# In Search of Chaos and Complexity of a Cognitive Language-Learning System

Authors: Gaofei Luo; Sayan Mukherjee
Journal: Complexity
 (2020)

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## Abstract

In this article, we investigate the long-term dynamics of a known cognitive-based language-learning system under the variation of a system parameter. Stability of the equilibrium points is studied. Period root to chaos is investigated by bifurcation analysis. A Lyapunov analysis is performed to verify the complex dynamics in the system. Existence of chaos is confirmed by 0-1 test. A noise-induced cognitive phenomenon is proposed under the effect of power noise. Chaotic and nonchaotic dynamics are explored in the noise-induced system. Furthermore, disorder as well as complexity, are investigated for both the systems using the concept of weighted recurrence. The whole analysis can be effective to understand the dynamical features and nonlinear structure of the cognitive language-learning model.


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## Body

## 1. Introduction

Human language is an expressible complex communication system which consists of phonemes, words, phrases, sentences, poetry, and publications [1–4]. Grammar is itself an internal computational system that is used to represent a sentence in the language-learning process [5–7]. People store this internal grammar in the subconscious mind and spontaneously explore it to develop linguistic behaviour. So, language changes over time and hence dynamics is an obvious property of it. Small changes in linguistic data accumulation and interaction with different languages can produce unpredictably complex dynamics in the learning system [1, 8, 9]. As data accumulation and its utilization depend on the human mind, complex dynamics is an effective tool to study the behaviour of cognitive-based language-learning (CBLL) system [10–12]. In [9], it has been already established that chaotic dynamics exist for a CBLL system. However, it does not contain “forgetting” property in the cognitive growth [9]. In this paper, the existence of deterministic and noise-induced chaos [13–19] in a known CBLL system was investigated for the first time. Moreover, the complexity of the respective dynamics has been studied which is not explored indeed.Chaos can be verified by measuring the exponential divergence between the phase space trajectories [14]. Lyapunov exponent (LE) is one of the powerful tools that can measure the exponential divergence properly [20, 21]. An alternative measure 0-1 test method [22, 23] was also proposed that can successfully characterize regular as well as chaotic behaviour of the system [22, 23]. Moreover, it is applicable for both deterministic and noise-induced system [24]. The invariant characteristic of phase space can be characterized by measuring the complexity of the corresponding system [15]. It can identify the disorder in the phase space trajectories by utilizing the concept of Shannon entropy [25]. Several entropy measures have been proposed to quantify the complexity [26–37]. Recurrence-based entropy is one of the effective measures that can be applied in any dimensional system [38, 39]. However, it does not correlate with the proper dynamics of the system due to the selection of incorrect threshold [39]. To overcome this situation, a new entropy measure has been proposed based on weighted recurrence plot [40].This manuscript is organized as follows. In Section2, a cognitive language-learning system is considered and illustrated by the schematic diagram. Complex dynamics is investigated for both the CBLL and its noise-induced system in Sections 3.1 and 3.2, respectively. In section 3.1, equilibrium points and the respective stability both are determined, and multiperiodicity and chaos are investigated by bifurcation analysis, Lyapunov study, and 0-1 test method. In Section 3.2, a noise-induced CBLL phenomenon is proposed and described by a schematic representation. Furthermore, chaos in the noise-induced CBLL is also investigated numerically by the 0-1 test method in the same section. The corresponding phase space behaviour is also investigated for the CBLL and its noise-induced systems in Sections 3.1 and 3.2, respectively. Section 4 discusses the complexity of both the systems using weighted recurrence entropy analysis. Section 5 is the conclusion.


## 2. Mathematical Model of a Cognition-Based Language Learning (CBLL) Phenomenon

We consider a cognitive language growth model proposed in [9]. In this model, the growth in language is subject to the constraint of forgetting or loss of proficiency in the human brain. A schematic diagram is given in Figure 1 to describe the CBLL model.Figure 1
Schematic diagram for CBLL Phenomenon.As cognitive learning is always dependent on some resource [9], we mentioned a set of language resources in Figure 1. The human brain collects the information from language resource and then processes it for learning. So, resource growth factor is one of the major parameters in cognitive learning. In Figure 1, r′ denotes the resource growth parameter. Furthermore, our brain is not capable to store all information with stabilized condition. Thus, forgetting or loss of information affects the learning process. In Figure 1, m′ denotes the forgetting parameters which have been incorporated with the CBLL system in [9]. Mathematically, the system can be expressed by Li+1=fLi,r′,m′, where Li and Li+1 represent input and output, respectively. In [9], the explicit form of the f is given by fLi,r′,m′=1+r′−m′Li. According to the study in [9], r′ depends on intrinsic growth rate r, carrying capacity K, and also the input Li by r′=rK−Li/K. Furthermore, it has been also established in [9] that the forgetting parameter m′ depends on intrinsic forgetting growth m, K, and Li by m′=mLi/K. Thus, the CBLL system is given by(1)Li+1=1+rLi−a+mKLi2,m<K,where a is known as braking parameter given by a=r/K. For our purpose, we fix the respective values of a,m, and K by 0.01,0.1,and0.5, respectively. By solving (1), we get an output which is called language growth oscillation or simply language growth of the system. In Figure 1, the language growth is shown at the extreme right panel.


