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This study aims to investigate a financial system consisting of four ordinary differential equations associated with the rate of interest, investment demand, price index, and the density of financial information gained by the population. The equilibrium and local stability of the system are investigated numerically. The impact of saving amounts and the rate of investment demand increases after getting financial information on the system are discussed. The findings of the study are verified graphically. It is found that the system becomes stable if the rate of investment demand increases after getting financial information kept at a certain level, such that the savings amount is maintained at a higher level. Also, the bifurcation diagrams of the system for various significant parameters that affect the system’s stability have been depicted.

This paper propose two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic equation and the logistic equation of arbitrary order respectively. It demonstrate the impact of chaos theory by integrating it with logistics equations and reveal how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. As well, this work presents BifDraw – a Python program for making bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system's response to the changes in the conditions. The research gives a pivotal role to the logistic equation's place in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigates dynamics nature of the equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic equations within chaos theory. This study not only deepens our comprehension of logistic equations and chaos theory but also introduces practical tools for visualizing and analyzing their behaviors.

In this paper, we consider two-dimensional periodic capillary-gravity waves traveling under the influence of a vertical electric field. The full system is a nonlinear, two-layered, free boundary problem. The interface dynamics are derived by coupling Euler equations for the velocity field of the fluid with voltage potential equations governing the electric field. We first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. We then prove the existence of small-amplitude electrohydrodynamic waves with constant vorticity using local bifurcation theory. Moreover, we show that these electrohydrodynamic waves are formally stable in the linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch, consisting of ripple solutions on the interface. As far as we know, such solutions in electrohydrodynamics are established for the first time. It is worth noting that the electric field E 0 plays a key role in controlling the shapes and types of waves on the interface. [ABSTRACT FROM AUTHOR]

This study investigates the influence of fear, refuge, and migration in a predator–prey model, where the interactions between the species follow an asymmetric function response. In contrast to some other findings, we propose that prey develop an anti-predator response in response to a concentration of predators, which in turn increases the fear factor of the predators. The conditions under which all ecologically meaningful equilibrium points exist are discussed in detail. The local and global dynamics of the model are determined at all equilibrium points. The model admits several interesting results by changing the rate of fear of predators and predator aggregate sensitivity. Numerical simulations have been performed to verify our theoretical findings. We found that under certain conditions, the system appears to be losing the stability to acquire the periodic attractor when the trait-mediated direct effect of one of each (prey growth, competition, fear, and prey death). [ABSTRACT FROM AUTHOR]

The bursting with long burst duration of the pancreatic β-cells related to the diabetes is helpful to maintain the normal blood glucose concentration. Then, nonlinear dynamics underlying burst duration is an important issue, which is investigated in a theoretical model containing two slow variables (s and z) in the present paper. For the deterministic case, different shapes and sizes of the bursting trajectory show that the slow bursting is modulated by s, z, and s-nullcline in wide ranges, the medium bursting by s and z in medium ranges, and the fast bursting mainly by s in narrow ranges. Through two-parameter bifurcation analysis, the burst of the three bursting patterns terminates at saddle homoclinic orbit (SH) bifurcation curve. For the slow and medium bursting patterns, the latter part of the burst mainly runs along z-direction and closely to the saddle surface. Then, noise can induce the burst to terminate earlier than the SH via passage through the saddle surface, resulting in decreased burst duration. Compared with the drastic decrease for the medium bursting, slow bursting exhibits small reduction, since the large size induces two phases insensitive to noise. One is the latter part of the quiescent state, which runs along the s-nullcline to exhibit extra-stable dynamics, and the other is the former part of the burst, which runs far from the saddle surface. However, the fast bursting mainly runs along s-direction and far from the saddle surface exclusive the SH, then, noise can only induce small changes. Then, the different changing trends of the stochastic slow, medium, and fast bursting patterns are well explained with the different stochastic responses around different bifurcation points or critical phases. The results present dynamical mechanism for the long burst duration which is favor for the maintenance of the normal blood glucose concentration. [ABSTRACT FROM AUTHOR]

In this work, we first discuss the positive equilibrium point of the continuous predator–prey system and its stability, and we discuss the parameter conditions under which the continuous system undergoes a cusp bifurcation (Bogdanov–Takens bifurcation) of codimension two bifurcation at the positive equilibrium point. Then, we provide an insightful study of discrete predator–prey systems by the use of Euler's method, which includes square-root function responses and nonlinear prey harvesting. By synthesizing the new standard form of differential-algebraic systems, the central manifold theorem, and the bifurcation theory, we identify the specific conditions under which the system may undergo flip bifurcation and Neimark–Sacker bifurcation. In addition, codimension-two bifurcations associated with 1:2 strong resonances are analyzed by using a series of affine transformations and bifurcation theory. Through numerical simulations, we not only verify the validity and correctness of our findings, but also elucidate the frequency of trajectory bifurcations in the intervals of 2, 4, and 8 and chaotic phenomena. These findings reveal a richer and more diverse dynamic behavior of discrete differential-algebraic bioeconomic systems, which is of great theoretical and practical significance to the fields of mathematics and biology. [ABSTRACT FROM AUTHOR]

