A GCM Neural Network with Piecewise Logistic Chaotic Map

Abstract

In order to explore dynamic mechanisms and chaos control of globally coupled map (GCM) chaotic neural networks, a new GCM model, called the PL-GCM model is proposed, of which a piecewise logistic chaotic map is used instead of a logistic map. As a result of the strong chaotic features of the map, the neurons’ period and chaotic characteristics over a wide range of parameters are discussed, the dynamic mechanism is demonstrated in detail, and the numerical simulations such as state evolution, the largest Lyapunov exponent (LLE), contour map, and so on are exhibited. Furthermore, chaos control of the proposed PL-GCM model is investigated by adopting two chaos control methods. It is shown that the network with conventional coupling or delay coupling can be precisely controlled to any specified periodic orbit by feedback control, and its dynamic associative memory is realized by the variable threshold parameter control method with external information. The results of simulations and experiments suggest that the network is controlled successfully and can output period patterns with a specified period that contains the stored pattern closest to the initial pattern. All features suggest that the network is fit for pattern recognition and information processing.

1. Introduction

Artificial neural network (ANN) is an algorithm model established by imitating the working principle of biological neurons. It can be used in the fields of information processing, pattern recognition, and so on. In 1986, Japanese scholar Aihara and his collaborators found chaos in an electrophysiological experiment of squid giant axons, and established a chaotic neural network (CNN) model [1,2,3]. It has both nonlinear characteristics of artificial neural networks and the ergodic characteristics of chaos, so that it can be used for intelligent information processing and has attracted extensive attention in recent decades. Since then, many chaotic neural network models have been successively presented according to different chaos mechanisms and have also been investigated, such as CNN on the experiment [1,2], the network introducing certain transformations to the Hopfield network created by Chen and Aihara [4,5,6], and the Inoue network composed of chaotic oscillators [7,8,9,10,11]. In 1990, Kaneko proposed a CNN based on the coupled mapping lattice model, which was named the globally coupled map (GCM) model [12]. As time goes on, each element of GCM network evolves according to a logistic map, which is expressed as a quadratic function with an extreme point, and appears as chaotic motion. Moreover, the network has interesting characteristics such as cluster frozen attractors and hidden coherence in turbulent states. With its two parameters specified, all units divide into several clusters, and the units falling into the same cluster will enter an identical orbit.

Associative memory is a vital function of the neural network. Its ability of storing and correctly recalling patterns and images has been widely investigated for different ANNs [13,14,15,16,17]. To CNNs, complex chaotic behavior is a double-edged sword. On the one hand, it makes CNNs fit for applications of associative memory in information processing; on the other hand, it restricts some applications. In fact, the associative memory states have aperiodicity and cannot converge to a stored pattern, thus the outputs cannot keep to a state among them or a periodic orbit and hover around all the stored patterns, and when to stop chaotic dynamics is hard to determined. Therefore, chaos control of CNNs for application in information processing is essential. The existing literatures on control methods for CNNs involve two kinds: with the control targets being specified at first (as in [18,19,20,21,22,23]), and without control target at the start to control CNNs and achieve associative memory, such as pinning control, phase space constraint control, and modulated parameter control as in [24,25].

For the GCM model, a modulated parameter control method with the outer information in the introduced parameter iteration was proposed by Ishii et al. in [26], which can implement both chaos control and information processing. Zheng and Tang improved its convergence rate in [27]. However, dynamic associative memory of the GCM network was still untouched. Based on the above work, we put forward several new GCM models and explored their chaos control and dynamic associative memory in [28,29,30]. The results show that these GCM networks can be controlled to the specified periodic orbit under parameter modulation control and can be used for information processing and pattern recognition.

In order to further understand the properties of GCM models, we make the dynamics of the network with different chaotic maps and its influence on the associative memory clear, and explore the chaos control effect of the GCM model under different control methods. We propose a new GCM chaotic neural network model named “globally coupled map with piecewise logistic chaotic map” (PL-GCM) model. With different ways of coupling and discontinuous piecewise logistic map, except at zero, it exhibits complex and rich dynamics. First, the dynamics analysis suggests that there exist cluster frozen attractors in a wide parameter field; thus, the information presents steadily, and the coupling mode does not affect the number of clusters. Second, feedback control and variable threshold parameter-modulated control are given to implement chaos control. Furthermore, compared with the controlled S-GCM in [26], under the condition of delayed coupling, the controlled PL-GCM by the variable threshold parameter-modulated control shows better performance, which manifests that the network can be controlled to the specified period orbit together with the cluster number controlled. Moreover, it can output not only a unique fixed pattern but also periodic pattern running, as in traditional neural networks or common CNNs. Finally, the effectiveness of the information processing applications of the PL-GCM model is verified with pattern recognition as an example in simulations.