## 3. Dynamics of the CBLL System

### 3.1. Nonnoisy Condition

We first investigated equilibrium points of (1) and the respective stabilities under the variation of r. To calculate equilibrium points of (1), we consider an equation: L=1+rL−a+m/KL2, where L represents equilibrium point in (1). The equation gives two solution of L, say L1andL2 given by L1=0 and L2=r/a+m/K. As we fixed a=0.01, m=0.1, and K=0.5, L2 can be considered as L2=ηr, where η=1/a+m/K=constant. Then, stability of L1 and L2 are investigated by perturbing system (1) with δLi=Li−L (perturbation). Then, by Taylor’s theorem, we can have δLi+1=dfL/dlδLi with the small perturbation. Hence, δLi+1/δLi=dfL/dl. It indicates that δLi+1<δLi if dfL/dl<1, i.e., the system is said to be stable and unstable if dfL/dl<1 and >1, respectively. We thus investigate stability region of L1andL2 by verifying dfL/dl<1 with the variation in r. As dfL/dl=1+r at the equilibrium points L1=0, we get dfL/dl≥1 for r≥0. As dfL/dl=1 for r=0, no conclusion can be inferred. Since dfL/dl>1 for all r>0, it implies L1 is unstable in nature. Same analysis is done for L2=ηr. The analysis shows dfL/dl<1 for 0<r<2. The stability regions for L1 and L2 are given in Figure 2.Figure 2
L vs. r plot represents regions of stability of the two fixed points, L1andL2, of system (1).We next investigate bifurcation phenomena of (1) for r∈1.5,3. The corresponding graph is shown in Figure 3.Figure 3
Li vs. r bifurcation diagram of system (1) with the initial condition L1=1 and r∈1.5,3. For each r, the roots are calculated by solving (1) with length N=500000. Here, 1, 2, and 3 indicate number of successive windows for r defined by 1.5,2.45,2.45,2.55,2.55,3, respectively.We divide the whole graph into three windows. From the first window, it can be observed that system (1) produces single and double periods for r∈1.5,2.45, whether the second window shows the existence of two as well as four periods in (1) for r∈2.45,2.55. On the other hand, multiple periods can be seen for the same system with r∈2.55,3 (see the third window in Figure 3). So, the entire bifurcation analysis indicates that the dynamics of (1) may be complex with the increase of r∈1.5,3. To confirm this, we have investigated Lyapunov exponent (LE) for the system (1) with r∈1.5,3. The LE is calculated by(2)λ=limn⟶∞1n∑i=1nlogdfLidLi,where n represents length of the Li.Figure4(a) shows fluctuation of λ over r∈1.5,3. It can be observed from Figure 4(a) that the oscillation of λ always fluctuates in [−4, 0] for r∈1.5,2.55. It assures nonchaotic dynamics in (1) over r∈1.5,2.55. Furthermore, λ<0,>0 and =0 can be seen for r>2.55. It indicates chaotic as well as nonchaotic dynamics exists in (1) for r∈2.55,3. However, it can be investigated that the values of positive λ≈0 for some r∈2.55,3 in Figure 4(a). In this case, chaos cannot be confirmed for the system as two different trajectories show very small divergence with the perturbation in initial condition. We thus investigate chaos in (1) by 0−1 test method. In this method, a solution component, say xnk=1N (N being the length of the component) of a system is considered [22, 23]. Then, xnk=1N decomposes into two components by the transformation:(3)pn,c=∑j=1nxjcosjc,qn,c=∑j=1nxjsinjc,c∈0,π.Figure 4
