Memory Storage Systems Utilizing Chaotic Attractor-Merging Bifurcation

In nonlinear dynamical systems with barriers/thresholds, the signal response against a weak external input signal is enhanced by an appropriate additive noise (stochastic resonance). In recent years, progress in the application of stochastic resonance shows that the existence of additive noise heightens the memory storage functions in memory elements using bistable oscillations even with extremely low power consumption. By not restricting the additive noise, the deterministic chaos (an internal fluctuation) induces a similar phenomenon known as chaotic resonance. Chaotic resonance appears in nonlinear dynamical systems and is accompanied by chaos–chaos intermittency, where the chaotic orbit intermittently transitions among separated attractor regions through attractor-merging bifurcation. Previously, a higher chaotic resonance sensitivity than that of stochastic resonance was reported in various types of systems. In this study, we hypothesize that chaotic-resonance-based memory devices can store information with lower power consumption than that of stochastic-resonance-based devices. To prove this hypothesis, we induced attractor-merging bifurcation in a cubic map system, which is the simplest model for the emergence of chaotic resonance. Thereafter, we adjusted the internal system parameter under a noise-free system as the chaotic resonance and applied stochastic noise similar to the condition for inducing stochastic resonance. The results of this study reveal that, even with weaker memory storage input signals, the former exhibits a higher memory storage capability than the latter. The approach using chaotic resonance could facilitate the development of memory devices that were hitherto restricted to the application of stochastic resonance.


I. INTRODUCTION
Considering recent developments in artificial intelligence (AI), neuromorphic computing, and big data analysis, the amount of data being stored is increasing exponentially [1]. To manage such large amounts of data, efforts are being made to develop data storage devices with high density and high-speed data transmission capability [2]. Such devices are typified by high-density/high-speed nonvolatile The associate editor coordinating the review of this manuscript and approving it for publication was Ye Zhou . memory and utilize phase-change random access memory (PCRAM) and a combination of PCRAM and neuromorphic computing [3]- [6]. However, they must also have low power consumption to achieve a low-carbon society [7]- [9]. Therefore, memory storage devices with low power consumption are needed.
To develop such a device, the stochastic resonance mechanism [10] (reviewed in [11]- [13]), in which synchronization in nonlinear systems possessing a barrier/threshold under a weak external input signal is strengthened using an additive stochastic noise with suitable strength, must be VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ utilized [14], [15]. In particular, Ibáñez et al. and Stotland and Di Ventra showed that the existence of additive noise heightens memory functions in memory elements with bistable oscillations, even under scenarios with extremely low power consumption [14], [15]. Consequently, studies have actively investigated stochastic resonance for realizing stochastic memory devices with low power consumption [16]. In memory devices, memory elements possesses bistable states corresponding to ''0'' and ''1.'' The elements are initialized to ''0.'' Then, they are changed to ''1'' during the storage phase. In memory devices that use the stochastic mechanism resonance, the amplitude of the input signal required for the state transition can be reduced by the effect of noise. Along with additive stochastic noise, the deterministic chaos of an internal system's fluctuations induces a phenomenon called chaotic resonance that is similar to stochastic resonance [17]- [25] (reviewed in [12], [13], [26]). In chaotic resonance, chaos-chaos intermittency (where the chaotic orbit hops intermittently among several attractors distributed in spatially different regions) synchronizes with a weak external input signal. The profile of synchronization degree follows a unimodal maximum peak around the attractor-merging bifurcation, which is the function that controls the internal system parameter for attractor-merging (reviewed in [12], [26]). The sensitivity against the weak input signal and synchronization degree in the chaotic resonance are superior to those of stochastic resonance [23], [27]. Although numerous studies have been conducted on chaotic resonance, investigations were limited to the evaluation of the response of steady periodic input signals [22], [23], [28]- [32]. Consequently, the transient response in chaotic resonance, such as the transition from ''0'' to ''1'' in memory devices, remains unclear.
In this study, we hypothesized that the state transition caused by chaotic resonance can realize a novel memory storage mechanism with lower power consumption than that of a mechanism based on the stochastic resonance effect. To prove this hypothesis, in our preliminary study [33], we confirmed that memory storage can be realized using the attractor-merging bifurcation in a cubic map system, which is the simplest model for the emergence of chaotic resonance [12], by adjusting the internal system parameter under a noise-free condition. However, the strength of the perturbation for transition in chaotic resonance, which relates to power consumption, has not been compared to that in the conventional approach that utilizes stochastic resonance [14], [15]. Additionally, the robustness of chaotic resonance to background noise and parameter setting error is not clearly understood. In this study, based on our preliminary study [33], we evaluated and compared the abilities of the chaotic and stochastic resonance approaches for memory storage.