The main content of the article is arranged as follows. A PL-GCM model is proposed and the dynamics analysis is demonstrated in detail, then the mechanism of the system’s chaotic behavior is illustrated from the angle of neurons in Section 2. After that, chaos control of the PL-GCM model is discussed, which involves feedback control and modulated parameter control with variable threshold, and further application and control effect are shown through the associative memory experiments in Section 3 and Section 4. Some conclusions are drawn in Section 5.

2. PL-GCM

2.1. PL-GCM Model

A single neuron of the PL-GCM model can be described as follows:

$$x(t+1)=(1-\epsilon )f\left(x\right(t\left)\right),$$

(1)

$$f\left(x\right)=\left\{\begin{array}{cc}1-4\mu {(x+0.5)}^2,& -1<x<0,\\ 4\mu {(x+0.5)}^2-1,& 0\le x<1,\end{array}\right.$$

(2)

$$y(t+1)=O\left(x\right(t+1\left)\right),$$

(3)

where $x\left(t\right)$ denotes the neuron’s internal state at time t, $y\left(t\right)$ denotes the neuron’s output at time t, $\epsilon$ is the coherent parameter indicating the coupling strength, $f(·)$ is a piecewise logistic chaotic map with bifurcation parameter $\mu$ and is shown in Figure 1a. It can be seen that the curve of the bimodal mapping has two extreme points in its domain. When $\mu =0.5$, it is discontinuous and jumps at $x=0$. When $\mu =1$, it is continuous for $\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$, so that the function is ergodic in the interval $[-1,1]$. When $\mu =2$, it is decomposed into two independent quadratic function images, each with a top, indicating that no matter how x is evaluated, the function can be searched twice on the top, which is the fundamental reason why the piecewise logistic map (2) is superior to the logistic map, based on the fact that Ishii once pointed out that bimodal mapping has better performance than single-modal mapping for information processing applications of the GCM networks.

Figure 2 shows the Lyapunov exponent (LE) of the piecewise logistic chaotic map with the variation of parameter $\mu$. It can be seen that map (2) has a very wide chaotic domain. When $\mu >0.522$, the system enters chaos from period doubling bifurcation; when $\mu \in (1.4012,2)$, the system has a stable chaotic state with the value of $x\left(t\right)$ traversing $[-1,1]$, and the parameter range of the map is much larger than that of logistic map. The above analysis shows that the chaotic characteristics of map (2) are stronger, and it is more difficult to analyze and predict the iterative sequence of the system. Therefore, from this angle, the performance of map (2) is also better than that of logistic map when used in the application of information processing.

In addition, the neuron $x\left(t\right)$ exhibits chaotic behaviour because each neuron’s motion characteristics is determined by the piecewise logistic map, which can be seen in Formula (1). For application, the neuron’s output function $O(·)$ can be selected as a binary representation:

$$O\left(x\right)=\left\{\begin{array}{cc}1,\hfill & x\ge x^{*},\hfill \\ -1,\hfill & x<x^{*}.\hfill \end{array}\right.$$

(4)

where $x^{*}=0$ because zero is the symmetric point of the piecewise logistic map.

A PL-GCM model made up of the above chaotic neurons was constructed. It had time-delay self-feedback and the spatiotemporal summation of the time-delay feedback inputs from other chaotic neurons, of which the ith chaotic neuron is expressed as:

$$x_i(t+1)=(1-\epsilon )f\left(x_i(t-\theta )\right)+\frac{\epsilon }{N}\sum _{j=1}^{N}f\left(x_j(t-\tau )\right),$$

(5)

where $f(·)$ is defined by Formula (2) and $x_i\left(t\right)$ expresses the ith unit’s state at discrete time t, N is the number of units, bifurcation parameter $\mu$, and coherent parameter $\epsilon$, respectively, denote the ith unit’s chaos strength and the coupling strength. $\theta ,\tau$ are time-delay parameters, and both equal to zero means PL-GCM with conventional coupling; otherwise, it means PL-GCM with delay coupling. The methods of delay coupling can be divided into three modes: the case of $\theta =0,\tau \ne 0$ is called delay coupling of the first kind; the case of $\theta =\tau \ne 0$ is called delay coupling of the second kind; and the case of $\theta \ne \tau$ is called delay coupling of the third kind, which includes the first case. It is noteworthy that $\theta ,\tau$ are set to be less than or equal to 10 in the following discussions on account of little overlarge time delay in practice. Additionally, symbols have the same meaning in the context unless otherwise specified.