(a)λ vs. r graph of system (1) for r∈1.5,3. (b, c) p,q-plots of (1) with r=2.5 and 3, respectively. To calculate p and q, we have considered c=2π/3. (d) Kc vs. r∈1.5,3 graph of the same system. In each case, Kc is calculated with ncut=N/10.
(a)(b)(c)(d)The chaotic and nonchaotic dynamics of a system can be identified by the nature ofp,q-plots. In fact, regular and Brownian motion-like structure of the p,q-plots corresponds with respective nonchaotic and chaotic dynamics of the system [22, 23]. We have investigated p,q-plots for system (1) with r∈1.5,3. Figures 4(b) and 4(c) show two p,q-plots for r=2.5 and 3, respectively.From Figure4(b), it can be seen that the p,q-plot contains a regular geometrical structure. It indicates nonchaotic dynamics in (1). On the contrary, Brownian motion-like structure can be observed in Figure 4(c). It corresponds to chaotic dynamics in (1). To quantify chaotic and nonchaotic dynamics, we compute fluctuation of Kcr,r∈1.5,3 given by(4)Kc=limn⟶∞logMcnlogn,where Mcn is defined as(5)Mcn=limN⟶∞1N∑j=1Npcj+n−pcj2+qcj+n−qcj.The values ofKc≈0 and 1 correspond nonchaotic and chaotic dynamics of the system. Figure 4(d) shows fluctuation of Kc over r∈1.5,3. From Figure 4(d), it can be observed that values of Kc≈0 for all r∈1.5,2.55. It assures nonchaotic dynamics in (1). It can be also observed that, Kc≈1, 0 and Kc≈1 may happens for the same system with r∈2.55,3. It implies chaotic as well as nonchaotic dynamics exists in (1) for r∈2.55,3.We further investigated oscillation ofLi and its corresponding phase space behaviour of system (1) over r∈1.5,3. Figures 5(a) and 5(c) show oscillation of Li for r=2.5 and 3, respectively. It can be observed from the figures that variation in Li for r=2.5 is lesser than that for r=3. It indicates system (1) reveals more complex language growth for r=3 compared to the same for r=2.5. Thus, nature of language growth can be investigated in (1) for the entire range of r. However, long-term dynamics in (1) cannot be possible to predict by investigating Li only. So, a phase space analysis is carried out for (1) with r∈1.5,3. In order to investigate phase space of (1), we construct Poincare’ plot for each r. Figures 5(b) and 5(d) represent two Poincare’ plots with r=2.5,3 respectively. In 5(b), the corresponding plot shows few cluster of Li,Li+1 clouds. As maximum number of Li,Li+1 points coverages to each another, it indicates stable dynamics in (1) for r=2.5. On the contrary, the Poincare plot in Figure 5(d) indicates the well-known chaotic logistic curve for r=3. It implies a strong correlation between the phase space and 0-1 test analysis. Moreover, existence of this correlation can also be verified for all r∈1.5,3. In this way, existence of chaotic as well as nonchaotic dynamics in (1) is identified.Figure 5
(a, c) representLi vs. i graphs with r=2.5 and 3, respectively. The corresponding Poincare plots are given in (b, d), respectively. In order to construct the Poincare plot, we have considered last 5000 points of the solution Li,i=1,2,…,50000.
(a)(b)(c)(d)