A. CUBIC MAP WITH INPUT SIGNALS
A discrete cubic map is a well-known and simple chaotic system with chaos-chaos intermittency, and it is used for evaluating chaotic resonance (reviewed in [12]). In this study, we used an assembly composed of N cubic maps with memory signals c i (t) (i = 1, 2, · · · , N ), a parameter setting error of a, and a Gaussian white noise of D a ξ (t). This is given by Here, x i (t) denotes the time series of the variable in the i-th cubic map, D a represents the noise strength (mean and standard deviation of ξ (t) are zero and 1.0, respectively; that is, ξ (t) is produced by a normalized Gaussian process), and a follows D c ζ , where D c and ζ are respectively the error strength and random values following a normalized Gaussian distribution. Owing to the increase in a, the attractor-merging bifurcation arises. Subsequently, a chaos-chaos intermittency appears [31]. The positive value of c i (t) above a certain strength causes the transition of x i (t) to the positive attractor region x i (t) > 0. In this study, b was set to 10.0 [31], and the size of the assembly was N = 512. The attractor-merging bifurcation is induced by adjusting the internal system parameter a in the chaotic resonance approach. When it is controlled by the additive stochastic noise as in the stochastic resonance approach, a is fixed.
In the memory-storage process for the assembly of cubic maps, the memory signals c i (t) are input in each cubic map. Before the memory-storage phase, 0 < t < t s : t s and t e represent the start and end times of the memory storage, respectively, and c i (t) is set to 0. Initially, x i (0) is set to a negative value. During memory storage (t s ≤ t ≤ t e ), the memory signals become c i (t) = Aη i (A > 0), where η i represents a binary memory pattern of ''0'' or ''1'' generated using the Poisson process. In the Poisson process, the occurrence probability of ''1'' corresponds to . After the memory-storage process (t > t e ), c i (t) becomes 0. An example of c i (t) is shown in Fig.1. The memory-storage patterns {X 1 , X 2 , · · · , X i · · · , X N } are produced through binarization (i.e., X i = 1 in x i (t e ) > 0 and X i = 0 in x i (t e ) ≤ 0). As the parameter set for this study, we used a start time of t s = 100 and end time of t e = 200 in the memory storage as well as = 0.1 for the Poisson process.

B. EVALUATION INDICES
To evaluate the dynamical system behaviors, we utilized the bifurcation diagram of x 1 . In this study, we assumed that the elements of the cubic map assembly exhibit similar temporal behaviors; therefore, we utilized the 1-st element as a typical example. Moreover, to detect the attractor-merging bifurcation and determine whether the attractor is separated or merged in the absence of external noise (D a = 0) and memory signals (A = 0), we used the values of F(f max,min ) − x d .
Here, x d exhibits the separation point for the attractors in the cubic map x d = 0. f max,min corresponds to the local maximum and minimum values around x = x d for the map function, respectively. F(f max,min ) − x d = 0 corresponds to the attractor-merging bifurcation [31]. Under the condition in the merged attractor, the orbit hops between the x i > x d and x i < x d regions. This indicates that chaos-chaos intermittency appears. The To determine the chaotic state of x i , the Lyapunov exponent was calculated as follows [34]: is a perturbation applied to the orbit. Further, λ > 0 and λ < 0 correspond to the chaotic and periodic states, respectively. In this study, the value of λ among N cubic maps is averaged.
To quantify the capability for memory storage, the bit error rate (BER) between the memory pattern η i and the stored pattern in the assembly of cubic maps X i (i = 1, 2, . . . , N ) was utilized.