We now review the network’s dynamical behaviors. First, the case of $\theta =\tau =0$ is considered. We chose a network of 100 neurons for our study, and the initial values were randomly selected. Figure 3 and Figure 4 describe the dynamic states of the system (5) under different parameters. When $\mu =0.1,\epsilon =0.9$, Figure 3 depicts the 2-cycle and 1-cluster orbit of the network. We set sixty of the initial values as positive and forty as negative in Figure 3a, then the network was stable at the point $0.9271$. Furthermore, changing positive and negative ratios of the initial values to 40:60 in Figure 3b, the network was stable at the point $-0.9271$. Moreover, it can be found in the experiment that the network clustering did not change with the change of the initial value in these two cases. However, when positive and negative ratios of the initial value changed and exceeded the critical ratio of 45:65, the network jumped from the 1-cluster state to the 2-cluster state. When $\mu ,\epsilon$ changed slightly, the network still remained in the 2-cycle and 1-cluster state with only the cluster point changing. Otherwise, the network jumped out of this state and entered the multi-period and multi-cluster state.

If $\mu$ is kept unchanged and $\epsilon$ is reduced, the network jumps into the 2-cycle and 2-cluster state, as shown in Figure 4. Moreover, it can be seen that the network’s coupling characteristics weaken with $\epsilon$ decreasing so that the two clusters separate from each other, and when $\epsilon \to 0$, the two clustering points are close to $-1$ and 1, respectively. In addition, when positive and negative ratios of the initial value are fixed, the change of the initial value does not affect the periodic orbit of the network. When the positive and negative ratio changes, only the sign of the cluster point changes, for example, given $\epsilon =0.7$, the orbit was $(0.1283,-0.4549)$ when the positive and negative ratio was 60:40, and it changed to $(-0.1283,0.4549)$ when the positive and negative ratio was 40:60.

If $\epsilon$ is kept unchanged and $\mu$ is appropriately increased, the network can still maintain 1-cluster state, but the number of cycles increases with the increase of $\mu$. For example, when $\epsilon =0.7$ remains unchanged and $\mu$ increases to $0.63$, the network enters a 4-period orbit; further when $\mu =0.69$, the network enters the 8-period orbit, and when $\mu =0.7$, the period of the orbit increases to 32 cycles. If $\mu$ continues to increase, the network leaves the periodic orbit and slowly enters a chaotic state. Another example, when $\epsilon =0.2$ remains unchanged and $\mu$ increases to $0.9$, the network enters the 4-period orbit, but at this time, the network is stable in different 4-period orbits with different initial values. When $\mu$ increases to $0.96$, the network operates randomly and is stable in the orbits with 2 periods and 2 clusters, 2 periods and 4 clusters, 4 periods and 2 clusters, or 8 periods and 2 clusters with different initial values. When $\mu$ continues to increase, the network becomes more unstable and gradually enters the chaotic state, and the network states of 3 clusters with 2 periods and 4 clusters with 4 periods under different parameters are shown in Figure 5.

It can be drawn from the above analysis that parameters $\mu ,\epsilon$ determine the network’s dynamic properties. Roughly speaking, with the increase of $\mu$, the chaotic characteristics of the network are enhanced, and the network finally falls into chaotic orbit after the increase of the running period number and the cluster number. Furthermore, the bonding property of the network is also enhanced, which shows the decrease of the running period number and the cluster number, and the network eventually falls into a 2-period orbit. Figure 6 exhibits the largest Lyapunov exponent (LLE) of the system (5) according to parameters $\mu ,\epsilon$ and the corresponding contour diagram. Red regions in contour diagram indicate the system is in a chaotic state, and other regions indicate the system is in a stable state. It can be concluded from these two figures that the system has a very wide periodic domain and chaotic domain.

