u}{{\rm{d}}t}={\rm{d}}^{\prime} u+\alpha \,\sin (ku)x.\end{array}\right.\end{eqnarray}$
The memristive ciruit in figure 1 has three energy storage elements, the field energy W and its dimensionless Hamilton energy H are given in equation ( $\begin{eqnarray}\left\{\begin{array}{c}W=\displaystyle \frac{1}{2}C{V}^{2}+\displaystyle \frac{1}{2}\varphi {i}_{M1}+\displaystyle \frac{1}{2}qV;\\ H=\displaystyle \frac{W}{C{V}_{0}^{2}}=\displaystyle \frac{1}{2}{x}^{2}+\displaystyle \frac{1}{2}a{w}^{2}x+\displaystyle \frac{1}{2}ux.\end{array}\right.\end{eqnarray}$
The Hamilton energy function H in equation ( $\begin{eqnarray*}\left(\begin{array}{l}\displaystyle \frac{{\rm{d}}x}{{\rm{d}}\tau }\\ \displaystyle \frac{{\rm{d}}w}{{\rm{d}}\tau }\\ \displaystyle \frac{{\rm{d}}u}{{\rm{d}}\tau }\end{array}\right)=\left(\begin{array}{l}r(x-{x}^{2})-awx-\alpha \,\sin (ku)x\\ cw+bx\\ {\rm{d}}^{\prime} u+\alpha \,\sin (ku)x\end{array}\right)={F}_{{\rm{c}}}+{F}_{{\rm{d}}}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{c}=\,\left(\begin{array}{c}-abwx-{\rm{d}}^{\prime} x\\ bx+0.5ab{w}^{2}+0.5bu\\ 2{\rm{d}}^{\prime} x+a{\rm{d}}^{\prime} {w}^{2}+{\rm{d}}^{\prime} u\end{array}\right)\\ +\,\left(\begin{array}{c}r(x-{x}^{2})-awx(1-b)+{\rm{d}}^{\prime} x-\alpha \,\sin (ku)x\\ cw-0.5ab{w}^{2}-0.5bu\\ -2{\rm{d}}^{\prime} x-a{\rm{d}}^{\prime} {w}^{2}+\alpha \,\sin (ku)x\end{array}\right)\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{c}=\,\left(\begin{array}{ccc}0 & -b & -2{\rm{d}}^{\prime} \\ b & 0 & 0\\ 2{\rm{d}}^{\prime} & 0 & 0\end{array}\right)\left(\begin{array}{c}x+0.5a{w}^{2}+0.5u\\ awx\\ 0.5x\end{array}\right)\\ +\,\left(\begin{array}{ccc}{A}_{11} & 0 & 0\\ 0 & {A}_{22} & 0\\ 0 & 0 & {A}_{33}\end{array}\right)\left(\begin{array}{c}x+0.5a{w}^{2}+0.5u\\ awx\\ 0.5x\end{array}\right);\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{A}_{11}\\ =\,\displaystyle \frac{r(x-{x}^{2})-awx(1-b)+d^{\prime} x-\alpha \,\sin (ku)x}{x+0.5a{w}^{2}+0.5u};\\ {A}_{22}=\displaystyle \frac{cw-0.5ab{w}^{2}-0.5bu}{awx};\\ {A}_{33}=\displaystyle \frac{-2d^{\prime} x-ad{w}^{2}+\alpha \,\sin (ku)x}{0.5x}.\end{array}\end{eqnarray}$
According to the Helmholtz's theorem, the Hamilton energy function H meets the following criterion $\begin{eqnarray}\begin{array}{c}{\rm{\nabla }}{H}^{{\rm{T}}}{F}_{{\rm{c}}}=\left(-abwx-{\rm{d}}^{\prime} x\right)\displaystyle \frac{\partial H}{\partial x}\\ +\,\left(bx+0.5ab{w}^{2}+0.5bu\right)\displaystyle \frac{\partial H}{\partial w}\\ +\,\left(2{\rm{d}}^{\prime} x+{\rm{d}}^{\prime} u+a{\rm{d}}^{\prime} {w}^{2}\right)\displaystyle \frac{\partial H}{\partial u}=0.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}\displaystyle \frac{{\rm{d}}H}{{\rm{d}}\tau }=\left[r(x-{x}^{2})-awx(1-b)+{\rm{d}}^{\prime} x\right.\\ \left.-\,\alpha \,\sin (ku)x\right](x+0.5a{w}^{2}+0.5u)\\ +\,awx(cw-0.5ab{w}^{2}-0.5bu)\\ +\,0.5x\left[-2{\rm{d}}^{\prime} x-a{\rm{d}}{w}^{2}+\alpha \,\sin (ku)x\right]={\rm{\nabla }}{H}^{{\rm{T}}}{F}_{{\rm{d}}}.\end{array}\end{eqnarray}$
Indeed, the energy function H in equation ( $\begin{eqnarray}\left\{\begin{array}{c}{y}_{{n}}=\displaystyle \frac{r{\rm{\Delta }}\tau }{1+r{\rm{\Delta }}\tau }{x}_{{n}},{z}_{{n}}=\displaystyle \frac{r{\rm{\Delta }}\tau }{1+r{\rm{\Delta }}\tau }{w}_{{n}},\\ {v}_{{n}}=\displaystyle \frac{r{\rm{\Delta }}\tau }{1+r{\rm{\Delta }}\tau }{u}_{{n}},\lambda =1+r{\rm{\Delta }}\tau ;\\ b=\displaystyle \frac{1+r{\rm{\Delta }}\tau }{r},\beta =\alpha {\rm{\Delta }}\tau ,\delta =k\displaystyle \frac{1+r{\rm{\Delta }}\tau }{r{\rm{\Delta }}\tau },\\ a=c{\rm{\Delta }}\tau +1,c=b{\rm{\Delta }}\tau ,d=d^{\prime} {\rm{\Delta }}\tau +1.\end{array}\right.\end{eqnarray}$
These renewed discrete variables accompanied with updated parameters can be used to define a new memristive map as follows $\begin{eqnarray}\left\{\begin{array}{c}{y}_{{n}+1}=\lambda \left({y}_{{n}}-{y}_{{n}}^{2}\right)-b{y}_{{n}}{z}_{{n}}-\beta \,\sin \left(\delta {v}_{{n}}\right){y}_{{n}};\\ {z}_{{n}+1}=a{y}_{{n}}+c{z}_{{n}};\\ {v}_{{n}+1}=d{v}_{{n}}+\beta \,\sin \left(\delta {v}_{{n}}\right){y}_{{n}}.\end{array}\right.\end{eqnarray}$