### 3.2. Noisy Condition

To study the dynamics of the CBLL system under the effect of noise, we incorporate a multiplicative noiseξ in (1) with the noise strength c. The whole phenomenon is described in a schematic diagram given by Figure 6. In Figure 6, ⋅ indicates multiplication. The green arrow denotes the incorporation of ξ with the intrinsic parameter r. The functional form of the resultant CBLL system is given by fLi,r⋅cξ,m (see Figure 6).Figure 6
Schematic diagram for noise-induced CBLL phenomenon.The noise-induced system is thus given by(6)Li+1=1+r⋅cξLi−a+mKLi2,m<K,where r,K,anda represent the same as mentioned for the nonnoisy case. For our purpose, ξ is taken as 1/fβ-noise with β=1.In this case, we have investigated chaotic and nonchaotic dynamics in (6) under both the variation of r∈1.5,3 and c∈0.1,1, respectively. To do so, we have applied the 0-1 test method in (6). Figures 7(a) and 7(b) show respective p,q-plots for r=1.5 and r=3 with the fixed c=0.5. From the respective p,q plots, a regular geometric structure and Brownian motion-like movement can be observed for r=1.5,c=0.5 and r=3,c=0.5, respectively. It indicates nonchaotic and chaotic dynamics in (6), respectively. These can be also identified for all r∈1.5,3 and c∈0.1,1.Figure 7
(a, b)p,q-plots of (6) with r=1.5c=0.5 and 3 c=0.5, respectively. To calculate p and q, we have considered c=2π/3. Kc vs. r∈1.5,3c=0.5 graph for Li obtained from (6) given in (c). (d) Kcr,c matrix plot for r,c∈1.5,3×0.1,1. In each case, Kcs are calculated with ncut=N/10. The color bar indicates values of Kc.
(a)(b)(c)(d)In the next, fluctuation ofKc is also measured for the same r∈1.5,3 with c=0.5. The corresponding graph is given in Figure 7(c). It can be observed from Figure 7(c) that Kc≈1 for all r≥2.4 and Kc≈1 for r<2.4. It assures chaotic and nonchaotic dynamics in (6) for r≥2.4 and r<2.4, respectively. Hence, the method of 0-1 test can classify chaotic as well as nonchaotic phenomena in (6) over r∈1.5,3 with fixed c=0.5. Furthermore, fluctuation of Kc is investigated over r,c∈1.5,3×0.1,1. Figure 7(d) shows the corresponding contour diagram. From the figure, it can be seen that yellow, green, blue, pink, and red regions are comparatively smaller than the dark red region. As dark red region corresponds Kc≈1 and other regions correspond Kc≈1, respectively, it implies chaos in (6), which can be observed for almost all r,c∈1.5,3×0.1,1. It can be also observed that Kc∈0.58,0.9 for r>1.5 and c∈0.1,0.3. It follows nonchaotic dynamics in (6) with the increasing r for the noise strength c∈0.1,0.3. Furthermore, an increasing trend in Kc with the increase of r∈1.5,3 with c∈0.3,0.5 can be observed for (6). As Kc attains its maximum value (i.e., 1) in this case, chaotic dynamics in (6) can be first observed in the region 1.5,3×0.3,0.5. We call it first chaotic paradigm for system (6). For the remaining region, i.e., 1.5,3×0.5,1, dark red color confirms chaotic dynamics in (6). It establishes the existence of complex dynamics in the noise-induced CBLL system. Moreover, analysis of the 0-1 test method also differentiates the chaotic and nonchaotic paradigm for the same system.We also investigated oscillation ofLi and its corresponding phase space nature of system (6) over r∈1.5,3 and c∈0.1,1. Figures 8(a) and 8(c) show oscillation of Li for r=1.5c=0.5 and r=3c=0.5, respectively. It can be observed from Figures 8(a) and 8(b) that abrupt changes in Li for r=3c=0.5 is higher than that for r=1.5c=0.5. It indicates, more complex oscillation in Li for r=3c=0.5 compared to the same for r=1.5c=0.5. By comparing the Lis in (1) and (6), it can be observed that Li=0 for some i in the deterministic chaos whether in the noise-induced chaos, Li does not vanishes for any i, r∈1.5,3, c∈0.5,1 (see Figures 5(a) and 5(c) and Figures 8(a) and 8(c)). To investigate long-term behaviour of (6), Poincare plots are constructed for the same r and c, respectively. Figures 8(b) and 8(d) represent two Poincare plots with r=1.5c=0.5 and r=3c=0.5, respectively. It can be seen that the corresponding Poincare cloud for r=1.5c=0.5 is less distorted compare with the same for +. The oscillation in Li and its corresponding phase space behaviour can also be investigated for r,c∈1.5,3×0.1,1. It gives a strong correlation with the fluctuation in Kc, as shown in Figure 7(d).Figure 8
(a, c) representLi vs. i graphs with r=1.5c=0.5 and 3 c=0.5, respectively. The corresponding Poincare’ plots are given in (b) and (d), respectively. To construct each Poincare plot, we have considered last 5000 points of Li,i=1,2,…,50000.
(a)(b)(c)(d)So, complex dynamics in both systems (1) and (6) are identified successfully over the feasible parametric region under the variable noise strength. However, it does not reflect complexity in the corresponding long-term dynamics of the systems. A quantification is thus needed to measure the complexity of the respective asymptotic dynamics. In the following section, the complexity of (1) and (6) are investigated by weighted recurrence entropy [40].