Next, the case of at least one of $\theta ,\tau$ being nonzero is discussed. The introduction of time delay greatly changes the dynamic properties of the network. The temporal and spatial characteristics of the delayed network is not only determined by bifurcation parameter $\mu$ and coherent parameter $\epsilon$, but also affected by delay parameters. The network will evolve into different clusters with different parameter values, and the cluster number in each evolution and the periodic orbits corresponding to the clusters are different. For example, when $\theta =\tau =1$, if $\mu =0.8,\epsilon =0.2$ are set, the network will run in the 4-period 2-cluster orbits, and if $\mu =0.8,\epsilon =0.1$, the network will operate in the 4-period 4-cluster orbits. Through the previous similar analysis, it can be found that the delayed network with $\theta =\tau =1$ and the conventional network have similar properties that both have cluster freezing attractors, and the network state will evolve into chaotic orbit from the unstable cluster. Furthermore, if delay parameter $\tau$ is increased and $\theta$ is kept unchanged, it can be observed that the system is sensitive to the change of parameters. In the case of $\mu =0.7,\epsilon =0.2$, all neuronal states enter 2-period orbit when $\tau$ is even and they enter $2(\tau +1)$-period orbit when $\tau$ is odd, and the delay parameters making the network run stably also change with the variation of $\mu ,\epsilon$. For instance, if the case is changed to $\mu =0.8,\epsilon =0.2$, the network will still converge to the corresponding period orbit when $\tau \le 14$. Further experiments suggest that the delay parameters can only affect the network’s operation cycle not cluster. When $\tau =9$, the network runs in the 20-cycle 2-cluster orbit, and it runs in 2-cluster orbit with 24 period when $\tau =11$. For too large delay parameters, the network’s running states become uncertain and change randomly with the initial input. Another example, when $\theta =0,\tau \ne 0$, the network operation is different from that of the delay coupling mode of the second type. If $\mu =0.7,\epsilon =0.2$, the network will run in different 2-period orbits with different initial values whether the delay parameter is odd or even, and it will keep this property in a large parameter domain such as $\mu =0.7,\epsilon \in (0.1,0.5)$ and $\mu =0.2,\epsilon \in (0.1,0.9)$. For excessively large delay parameters, the network operating state will also become uncertain and change randomly with the initial input.

The network with delay coupling of the third kind is first considered by setting $\theta =1$. It can be found that when $\mu =0.2,\epsilon =0.2$, no matter what $\tau$ is, the network runs in the 4-period 2-cluster orbit. If $\mu$ is kept unchanged and $\epsilon$ is increased to $0.8$, the network will enter the 2-period 2-cluster orbit. Further, keeping $\epsilon$ unchanged while changing $\mu \in (0.2,0.8)$, the network keeps its operation characteristics unchanged. For too large $\mu$, it jumps out of the periodic state. Therefore, it indicates that $\tau$ does not affect the network operation period and cluster, and the change of coherent parameter $\epsilon$ only affects the orbit period not cluster. Next $\theta =2$ is fixed, the network’s property is similar to that of the analysis above, except that it runs in a 2-period 2-cluster orbit. Further experiments suggest that when $\theta$ is an odd number, no matter what $\tau$ is, the network will run in $2(\tau +1)$-period orbit; when $\theta$ is an even number, no matter what $\tau$ is, it will be in a 2-period orbit. It is the same as the delay coupling mode of the first type, which indicates that in the process of network operation, delay parameter $\theta$ plays a major role, while $\tau$ plays a small role.

2.2. Analysis of Neurons

The dynamic behavior of a single neuron in the coupling network at time $t+1$ is determined by two factors, one is the iteration of the neuron’s own signal at time t, the other is the feedback signal of other neurons to it at time t. These two parts work together to form its complex dynamic behavior. The role of feedback items in neuron movement will be discussed in the following.

Without loss of generality, the dynamical mechanism on neurons of PL-GCM with conventional couplings is investigated, of which the ith neuron is expressed as:

$$x_i(t+1)=(1-\epsilon )f\left(x_i\left(t\right)\right)+\frac{\epsilon }{N}\sum _{j=1}^{N}f\left(x_j\left(t\right)\right),$$

(6)

and it is the same as that of PL-GCM with delay couplings. Formula (6) is rewritten as:

$$x_i(t+1)=(1-\epsilon )f\left(x_i\left(t\right)\right)+\epsilon m_N\left(t\right),$$

(7)

where, for convenience,

$$m_N\left(t\right)=\frac{1}{N}\sum _{j=1}^{N}f\left(x_j\left(t\right)\right).$$

(8)