Under the same weights for two terms in the discrete energy function for equation ( $\begin{eqnarray}{H}_{n}=\displaystyle \frac{1}{2}{y}_{n}+\displaystyle \frac{1}{2}b{z}_{n}^{2}{y}_{n}+\displaystyle \frac{1}{2}{v}_{n}{y}_{n}.\end{eqnarray}$
As suggested in the recent works [44, 45], membrane parameters or memristive parameters can be controlled by the energy flow to induce a mode transition when the inner energy level is beyond a threshold. To investigate the self-adaption in the dual memristive map, it requires that parameter a for the dual memristive map can be controlled by the energy flux in an adaptive way as follows $\begin{eqnarray}\left\{\begin{array}{l}{a}_{n+1}={a}_{n}+k\cdot \theta (p-{H}_{n}),\theta (\cdot )\\ =1,p\geqslant 0,\theta (\cdot )=0,p\lt 0;\\ or,\,a=k\cdot \theta (p-{H}_{n}),\theta (\cdot )\\ =1,p\geqslant 0,\theta (\cdot )=0,p\lt 0.\end{array}\right.\end{eqnarray}$
Where the energy threshold 0 < p < 1 determines the energy level of a memristive map in the nth iteration. an denotes value for the map parameter in the nth iteration, parameter k means growth step, and the Heaviside function θ(*) is used to control parameter growth to a saturation value.3. Results and discussion
Similar to most of the memristive oscillators [20, 21, 46-50], numerical solutions for equation (5 ) can be obtained and additive noise can be applied to induce nonlinear resonance and mode control in neural activities. Numerical results are mainly explored in the memristive map under the control of adaptive law in equation (13 ). To investigate the complex dynamic characteristics of the dual memristive map presented in equation (11 ), the initial values of the memristive map are selected as (0.1, 0.1, 0). The parameters of the memristive map are fixed as λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1. The bifurcation diagram and the Lyapunov exponents (LEs) spectrum are calculated by changing the parameter a carefully, and the results are plotted in figure 4.
Figure 4. (a) Bifurcation diagram and (b) the distribution of the LEs for the memristive map by changing parameter a. Setting λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1. The initial value is (0.1, 0.1, 0). |
The results in figure 4 show that the dual memristive map triggers a mode transition with the increase of the parameter a, such as period-1, period-2, period-4, period-8 and chaotic behaviors are generated. The positive Lyapunov exponent becomes negative and the inverse double period bifurcation occurs by increasing parameter a in a continuous way, and chaos is suppressed effectively. Furthermore, phase portraits of the memristive map with different values for parameter a are shown in figure 5.
Figure 5. Phase portraits of the memristive map by changing parameter a. For (a) a = 2; (b) a = 1.5; (c) a = 1.4; (d) a = 0.5. The parameters are λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1. The initial value is (0.1, 0.1, 0). |
In figure 5, the profile of the attractors shows period-2, period-4, period-8 and chaotic characteristics. According to the Hamilton energy function presented in equation (12 ), the relation between the Hamilton energy and the oscillatory states in the memristive map is calculated, and the results are shown in figure 6.
Figure 6. Evolution of energy function and its average value in the memristive map by changing parameter a. For (a) a = 2; (b) a = 1.5; (c) a = 1.4; (d) a = 0.5. Setting λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1. The initial value is (0.1, 0.1, 0). <Hn> denotes the average value of the Hamilton energy. |
The results in figure 6 illustrate that the dual memristive map has a higher mean value of Hamilton energy with period state, while it has a lower mean value of Hamilton energy with a chaotic pattern. In addition, the mean value of Hamilton energy decreases with the increase of a period number.