## 3.1. Nonnoisy Condition

We first investigated equilibrium points of (1) and the respective stabilities under the variation of r. To calculate equilibrium points of (1), we consider an equation: L=1+rL−a+m/KL2, where L represents equilibrium point in (1). The equation gives two solution of L, say L1andL2 given by L1=0 and L2=r/a+m/K. As we fixed a=0.01, m=0.1, and K=0.5, L2 can be considered as L2=ηr, where η=1/a+m/K=constant. Then, stability of L1 and L2 are investigated by perturbing system (1) with δLi=Li−L (perturbation). Then, by Taylor’s theorem, we can have δLi+1=dfL/dlδLi with the small perturbation. Hence, δLi+1/δLi=dfL/dl. It indicates that δLi+1<δLi if dfL/dl<1, i.e., the system is said to be stable and unstable if dfL/dl<1 and >1, respectively. We thus investigate stability region of L1andL2 by verifying dfL/dl<1 with the variation in r. As dfL/dl=1+r at the equilibrium points L1=0, we get dfL/dl≥1 for r≥0. As dfL/dl=1 for r=0, no conclusion can be inferred. Since dfL/dl>1 for all r>0, it implies L1 is unstable in nature. Same analysis is done for L2=ηr. The analysis shows dfL/dl<1 for 0<r<2. The stability regions for L1 and L2 are given in Figure 2.Figure 2
L vs. r plot represents regions of stability of the two fixed points, L1andL2, of system (1).We next investigate bifurcation phenomena of (1) for r∈1.5,3. The corresponding graph is shown in Figure 3.Figure 3
Li vs. r bifurcation diagram of system (1) with the initial condition L1=1 and r∈1.5,3. For each r, the roots are calculated by solving (1) with length N=500000. Here, 1, 2, and 3 indicate number of successive windows for r defined by 1.5,2.45,2.45,2.55,2.55,3, respectively.We divide the whole graph into three windows. From the first window, it can be observed that system (1) produces single and double periods for r∈1.5,2.45, whether the second window shows the existence of two as well as four periods in (1) for r∈2.45,2.55. On the other hand, multiple periods can be seen for the same system with r∈2.55,3 (see the third window in Figure 3). So, the entire bifurcation analysis indicates that the dynamics of (1) may be complex with the increase of r∈1.5,3. To confirm this, we have investigated Lyapunov exponent (LE) for the system (1) with r∈1.5,3. The LE is calculated by(2)λ=limn⟶∞1n∑i=1nlogdfLidLi,where n represents length of the Li.Figure4(a) shows fluctuation of λ over r∈1.5,3. It can be observed from Figure 4(a) that the oscillation of λ always fluctuates in [−4, 0] for r∈1.5,2.55. It assures nonchaotic dynamics in (1) over r∈1.5,2.55. Furthermore, λ<0,>0 and =0 can be seen for r>2.55. It indicates chaotic as well as nonchaotic dynamics exists in (1) for r∈2.55,3. However, it can be investigated that the values of positive λ≈0 for some r∈2.55,3 in Figure 4(a). In this case, chaos cannot be confirmed for the system as two different trajectories show very small divergence with the perturbation in initial condition. We thus investigate chaos in (1) by 0−1 test method. In this method, a solution component, say xnk=1N (N being the length of the component) of a system is considered [22, 23]. Then, xnk=1N decomposes into two components by the transformation:(3)pn,c=∑j=1nxjcosjc,qn,c=∑j=1nxjsinjc,c∈0,π.Figure 4
(a)λ vs. r graph of system (1) for r∈1.5,3. (b, c) p,q-plots of (1) with r=2.5 and 3, respectively. To calculate p and q, we have considered c=2π/3. (d) Kc vs. r∈1.5,3 graph of the same system. In each case, Kc is calculated with ncut=N/10.
(a)(b)(c)(d)The chaotic and nonchaotic dynamics of a system can be identified by the nature ofp,q-plots. In fact, regular and Brownian motion-like structure of the p,q-plots corresponds with respective nonchaotic and chaotic dynamics of the system [22, 23]. We have investigated p,q-plots for system (1) with r∈1.5,3. Figures 4(b) and 4(c) show two p,q-plots for r=2.5 and 3, respectively.From Figure4(b), it can be seen that the p,q-plot contains a regular geometrical structure. It indicates nonchaotic dynamics in (1). On the contrary, Brownian motion-like structure can be observed in Figure 4(c). It corresponds to chaotic dynamics in (1). To quantify chaotic and nonchaotic dynamics, we compute fluctuation of Kcr,r∈1.5,3 given by(4)Kc=limn⟶∞logMcnlogn,where Mcn is defined as(5)Mcn=limN⟶∞1N∑j=1Npcj+n−pcj2+qcj+n−qcj.The values ofKc≈0 and 1 correspond nonchaotic and chaotic dynamics of the system. Figure 4(d) shows fluctuation of Kc over r∈1.5,3. From Figure 4(d), it can be observed that values of Kc≈0 for all r∈1.5,2.55. It assures nonchaotic dynamics in (1). It can be also observed that, Kc≈1, 0 and Kc≈1 may happens for the same system with r∈2.55,3. It implies chaotic as well as nonchaotic dynamics exists in (1) for r∈2.55,3.We further investigated oscillation ofLi and its corresponding phase space behaviour of system (1) over r∈1.5,3. Figures 5(a) and 5(c) show oscillation of Li for r=2.5 and 3, respectively. It can be observed from the figures that variation in Li for r=2.5 is lesser than that for r=3. It indicates system (1) reveals more complex language growth for r=3 compared to the same for r=2.5. Thus, nature of language growth can be investigated in (1) for the entire range of r. However, long-term dynamics in (1) cannot be possible to predict by investigating Li only. So, a phase space analysis is carried out for (1) with r∈1.5,3. In order to investigate phase space of (1), we construct Poincare’ plot for each r. Figures 5(b) and 5(d) represent two Poincare’ plots with r=2.5,3 respectively. In 5(b), the corresponding plot shows few cluster of Li,Li+1 clouds. As maximum number of Li,Li+1 points coverages to each another, it indicates stable dynamics in (1) for r=2.5. On the contrary, the Poincare plot in Figure 5(d) indicates the well-known chaotic logistic curve for r=3. It implies a strong correlation between the phase space and 0-1 test analysis. Moreover, existence of this correlation can also be verified for all r∈1.5,3. In this way, existence of chaotic as well as nonchaotic dynamics in (1) is identified.Figure 5
(a, c) representLi vs. i graphs with r=2.5 and 3, respectively. The corresponding Poincare plots are given in (b, d), respectively. In order to construct the Poincare plot, we have considered last 5000 points of the solution Li,i=1,2,…,50000.
(a)(b)(c)(d)