As can be seen from Section 2.1, taking the number of network neurons to 100, when $\mu =0.2$, $\epsilon =0.2$, the PL-GCM network will enter a 2-period orbit for any initial value, and all states are divided into a 2-period cluster; that is, the whole network runs in two 2-period orbits. In fact, according to Formula (8) and experimental analysis, each neuron is affected by the gain $m_N\left(t\right)$ during each iteration, the number of positive and negative initial values of the system (5) in $[-1,1]$, and the magnitude of their absolute values determine the neuron’s motion direction so that $m_N\left(t\right)$ may be a constant positive or constant negative. However, due to the existence of a reduction coefficient (coherent parameter), the order of the second term magnitude is one order of magnitude smaller than that of the first term in Equation (5), thus the second term will not affect the sign of each iteration. Further analysis suggests that due to the intrinsic effect of the network structure, $m_N\left(t\right)$ makes each neuron move towards the network’s attractor direction. Once the neuron reaches the network attractor, $m_N\left(t\right)$ is no longer changed, which indicates the function of constant gain is to keep the neuron stable at the attractor. The final running orbit of $m_N\left(t\right)$ is affected by the initial value and parameters. For example, when $\mu =0.4$, $\epsilon =0.2$, the network runs in the 2-period 2-cluster orbit, and $m_N\left(t\right)$ runs in a 2-period orbit $(0.0334,-0.0334)$ for any initial value. If $\mu =0.8,\epsilon =0.2$, $m_N\left(t\right)$ runs in a 2-period orbit $(0.0096,0.0639)$ or $(-0.0096,-0.0639)$ according to the initial value. It is worth noting that when $\epsilon =0.2,\mu \in (0.1,0.5)$, in the process of initial value random selection, there is one situation that will make the network’s coupling structure “invalid”. When the initial values are randomly selected as 50 positive numbers and 50 negative numbers, after the coupling iteration in the initial stage, $m_N\left(t\right)$ finally stabilizes at 0; that is, 50 neurons run on one track, while the other 50 neurons run on another track with the opposite sign, so that $m_N\left(t\right)$ is always 0 after each iteration. The situation is not affected by the specific value of the initial state, and the absolute value of the initial value only insensitively affects the network’s final stable orbit. When $\epsilon =0.2,\mu =0.4$, the neuron will enter the orbit with opposite sign after the network is stabilized, and the initial value completely determines which orbit the neuron finally runs on. At this time, even though $m_N\left(t\right)$ is not stabilized to zero, it is in a symmetric orbit with the odd step at $-0.7314$ and the even step at $0.7314$, and the neuron runs on two 2-period orbits $(0.6339,0.7569)$ and $(-0.6339,-0.7569)$. The symmetry of the orbit makes the coupling ineffective, each neuron runs in the same (opposite) orbit and does not interfere with each other.

If the positive and negative ratio of the initial value is 1:1, when $\mu =0.6,\epsilon =0.2$, the network enters the 2-period 2-cluster orbit. $m_N\left(t\right)$ will jump back and forth between two fixed points $0.0132$ and $-0.0132$ according to different initial values. When the bifurcation parameter $\mu$ increases further, the network’s chaotic characteristics enhance, while the coupling characteristics weaken, and $m_N\left(t\right)$ transforms from periodic to random gradually. The law remains unchanged now, and it will run on a symmetric orbit in most cases and the neuron’s operating period will gradually increase until the network enters chaos. After the network enters the chaotic state, it is natural to think that $m_N\left(t\right)$ as a mean value term is also running in chaotic orbit since every neuron is running in chaotic orbit. However, it is found through experiments that when the network dimension is sufficiently large, the value of $m_N\left(t\right)$ represents a certain statistical regularity. Figure 7 shows the density histogram of $m_N\left(t\right)$ in the condition of network dimension ${10}^2$ and ${10}^3$, when $\mu =1.8,\epsilon =0.2$. It can be seen that the density is approximately a Gaussian distribution; that is, the distribution of $m_N\left(t\right)$ approximately obeys the central limit theorem. It can be analyzed that although each neuron runs in chaotic orbit, the proportion of each neuron’s value in the sum is sufficiently small due to the large number of neurons. Moreover, the chaotic characteristics are further weakened after being averaged by the number of neurons, thus, the variation of $m_N\left(t\right)$ shows statistical regularity.

3. Chaos Control of PL-GCM

The spatiotemporal chaos control is indispensable to the chaotic system and different chaos control methods can be applied to realize different goals. For the system composed of coupled chaotic neurons, chaos can be eliminated by controlling it to the periodic orbit, and the corresponding control methods involved in feedback control, phase space compression control, perturbation control, and so on. Among them, the control method with control signals no external information contained and the unchanged parameter domain is usually adopted, such as the state feedback control method [25]. Besides, the control method with the external information injected in the process, such as modulated parameter control put forward by Ishii [26] to implement successful control of S-GCM and CL-GCM, can be used in pattern recognition and associative memory. In the following, feedback control and modulated parameter control of PL-GCM is discussed in detail.

3.1. Feedback Control of PL-GCM

Adding feedback control signals to each neuron for iteration, the ith neuron state equation of PL-GCM is expressed as:

$$x_i(t+1)=f\left(x_i(t-\theta )\right)+\frac{\epsilon }{N}\sum _{j=1}^{N}f\left(x_j(t-\tau )\right)-\rho C,$$

(9)

$$C=f\left(x_i(t-\alpha )\right)-x_i(t-\alpha ),$$

(10)

where C is the feedback item, which has different forms, $\theta ,\tau ,\alpha$ are delay parameters defined in (5), $\rho$ is the control parameter. The PL-GCM network can be controlled by using the control method borrowed from that of a low-dimensional chaotic systems to achieve designated cycle orbit and cluster with specific cluster number by adjusting parameter $\rho$.