A. DEPENDENCY OF SYSTEM BEHAVIOR ON INTERNAL PARAMETER OF CUBIC MAP
First, we demonstrate the dependency of the system behavior on the internal parameter of cubic map a. A bifurcation

FIGURE 3.
Dependency of the capability to store bit-series on the internal parameter a when the input signal strength is A = 0.01. The BER between the input bit-series {η 1 , η 2 , · · · , η N } and stored bit-series diagram using positive and negative initial values of x 1 (0), Lyapunov exponent λ, and the attractor-merging conditions F(f max,min )−x d is presented in Fig. 2. In a 2.84, the condition for attractor-separation (F(f max )−x d > 0, F(f min )−x d < 0) is satisfied. Consequently, the chaotic attractor (λ > 0) separates either the positive or negative x(n) region depending on the sign of the initial value x(0). While this separated attractor merges in a 2.84 through the attractor-merging bifurcation (F(f max,min )−x d = 0), chaos-chaos intermittency     Fig. 3). By increasing the noise strength D a , the lower limit of the input signal strength A, where BER ≈ 2.0 × 10 −3 is achieved, increases (b = 10, a = 0).

B. CAPABILITY FOR MEMORY STORAGE BY ADJUSTING INTERNAL CUBIC MAP PARAMETER 1) CAPABILITY UNDER NOISE-FREE AND NO-PARAMETER-SETTING ERROR
We demonstrate the capability of storing the bitseries {η 1 , η 2 , · · · , η N } for controlling the attractor-merging  Fig. 3). When the strength of the parameter-setting error D c is increased, the lower limit of the input signal strength A, where BER ≈ 2.0 × 10 −3 is achieved, increases (b = 10, D a = 0). bifurcation by adjusting the internal parameter a under a noise-free condition D a = 0 and a non-parameter-setting error D c = 0. In the upper panel of Fig. 3, the BER between the input bit-series {η 1 , η 2 , · · · , η N } and stored bit-series {X 1 , X 2 , · · · , X N } as a function of the parameter a with an input signal strength of A = 0.01 is shown. The lower panel in Fig. 3 shows the attractor-merging conditions F(f max,min )−x d corresponding to the parameter value a in the upper panel of Fig. 3. The result is shown around the attractor-merging bifurcation F(f max,min ) − x d = 0 at a ≈ 2.839, and the BER minimizes to ≈ 4 × 10 −3 . This minimization can be construed as a chaotic resonance, where there is a response of the chaos-chaos intermittency against the weak input signal for storing the bit-series. In Fig. 4, a typical example of the time-series of x i and c i at the internal parameter a = 2.839, where the BER is minimum, is represented (the corresponding BER is shown in the upper panel of Fig.3). In the 1st and 3rd cubic maps, x i (t) follows the rising behavior c i (t) = 0.01 when t > 100. The other cubic maps remain in the negative x i (t) region set as the initial state of x i (0) according to c i (t) = 0.
To investigate the sensitivity of adjusting the internal parameter a, the dependence of the BER on the input signal strength of A and the internal parameter a is illustrated in Fig. 5. At approximately a = 2.839, the attractor-merging bifurcation, as shown in Fig.2, in A 4.0 × 10 −3 , BER 2.0 × 10 −3 is achieved.

2) CAPABILITY UNDER NOISE AND PARAMETER-SETTING ERROR
We investigated the robustness against the additive stochastic noise for high memory storage capability around the attractormerging bifurcation. Figure 6 shows the dependency of BER on the input signal strength of A and the strength of the additive stochastic noise of D a at a = 2.839, where BER minimizes under a noise-free condition (corresponding to Fig. 3). Increasing the noise strength D a , the lower limit of the input signal strength A where BER ≈ 2.0 × 10 −3 is achieved increases. Subsequently, the A − D a region with BER ≈ 2.0 × 10 −3 disappears in D a 2.0 × 10 −4 .
In addition to the influence of additive stochastic noise, the robustness against the parameter setting error for a was evaluated. Figure 7 shows the dependency of BER on the input signal strength of A and the strength of the parameter-setting error D c at a = 2.839, where BER minimizes under the noise-free (D a = 0) and non-parametersetting error ( a = 0) condition (shown in Fig. 3). When the strength of the parameter-setting error D c is increased, the lower limit of the input signal strength A, where BER ≈ 2.0 × 10 −3 is achieved, increases. Eventually,  Fig. 8. In contrast to the case where the attractor-merging bifurcation is induced by adjusting the internal system parameter a as shown in Fig.4, the noise-induced chaos-chaos intermittency of x i (t ) does not dominantly follow the rising behavior and remains for c i (t ) when t > 100. In this trial for memory-storage patterns, η 1,3 = 1, η 2,4,5,6 = 0 (D = 2.8, b = 10, A = 0.03, a = 0).