## 3.2. Noisy Condition

To study the dynamics of the CBLL system under the effect of noise, we incorporate a multiplicative noiseξ in (1) with the noise strength c. The whole phenomenon is described in a schematic diagram given by Figure 6. In Figure 6, ⋅ indicates multiplication. The green arrow denotes the incorporation of ξ with the intrinsic parameter r. The functional form of the resultant CBLL system is given by fLi,r⋅cξ,m (see Figure 6).Figure 6
Schematic diagram for noise-induced CBLL phenomenon.The noise-induced system is thus given by(6)Li+1=1+r⋅cξLi−a+mKLi2,m<K,where r,K,anda represent the same as mentioned for the nonnoisy case. For our purpose, ξ is taken as 1/fβ-noise with β=1.In this case, we have investigated chaotic and nonchaotic dynamics in (6) under both the variation of r∈1.5,3 and c∈0.1,1, respectively. To do so, we have applied the 0-1 test method in (6). Figures 7(a) and 7(b) show respective p,q-plots for r=1.5 and r=3 with the fixed c=0.5. From the respective p,q plots, a regular geometric structure and Brownian motion-like movement can be observed for r=1.5,c=0.5 and r=3,c=0.5, respectively. It indicates nonchaotic and chaotic dynamics in (6), respectively. These can be also identified for all r∈1.5,3 and c∈0.1,1.Figure 7
(a, b)p,q-plots of (6) with r=1.5c=0.5 and 3 c=0.5, respectively. To calculate p and q, we have considered c=2π/3. Kc vs. r∈1.5,3c=0.5 graph for Li obtained from (6) given in (c). (d) Kcr,c matrix plot for r,c∈1.5,3×0.1,1. In each case, Kcs are calculated with ncut=N/10. The color bar indicates values of Kc.
(a)(b)(c)(d)In the next, fluctuation ofKc is also measured for the same r∈1.5,3 with c=0.5. The corresponding graph is given in Figure 7(c). It can be observed from Figure 7(c) that Kc≈1 for all r≥2.4 and Kc≈1 for r<2.4. It assures chaotic and nonchaotic dynamics in (6) for r≥2.4 and r<2.4, respectively. Hence, the method of 0-1 test can classify chaotic as well as nonchaotic phenomena in (6) over r∈1.5,3 with fixed c=0.5. Furthermore, fluctuation of Kc is investigated over r,c∈1.5,3×0.1,1. Figure 7(d) shows the corresponding contour diagram. From the figure, it can be seen that yellow, green, blue, pink, and red regions are comparatively smaller than the dark red region. As dark red region corresponds Kc≈1 and other regions correspond Kc≈1, respectively, it implies chaos in (6), which can be observed for almost all r,c∈1.5,3×0.1,1. It can be also observed that Kc∈0.58,0.9 for r>1.5 and c∈0.1,0.3. It follows nonchaotic dynamics in (6) with the increasing r for the noise strength c∈0.1,0.3. Furthermore, an increasing trend in Kc with the increase of r∈1.5,3 with c∈0.3,0.5 can be observed for (6). As Kc attains its maximum value (i.e., 1) in this case, chaotic dynamics in (6) can be first observed in the region 1.5,3×0.3,0.5. We call it first chaotic paradigm for system (6). For the remaining region, i.e., 1.5,3×0.5,1, dark red color confirms chaotic dynamics in (6). It establishes the existence of complex dynamics in the noise-induced CBLL system. Moreover, analysis of the 0-1 test method also differentiates the chaotic and nonchaotic paradigm for the same system.We also investigated oscillation ofLi and its corresponding phase space nature of system (6) over r∈1.5,3 and c∈0.1,1. Figures 8(a) and 8(c) show oscillation of Li for r=1.5c=0.5 and r=3c=0.5, respectively. It can be observed from Figures 8(a) and 8(b) that abrupt changes in Li for r=3c=0.5 is higher than that for r=1.5c=0.5. It indicates, more complex oscillation in Li for r=3c=0.5 compared to the same for r=1.5c=0.5. By comparing the Lis in (1) and (6), it can be observed that Li=0 for some i in the deterministic chaos whether in the noise-induced chaos, Li does not vanishes for any i, r∈1.5,3, c∈0.5,1 (see Figures 5(a) and 5(c) and Figures 8(a) and 8(c)). To investigate long-term behaviour of (6), Poincare plots are constructed for the same r and c, respectively. Figures 8(b) and 8(d) represent two Poincare plots with r=1.5c=0.5 and r=3c=0.5, respectively. It can be seen that the corresponding Poincare cloud for r=1.5c=0.5 is less distorted compare with the same for +. The oscillation in Li and its corresponding phase space behaviour can also be investigated for r,c∈1.5,3×0.1,1. It gives a strong correlation with the fluctuation in Kc, as shown in Figure 7(d).Figure 8
(a, c) representLi vs. i graphs with r=1.5c=0.5 and 3 c=0.5, respectively. The corresponding Poincare’ plots are given in (b) and (d), respectively. To construct each Poincare plot, we have considered last 5000 points of Li,i=1,2,…,50000.
(a)(b)(c)(d)So, complex dynamics in both systems (1) and (6) are identified successfully over the feasible parametric region under the variable noise strength. However, it does not reflect complexity in the corresponding long-term dynamics of the systems. A quantification is thus needed to measure the complexity of the respective asymptotic dynamics. In the following section, the complexity of (1) and (6) are investigated by weighted recurrence entropy [40].