The feedback control of the conventional coupled PL-GCM with $\theta =\tau =\alpha =0$ was investigated first. Assume the network condition is $N=100,\epsilon =0.2$ and the initial value generates randomly, the network is chaotic when taken $\mu =1.8$. After 100 steps of network operation, control signals were added to the system. Given $\rho =0.3$, Figure 8 describes the neurons’ state changes before and after time control. It can be seen that the system is chaotic before control, and the system motion changes from chaos to order, which represents the network’s convergence to the 2-period and 1-cluster state after control. When $\rho =0.2$, the controlled network enters the 2-period 2-cluster orbit. Further experiments suggest that the network can be controlled stably when $\rho \in (0.1,0.3)$ and the control becomes unstable if $\rho$ continues to decrease or increase. Moreover, the networks with different initial inputs will run in different periodic orbits after control. For example, when $\rho \in (0.08,0.1)$, the controlled network’s operation cycle changes randomly at $2,4,8$, while the number of clusters changes randomly at $2,3,4$.

In addition, according to the change of coherent parameter $\epsilon$, the control effect remains unchanged while the network’s stability control domain changes.

Table 1. The control parameter fields of $\rho$ corresponding to different $\epsilon$.

Coherent Parameter	Control Parameter Fields
	6 Cycle and 1 Cluster	4 Cycle and 1 Cluster	2 Cycle and 1 Cluster
$\epsilon =0.4$	$\rho =0.15$	$\rho \in (0.19,0.2)$	$\rho \in (0.2,0.4)$
$\epsilon =0.5$	$\rho =0.15$	$\rho \in (0.19,0.2)$	$\rho \in (0.2,0.41)$
$\epsilon =0.6$	$\rho =0.15$	$\rho \in (0.19,0.21)$	$\rho \in (0.21,0.41)$
$\epsilon =0.8$	$\rho =0.15$	$\rho \in (0.19,0.21)$	$\rho \in (0.21,0.41)$

lists the control parameter fields corresponding to different $\epsilon$ values. It can be further found that when $\epsilon >0.6$, the value of the control parameter remains unchanged, which indicates that the change of coherent parameter is insensitive to the influence of the control effect.

Next, feedback control is adopted in the delay network with $\theta \ne 0$ or $\tau \ne 0$. The network with the first delay coupling mode $\theta =0,\tau \ne 0$ can be controlled by 2-cycle 1-cluster orbit and 2-cycle 2-cluster orbit under ordinary feedback control when $\epsilon =0.2,\tau =1$, and the change of $\epsilon$ only affects the corresponding control parameter domain not the control effect. Under the same parameters, $\tau$ does not affect the control effect either; that is, the stable running track of the controlled network is completely determined by $\epsilon ,\mu$. However, the network can not be controlled by the delay feedback control, and it cannot enter the periodic orbit no matter what the feedback delay $\alpha$ is.

Now, the feedback control of the network with the second delay coupling mode $\theta =\tau \ne 0$ is considered. The delay feedback control is adopted here because the network with this kind of coupling can not be controlled by conventional feedback control. When $\theta =\tau =1$, the network can be controlled by different 4-period 1-cluster orbits according to different initial values under $\epsilon =0.2,\rho \in (0.25,0.34)$. The system’s dynamic behavior becomes more complex with the increase of the delay parameters, the network can be controlled stably only when the coherent parameter increases. For example, when $\epsilon =0.5$, the network can be controlled by the corresponding $2(\tau +1)$ orbit according to different delay parameters, and the cluster number remains unchanged while the cluster point changes randomly under the influence of the initial value. Figure 9 shows the 8-period orbits of the network with $\theta =\tau =3$, and the corresponding three clusters are represented by “∗”, “+” and “∆”, respectively. It is worth noting that the value of the delay parameter in the delay feedback control should be consistent with that in the system, otherwise the control system cannot be stabilized, and the parameter domain of the network stability control decreases with the increase of the delay.

As for the feedback control of the network with the third delay coupling mode $\theta \ne \tau$ and $\theta \ne 0,\tau \ne 0$, the system can be controlled stably by delay feedback control and run in $2(\tau +1)$ periodic orbits by choosing the value of parameters appropriately. For example, when $\theta =3,\alpha =3,\epsilon =0.5$, the controlled network runs stably under $\tau \in (4,15)$; when $\theta =4,\alpha =4,\epsilon =0.5$, the network can be stably controlled under $\tau \in (5,15)$. Furthermore, when $\tau >15$, the feedback control fails.

3.2. Modulated Parameter Control of PL-GCM

When considering network applications such as pattern recognition or associative memory, external input signals must be required. Besides, the difference between the GCM network and traditional neural network is that the network’s connection weight is constant $\epsilon /N$ and the information cannot be stored on the connection weight, so the external input signal is necessary. Ishii proposed a parameter threshold control method that can bring external information into the network [26], and the method was improved and applied to the control of the CL-GCM network to achieve good results [27]. Motivated by the above points, we will discuss the dynamical characteristics of the PL-GCM network under this control method.