C. CAPABILITY TO STORE BIT-SERIES BY ADDITIVE STOCHASTIC NOISE
After evaluating the control of the attractor-merging bifurcation by adjusting the internal parameter a, we investigated the memory storage capability when the attractor-merging bifurcation was induced by an additive stochastic noise (D a > 0). This is similar to the case of the conventional approach using the stochastic resonance mechanism [14], [15]. Considering the evaluation of the dependency of the capability to store bit-series on the noise strength D a , the upper part of Fig. 8 shows the BER between the input bit-series {η 1 , η 2 , · · · , η N } and the stored bit-series {X 1 , X 2 , · · · , X N } with a signal strength of A = 0.01 (corresponding to the input strength A, as shown in Fig.3). 0.02 and 0.03 are the functions of the noise strength D a . The lower part of Fig. 8 demonstrates the bifurcation diagram of x 1 as a function of the noise strength D a under a signal-free condition (A = 0). In this evaluation, the parameter-setting error was absent ( a = 0). For A = 0.01 and 0.02, the BERs monotonously degenerate a noise of increasing strength. When A = 0.03, the BER minimizes at D a ≈ 7.0 × 10 −3 around the noise-induced attractor-merging bifurcation. Nevertheless, this BER becomes higher (BER≈ 0.02) in comparison with the case where the attractor-merging bifurcation is adjusted by the internal parameter a, as shown in Fig.3 (BER≈ 4.0 × 10 −3 ). Figure 9 shows a typical example of the time-series of x i and c i at the noise strength of D a = 7.0 × 10 −3 , where the BER is minimum, as shown in Fig.8 (A =  0.03). In contrast to the case where the attractor-merging bifurcation is induced by adjusting the internal system parameter a, as shown in Fig.4, the noise-induced chaos-chaos intermittency of x i (t) does not dominantly follow the rising behavior c i (t) = 0.03 (i = 1, 3) and remains at c i (t) = 0 (i = 2, 4, 5, 6) when t > 100.
To further study the sensitivity of this memory storage to additive stochastic noise, the dependence of the BER on the input signal strength of A and the strength of the additive stochastic noise D a is illustrated in Fig.10(a). At approximately D a = 7.0 × 10 −3 , where the attractor-merging bifurcation occurs (as shown in Fig.8), in A 0.03, BER 2.0×10 −2 is achieved. According to the values of BER shown in Fig.10(a), the minimum BER between 10 −4 ≤ D a ≤ 10 −2 at each input signal strength of A are represented in Fig.10(b). In comparison with the case where an adjustment is made to the internal parameter a under a noise-free condition where BER ≈ 4.0 × 10 −3 in A = 10 −3 , to achieve the same amount of BER, a higher A is needed (A 0.4).