## 4. Recurrence-Based Complexity Analysis

The weighted recurrence entropy (WRE) is proposed by utilizing Shannon entropy based on the weighted recurrence of phase space. An weighted recurrence for a givenn-dimensional phase space P=xi∈Rn is denoted by wi,j and defined as(7)wi,j=eri,j,i,j=1,2,…,N,where N being the length of the trajectory of the phase space and ri,j is given by ri,j=−xi−xj. As difference between xi,xj indicates separation between two points, it implies ri,j can quantify dispersion between the trajectories. Hence, disorder in P can be measured by ri,j matrix plot. Furthermore, wi,j can measure exponential decay of ri,j. So, values of wi,j always lies between 0 and 1. It indicates disorder in the phase space can be sampled within 0,1. Then, weight recurrence entropy Sw is defined as(8)Sw=−∑sk∈Spsklogpsk,where psk denotes probability of sk∈S=sk:sk=1/N∑j=1Tωkj,1≤k≤T (T being number of events).We investigate disorder and complexity for both systems (1) and (6), respectively. Only two ri,j matrix plots are shown in Figures 9(a) and 9(c) for system (1) with r=2.5 and 3, respectively.Figure 9
(a, c)ri,j-matrix plot of (1) with r=2.5 and 3 respectively. Here, ri,j=−xi−xj. The color bars indicate values of ri,j for i,j=1,2,…,100. The corresponding Psk vs. sk plots are given in (b) and (d), respectively. In each case, probabilities are calculated by counting 50 bins. (e) Represents Sw vs. r graph of (1) with r∈1.5,3. To calculate Sw by (8), the probabilities are calculated by the aforementioned bin.
(a)(b)(c)(d)(e)From both the figures, it can be observed that greater variation in Figure9(c) exists compare to the same in Figure 9(a). It indicates disorder in the phase space of (1) for r=3 is greater than that for r=2.5. In this way, disorder and hence complexity in the phase space of (1) for the entire range of r can be recognized. To quantify complexity, we further calculate the probabilities psk using wi,j for r∈1.5,3. Figures 9(b) and 9(d) represent probability curves of the respective disorder given in Figures 9(a) and 9(c). In Figure 9(b), only two sharp peaks can be observed. It signifies that there exists only two variations in Figure 9(a). On the contrary, large number of probabilities in Figure 9(d) indicates that the corresponding phase space is more complex than for r=2.5. To verify this, we have calculated respective Sw using (8). It has been seen that Sw/r=3=3.821>Sw/r=1.5=0.1. It confirms existence of higher complexity in (1) with r=3 than that in same for r=1.5. We also investigated complexity in (1) for the entire range of r. To do this, fluctuation in Sw is measured over r∈1.5,3. The corresponding curves are shown in Figure 9(e). From the figure, it can be observed that a strong correlation exists between Sw and the Kc (given in Figure 4(d)).Similar analysis is carried out for system (6) with the variation of r∈1.5,3 and c∈0.1,1. Figures 10(a) and 10(c) show corresponding ri,j matrix plots with r=1.5,3(c = 0.5), respectively.Figure 10
(a, c)ri,j-matrix plot of (6) with r=1.5 and 3 (fixed c=0.5), respectively. Here, ri,j=−xi−xj. The color bars indicate values of ri,j for i,j=1,2,…,100. The corresponding Psk vs. sk plots are given in (b, d), respectively. In each case, probabilities are calculated by counting 50 bins. (e) Sw vs. r graph of (1) with r∈1.5,3. To calculate Sw by (8), the probabilities are calculated by aforementioned bin, keeping fixed c=0.5. (f) Swr,c matrix plots with r,c∈1.5,3×0.1,1. The associated color bar indicates values of Sw.