The bifurcation parameter of the ith neuron is controlled as follows:

$${\mu }_i=\left\{\begin{array}{cc}{\mu }_{min},& {\mu }_i<{\mu }_{min},\\ {\mu }_i+\left({\mu }_i-{\mu }_{min}\right)tanh\left(\beta E_i\right),& {\mu }_{min}<{\mu }_i<{\mu }_{max},\\ {\mu }_{max},& {\mu }_i>{\mu }_{max},\end{array}\right.$$

(11)

$$E_i=-x_i\sum _{j=1}^N{\sigma }_{ij}{x}_j,i=1,2,\cdots ,N,$$

(12)

$${\sigma }_{ij}=\frac{1}{N}\sum _{k=1}^N{\omega }_i^k{\omega }_j^k,i,j=1,2,\cdots ,N,$$

(13)

where ${\sigma }_{ij}$ is the crossing element of the ith row and the jth column of the matrix generated by the tensor product, ${\omega }^k=({\omega }_1^k,\cdots ,{\omega }_N^k)\in {\{-1,1\}}^N$ is a storage memory mode for $k=1,2,\cdots ,N$, $x_i$ is the state of the ith neuron, $E={\sum }_{i=1}^N{E}_i$ is the conventional energy function and $E_i$ is the ith partial energy. $\beta$ is a constant weighted parameter whose role is to enlarge or shrink the energy function and it indirectly influences the iteration speed of $\mu$. In addition, the value of ${\mu }_{max}$ depends on $f\left(x\right)$ and ${\mu }_{min}$ is a modulated parameter.

The network condition is given as $N=100,\epsilon =0.1$. Selecting the control parameter ${\mu }_{min}=0.5,{\mu }_{max}=2,\beta =0.5$ and the initial values of ${\mu }_i$ to be $1.8$ for all i, the network is chaotic according to the analysis in the previous section. For the application of the controlled network to information processing and its fast convergence, the following input coding is adopted as:

$$I\left(x\right)=\left\{\begin{array}{cc}{x}^{+},\hfill & x>0,\hfill \\ x^{-},\hfill & x<0,\hfill \end{array}\right.$$

(14)

where $x^{+},x^{-}$ are points in two 2-period orbits of the PL-GCM network. Under the control strategy (11), the chaotic behavior of the conventional coupled network is well controlled. Figure 10 shows that the system is controlled by the 2-period 2-cluster and 2-period 3-cluster orbits, respectively, corresponding to ${\mu }_{min}=0.2$ and ${\mu }_{min}=0.65$. The results of the experiment suggest that when ${\mu }_{min}\in (0.01,0.5)$, the system can be controlled by the 2-period 2-cluster orbit and it can be controlled by a 2-period 3-cluster orbit when ${\mu }_{min}\in (0.65,0.69)$. If the value of ${\mu }_{min}$ continues to increase, the control will become unstable. For example, the controlled network will be randomly stabilized at a 2-period 3-cluster orbit and 2-period 4-cluster orbit when ${\mu }_{min}=0.7$. The control will fail as ${\mu }_{min}$ continues to increase.

For the time-delayed coupled network with $\theta =0,\tau \ne 0$, given $\tau =1$, the network can be stably controlled in a 2-period 2-cluster orbit if ${\mu }_{min}\in (0.01,0.55)$; it can be controlled by a 2-period 3-cluster orbit when ${\mu }_{min}\in (0.56,0.69)$. The variance of $\tau$ does not affect the control effect of the network, and the corresponding delay network can all be controlled by the 2-period orbit under the appropriate parameter domain. In addition, the network’s final stable cluster number will change as the coherent parameter $\epsilon$ decreases. For example, when $\epsilon =0.05$, the network will be controlled in the 2-period 4-cluster orbit. However, the control becomes unstable and eventually fails as $\epsilon$ increases.

For the time-delayed coupled network with $\theta =\tau \ne 0$, the system will be controlled in the $2(\tau +1)$ orbit with different delays when ${\mu }_{min}\in (0.01,0.6)$, which is the same as the conclusion of the network’s feedback control. For the time-delayed coupled network with $\theta \ne \tau \ne 0$, when $\theta$ is fixed, the network will be controlled in the $2(\theta +1)$ periodic orbit, and the value of $\tau$ does not affect the operation period of the network. In conclusion, it can be seen that the first term in the neuron equation, namely the iterative term of PL function, plays a major role in the operation of neurons, which is consistent with the previous conclusion in Section 2.1.