IV. DISCUSSION AND CONCLUSION
This study proposed a memory storage mechanism using the high sensitivity of the signal response around the attractor-merging bifurcation in the assembly of cubic maps with chaos-chaos intermittency and evaluated its capability. To control the attractor-merging bifurcation, two approaches were utilized: one was for adjusting the internal cubic map parameter and the other, for applying stochastic noise similar to the conventional approach using the mechanism of stochastic resonance. After comparing the capabilities of both approaches, we confirmed that the former exhibits a high memory storage capability even under weak memory storage input signals. Moreover, this high memory storage ability was maintained under certain levels of existence of stochastic noise and parameter-setting error.
First, we examined the reason why the memory storage capability exhibits a peak around the attractor-merging bifurcation. Under the condition of the attractor-merging bifurcation, the state transition frequency in the autonomous chaos-chaos intermittency is extremely low [31]. The application of an external input signal to the system having low autonomous chaos-chaos intermittencies plays a role in the perturbation for inducing exogenous chaos-chaos intermittencies even under a weak applied signal. Consequently, the state transition that corresponds to memory storage is achieved through the synchronization of chaos-chaos intermittency. This tendency of high synchronization of the chaos-chaos intermittency is congruent with previous findings in other systems with attractor-merging bifurcation [12], [22], [23]. Therefore, dynamical systems with locally cubic map structures can be utilized to achieve the memory storage systems (reviewed in [12], [26]).
Second, we discussed the memory storage capability when the attractor-merging bifurcation is induced by adjusting the internal parameter a (see Fig. 3) and examined why it is higher than that of the case where the attractor-merging bifurcation is controlled by stochastic noise (see Fig. 8). The additive stochastic noise induces attractor-merging [12]. Through the abovementioned perturbation, the chaos-chaos intermittency, which is unrelated to the input signal for memory storage, also emerges owing to additive noise. Subsequently, the input signal strength A must be set sufficiently higher than the noise strength of D a . Therefore, the memory storage capability in the case with noise is inferior to the case with internal system parameters. This tendency was also reported in studies that compared the chaotic and stochastic resonances [23], [27]. Utilizing this high memory storage capability in the case with control through the internal parameter a might enable the realization of memory devices with low power consumption.
Third, in contrast, because the approach utilizes stochastic noise, which is similar to stochastic resonance, background noise can be employed. Consequently, the use of background noise leads to low power consumption [14], [15]. However, recent studies on logical circuits used for memory devices driven by the effects of stochastic resonance have indicated that the range of the noise strength required to induce stochastic resonance is narrow [35]. Consequently, because the noise strength required for the induction of stochastic resonance is weak, an approach that applies compensated nonstochastic perturbation was proposed to achieve an adequate level of perturbation for stochastic resonance [35], [36]. The generation of an external perturbation increases power consumption. Therefore, when the noise strength is weak, an alternative solution, which uses an approach for the chaotic resonance controlled by an internal system parameter, should be utilized to overcome the issue related to stochastic resonance, rather than additionally applying the perturbation; as a result, this alternative solution achieves low power consumption.
The limitations of this study are discussed as follows. We investigated the capability of memory storage with chaotic resonance. However, to reveal the relationship between power consumption and storage capability in actual application environments, the electronic circuit of this mechanism should be implemented. Our previous study demonstrated the electronic circuit implementations for chaotic resonances in Chua's circuit [37]; the electronic circuit systems used in the conventional approach that utilizes stochastic resonance [14], [15] might be utilized for the approach for chaotic resonance under the parameter set for chaos-chaos intermittency. The outcome obtained in the aforementioned study can be used in the application of the electronic circuit for storing memory with chaos-chaos intermittency. In the actual circuit implementations using an approach that controls the attractor-merging bifurcation by adjusting the internal system parameter, the memory storage capability is influenced by the stochastic noise and memory-setting error (see Figs. 6 and 7). This influence has also been noted in various types of chaotic systems where chaotic resonance appears [24]. Therefore, the enhancement of its robustness is important for circuit implementation. Moreover, the chaos-chaos intermittency induced by attractor-merging bifurcation is widely observed in various systems, especially neural systems [22]- [24], [32]. Therefore, this approach for chaotic resonance might be applied to neuromorphic computing. Additionally, we proposed the ''reduced region of orbit'' (RRO) feedback method as an alternative approach for the emergence of attractor-merging bifurcation. The chaotic resonance induced by this method has demonstrated high sensitivity against a weak external input signal [22], [23], [31], [38], [39]. Therefore, the capability of the RRO feedback signal must be investigated for memory storage. In future studies, to realize a memory device based on chaotic resonance with low power consumption, the aforementioned aspects must be evaluated.
In summary, this study demonstrated that the chaotic resonance induced by adjusting internal system parameters for the emergence of attractor-merging can detect memory storage signals, even the strength is weak, until the noise-and parameter-setting errors reach a certain level. Moreover, this detection capability is superior to that in the case of the approach induced by additive stochastic noise, which aids the realization of memory devices with extremely low power consumption. Although several limitations remain, our proposed method, which utilizes chaotic resonance, could facilitate the development of memory devices hitherto restricted to the application of stochastic resonance.