(a)(b)(c)(d)(e)(f)From Figures10(a) and 10(c), similar kind of patterns can be observed in the respective ri,j matrix, except the values of ri,j. From the color bar, it can be seen that range of ri,j∈−2,0 for r=1.5, whether ri,j∈−4,0 for r=3. It indicates greater dispersion between the trajectories for r=3 compared to the same for r=1.5. It signifies more disordered structures in the corresponding phase space with r=3 can be observed compare to the other. In order to verify this, we further calculate respective probability distribution, which is given in Figures 10(b) and 10(d). The distribution for r=3 shows variation in psk for larger range compare to the same for r=1.5. Finally, using (8), we have found that Sw/r=1.5=3.4201<Sw/r=3=3.58. It confirms that complexity in (6) for r=3 is greater than that for r=1.5. Since difference ΔSw=Sw/r=3−Sw/r=3=0.1599, it corresponds difference between the disorder in the phase spaces are very small. Furthermore, complexity in (6) is studied over r∈1.5,3 with c=0.5. Figure 10(e) shows the corresponding Sw vs. r graphs. It is observed that there exists a strong correlation between the trend in Sw and Kc. It is noted that system (6) can possesses higher complexity compare to the same in (1) at the nonchaotic state.Finally, we have investigated variation inSw over r,c∈1.5,3×0.1,1. The corresponding contour is shown in Figure 10(f). From the figure, it is observed that values of Sw∈2,3 can be obtained for a very small range of r,c. In fact, values of Sw lies in 2,4 for almost all region.


## 5. Conclusions

In this article, dynamics and complexity are explored for a CBLL and its noise-induced system. The dynamics of the CBLL system have been investigated by equilibrium and its stability, bifurcation, Lyapunov, and 0-1 test analysis. Bifurcation analysis shows periodic as well as multiperiodic solution can be obtained with the variation of intrinsic growth parameterr. It has been also observed that multiperiodicity increases with an increase of r. It initially indicates the existence of complex dynamics in the CBLL. To verify this, Lyapunov analysis has been performed under the same variation in r. Our investigation confirms that the CBLL can have chaotic as well as multiperiodic features. As some of the positive λ have been obtained very close to zero, we have carried out 0-1 test analysis. It successfully quantified the chaotic paradigm for the same r. Later on, oscillation in language growth and the phase space behaviour have been investigated. It assures that the CBLL system can possess unstable oscillations in Li and hence chaotic dynamics with the various r. For the noise-induced CBLL, the 0-1 test method is applied. The effect of noise strength and the variation of r both have been investigated in the noise-induced system. It assures that chaos can be enhanced in the CBLL under the effect of power noise. It has been remarked that deterministic SBLL may contain zero level language growth in some chaotic state. It is not feasible in any real-world cognitive learning phenomena. On the contrary, noise-induced chaotic language growth does not vanish at any condition. It indicates the effectiveness of noise-induced chaos in cognitive language learning. The complexity of both the systems has been quantified using WRE. The corresponding analysis shows that complexity in the CBLL increases with an increase of r. On computing complexity, it has been notified that the noise-induced CBLL can produce highly complex phenomena even if its corresponding dynamics was nonchaotic. This phenomenon does not exist for the deterministic case.



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