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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.

Human language is an expressible complex communication system which consists of phonemes, words, phrases, sentences, poetry, and publications [

Chaos can be verified by measuring the exponential divergence between the phase space trajectories [

This manuscript is organized as follows. In Section

We consider a cognitive language growth model proposed in [

Schematic diagram for CBLL Phenomenon.

As cognitive learning is always dependent on some resource [

We first investigated equilibrium points of (

We next investigate bifurcation phenomena of (

We divide the whole graph into three windows. From the first window, it can be observed that system (

Figure

(a)

The chaotic and nonchaotic dynamics of a system can be identified by the nature of

From Figure

The values of

We further investigated oscillation of

(a, c) represent

To study the dynamics of the CBLL system under the effect of noise, we incorporate a multiplicative noise

Schematic diagram for noise-induced CBLL phenomenon.

The noise-induced system is thus given by

In this case, we have investigated chaotic and nonchaotic dynamics in (

(a, b)

In the next, fluctuation of

We also investigated oscillation of

(a, c) represent

So, complex dynamics in both systems (

The weighted recurrence entropy (WRE) is proposed by utilizing Shannon entropy based on the weighted recurrence of phase space. An weighted recurrence for a given

We investigate disorder and complexity for both systems (

(a, c)

From both the figures, it can be observed that greater variation in Figure

Similar analysis is carried out for system (

(a, c)

From Figures

Finally, we have investigated variation in

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 parameter

All the data used in this study are included within the text.

The authors declare that they have no conflicts of interest.