When the parameters in equation (11 ) are fixed at d = 1 and δ = 1, the memristive map exits the fixed point (0, 0, 0±hπ) (h = 0, ±1, ±2, ±3…), and indicates that the memristive map can be initially boosted to give extreme multistability. The parameters are λ = 3.9, a = 0.5, b = 0.2, c = 0.3, d = 1, β = 0.1, δ = 1, by changing the initial value v(0), the phase portraits are shown in figure 7.
Figure 7. Phase portraits of the memristive map by changing initial value v(0). For (a) initial value is (0.1, 0.1, 0-hπ), (h = 0, 1, 2, 3, 4); (b) initial value is (0.1, 0.1, 0+hπ), (h = 0, 1, 2, 3, 4). The parameters are λ = 3.9, a = 0.5, b = 0.2, c = 0.3, d = 1, β = 0.1, δ = 1. |
The results in figure 7 confirm that the memristive map will produce an extreme multistable phenomenon by changing the initial value for the variable v. According to the criterion of the growth for parameter a in equation (13 ), it sets the threshold p = 0.35, parameters are fixed at λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1, and initial values are selected as (0.1, 0.1, 0). The initial value for parameter a = 0.5, gains k = 0.005 and k = 0.007, the memristive map in equation (11 ) is iterated 1200 times, respectively. The growth of parameter a, changes of Hamilton energy and phase diagram are shown in figure 8.
Figure 8. Growth of parameter a, Hamilton energy and phase diagram for gain k. For (a) k = 0.005; (b) k = 0.007. Setting λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1. Setting initial values for variables (0.1, 0.1, 0). |
It is confirmed that parameter a of the memristive map relative to when the MFCM reaches a stable value of 2.39 after 800 iterations. The Hamilton energy shows a distinct transition when the memristive maps change the chaotic patterns to periodic states, and a similar shift occurs in the phase diagram. By comparing figures 8(a) and (b), it is found that with the increased value of gain k, parameter a needs fewer iterations to reach a saturation value. The result indicates that the dynamics for the memristive map can be controlled by the energy flow in an adaptive way. Considering that the chaotic systems are extremely sensitive to the parameters and initial values, the map attractor basins on y(0)-v(0) and z(0)-v(0) are shown in figure 9.
Figure 9. Map attractor basins on the plane. For (a) initial value (y(0), 0.1, v(0)); (b) initial value (0.1, z(0) , v(0)). Setting λ = 3.9, b = 0.2, c = 0.3, d = 0.4, β = 0.1, δ = 0.1, a = 0.5. |
It is found that the initial values affect the chaotic region (blue region) and the periodic region (red region) at different initial values. As a result, the stochastic and continuous switch of the initial values for the memristive variables will induce a distinct mode transition and energy shift.
From the memristive oscillator in equation (5 ) to the memristive map in equation (11 ), perfect covariance in the formulas is confirmed. The continuous energy function in equation (6 ) is relative to the state variables and bifurcation parameter a directly, while the discrete energy function for the map in equation (12 ) is dependent on the discrete variables and parameter b completely. From a dynamical viewpoint, the energy level for both the memristive oscillator and the memristive map can be calculated by adjusting one bifurcation parameter carefully. In fact, the two curves for the energy function in the oscillator and the map seldom intersect, keeping the same energy value due to the involvement of the time step when the memristive oscillator is obtained with numerical solutions. In an experimental approach, analog signals from the nonlinear circuit can be filtered and adjusted to present discrete signals, and the memristive map provides fast computing without further estimating the time step. Our scheme provides a reference to verify the reliability of the mathematical maps. As a result, these physical maps can be connected to build an artificial array for signal processing in parallel.
4. Conclusion
In this paper, a dual memristive circuit is built to develop a memristive oscillator for discerning the effect of the magnetic field and the electric field synchronously. The energy function for the dual memristive oscillator is obtained in a theoretical way. By applying a linear transformation on the sampled variables for the memristive oscillator, a dual memristive map is designed with a clear definition of the energy function and adaptive growth law for one bifurcation parameter. The results indicate that the dual memristive map has rich dynamical behaviors, and the dynamics can be adjusted in an adaptive way under energy flow. The memristive map has a smaller average Hamilton energy value in the chaotic state, while it prefers to keep a higher mean Hamilton energy value in the periodic state. In addition, the mean Hamilton energy value of the memristive map decreases with the increase of the period number. The results confirm the reliability of discrete maps from nonlinear circuits. The modeling method of the discrete map proposed in this paper can be applied to other high-dimensional discrete maps including discrete memristive maps and neurons. The Hamilton energy calculation and adaptive control method can be applied to the energy calculation and the adaptive control of other discrete scatter maps. In addition, the simplest map proposed in this paper can be used for pseudo-random sequence generators and image encryption fields.