4. Dynamic Associative Memory of PL-GCM

Because chaos has sensitive dependence on the initial value and orbit ergodicity, the correct pattern can be found through a chaos search, and the network can carry out normal associative memory through chaos control. In this section, the dynamic associative memory features of the PL-GCM network is investigated by using the control method presented in Section 3.2, and the sample patterns below are selected as storage patterns.

Here, each storage mode is represented by the $10\times 10$ dot matrix so that the network is composed of 100 neurons. The neuron’s output function is selected as Formula (4). It is represented by “.” or blank, respectively, if the output of the neuron is 1 or $-1$. According to the characteristics of the network, Formula (14) can be used for input encoding instead of “$-1$” or “1”, and the Hamming distance is used to measure whether the output is correct. For the PL-GCM network consisting of 100 neurons, when the output mode and input mode are the same, the Hamming distance takes the minimum value 0; when the two modes are completely opposite, the Hamming distance takes the maximum value 100.

Now, based on the above descriptions, we set the network parameter as $N=100,\epsilon =0.1$, sample mode as in Figure 11, the input function and output function as in Formulas (14) and (4), respectively, and the Hamming distance to describe the difference between the sample mode and the input mode, then the complete information processing can be shown as:

$$P1\stackrel{I\left(x\right)}{⟶}x\left(0\right)\stackrel{\mathrm{control}}{⟶}x\left(t\right)\stackrel{O\left(x\right)}{⟶}P2$$

where $P1,P2$ are initial input and stable output, respectively.

The dynamic associative memory of the PL-GCM network with conventional coupling is realized by controlling the bifurcation parameter $\mu$. We set the control parameters as ${\mu }_{max}=2$, $\beta =0.5$. Figure 12 shows the output response of the conventional coupled network with a disturbance mode when ${\mu }_{min}=0.5$. On the left is the input mode, and on the right is the stable output mode. It can be seen that the stable output is only the correct mode “J” and its opposite mode, and does not contain other storage modes, which indicates that the dynamic associative memory is successful. The change of threshold ${\mu }_{min}$ will affect the network’s associative memory performance. When the value of ${\mu }_{min}$ increases or decreases, the control is still successful; however, the associative success rate of the network decreases. The experiment shows that when ${\mu }_{min}\in (0.45,0.55)$ the network’s associative memory is successful and the associative correct rate is more than $99\%$.

Similarly, under threshold control (11), the time-delay coupled PL-GCM network also has dynamic associative memory capabilities. For PL-GCM with a delay coupling mode of the first type, its motion characteristics under control are the same as those of the conventional coupling network, and it eventually converges to the 2-period output. For PL-GCM with delay coupling mode of the second type, its output will be stable at the $2(\tau +1)$ period under control. Figure 13 exhibits the output characteristics of the system with different delays under the condition of ${\mu }_{min}=0.5$. The left side of the figure is the initial input, which manifests the fragmentary mode of four storage modes and it is a typical situation in practical application. The right end of the figure is the stable output of the system in three cycles, which is the output of four cycles with $\tau =1$ and the output of eight cycles with $\tau =2$, respectively. It can be seen that the output is consistent with the control results in the previous section; that is, the output and state of the system run in the same orbit. Besides, the network can carry out chaos search to realize correct associations, and the stable output contains only the storage mode with the minimum Hamming distance from the input, but not other storage modes, which is beneficial to practical application.

5. Conclusions

A new chaotic neural network called PL-GCM was proposed. It has richer dynamical properties on account of the introduction of the piecewise logistic map and different ways of coupling, which was exhibited by deeply analyzing the neurons’ behaviors and numerical simulations, such as state evolution, LLE and contour map to explain the running regularity clearly. It can be concluded that the system’s dynamical behaviors appear as clustering features and depend on both the neurons’ chaotic behaviors (which generate the system’s chaos) and their coupling which was denoted as $m_N\left(t\right)$. Moreover, by comparing with the existing literature, it can be found that GCM networks with different chaotic maps that have similar dynamic behaviors, except that the parameter domain varies with different chaotic maps. The analysis method can be used for all kinds of GCM modeling. In addition, the chaotic behavior of the PL-GCM model is successfully controlled by using feedback control or modulated parameter control, which shows that the model with different couplings can be controlled to any specified-periodic orbit and time delay only affects the stable running period of the system. The excellent dynamic associative memory ability to noise initial pattern was exhibited by simulation experiments on 100 pixel images, which suggests the network can be used in pattern recognition and information processing.

In future research, two aspects will be considered. On the one hand, whether the full connection structure of the GCM model can be simplified to improve the operation efficiency of the network. On the other hand, how the GCM model can play a role for more complex gray image recognition. We willll make efforts and look forward to breakthroughs.