Mitigating traffic jams is a critical step for the betterment of the urban transportation system, which comprises a large number of interconnected routes to form an intricate network. To study the distinct features and complexities of vehicular flow, a network consisting of multiple-loops with a single intersection is considered a directed weighted graph. The governing equations for individual loop densities are derived using the principle of mass conservation, considering a data-driven flow-density relationship. Firstly, we examine the stability of a double-loop network and explore the traffic behavior using the macroscopic fundamental diagram (MFD) for different sets of parameters. Utilizing popular techniques of nonlinear dynamics, the existence of bistability, bifurcations, oscillations, and switching dynamics is demonstrated in the case of the triple-loop network. Further, bistability is characterized by plotting the basin of attraction diagram for the coexisting attractors. Our study reveals that an increase in the number of loop lines enriches the dynamical properties of vehicular traffic flow. It is observed that depending upon the initial density configuration, a loop network can show various phases, namely free flow, stop-and-go traffic jams, and loop congestion. Additionally, we show the presence of period-2 orbits in the case of the quadruple-loop system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel two-delayed tri-neuron neural network (NN) with no connection between the first and third neurons. Neural networks with incomplete connections offer a range of advantages, including improved efficiency, generalisation, interpretability, and biological plausibility, making them useful in various applications across different domains. Such kinds of NNs exist in some diseases, such as epilepsy, Alzheimer's, and schizophrenia, where the neuron's connections can be broken. Our NN is defined in two different forms: one with integer-order derivatives and another with Caputo fractional derivatives. The fundamental results of existence, uniqueness, and boundedness of the solution for the proposed NN are derived. We perform the bifurcation analysis along with the stability of the initial state of the fractional-order NN, considering self-connection delay and communication delay as bifurcation parameters, respectively. The proposed NN is numerically solved by using a recently proposed L1-predictor-corrector method with its error analysis. The theoretical proofs are verified through graphical simulations. [ABSTRACT FROM AUTHOR]

In the study of continuous amensalism systems, it has been widely accepted that Michaelis–Menten type harvesting has a significant impact on the survival and extinction of species. However, scholars have not yet studied discrete amenslism models that include Michaelis–Menten type harvesting. To characterize such dynamics, a discrete amensalism system with Michaelis–Menten type harvesting for the second species is investigated. Firstly, we study the existence and stability of all possible equilibrium points. Under different parameters, there are two stable equilibria, which means that the model is not always globally stable. Then, the conditions of various types of bifurcations likely: pitchfork bifurcations, transcritical bifurcations, fold bifurcations, and flip bifurcations have been established. In addition, a global dynamics analysis of the model is also conducted. Finally, the significance of Michaelis–Menten type harvesting in species relationships is shown by numerical simulations. Although proper harvesting reduces the density of the second species, it favors the stable coexistence of both species and excessive harvesting leads directly to the extinction of the second species. Therefore, the results of this paper can provide a reference for research on how to maximize harvesting without destroying the ecological balance of the species. [ABSTRACT FROM AUTHOR]

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of 'simplest' diagrams proposed by Baesens and MacKay (2018 Nonlinearity 31 2928–81). This shows that the proposed class is realisable. [ABSTRACT FROM AUTHOR]

This paper proposed two new generalizations of the logistic function, each drawing on non-extensive thermodynamics, the q-logistic Equation and the logistic Equation of arbitrary order, respectively. It demonstrated the impact of chaos theory by integrating it with logistics Equations and revealed how minor parameter variations will change system behavior from deterministic to non-deterministic behavior. Moreover, this work presented BifDraw – a Python program for drawing bifurcation diagrams using classical logistic function and its generalizations illustrating the diversity of the system’s response to the changes in the conditions. The research gave a pivotal role to the place of the logistic Equation in chaos theory by looking at its complicated dynamics and offering new generalizations that may be new in terms of thermodynamic basic states and entropy. Also, the paper investigated dynamics nature of the Equations and bifurcation diagrams in it which present complexity and the surprising dynamic systems features. The development of the BifDraw tool exemplifies the practical application of theoretical concepts, facilitating further exploration and understanding of logistic Equations within chaos theory. This study not only deepens the comprehension of logistic Equations and chaos theory, but also introduces practical tools for visualizing and analyzing their behaviors. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the free boundary problem in three dimensions describing the growth of tumor cords. The model consists of a reaction-diffusion equation describing the concentration $ \sigma $ of nutrients and an elliptic equation describing the distribution of the internal pressure $ p $. The model is defined in a bounded domain in $ \mathbb{R}^{3} $ whose boundary consists of two disjoint closed curves, the known interior part and the unknown exterior part. The concentration of nutrients outside the tumor region is denoted by $ \bar{\sigma} $. We shall show that there is a positive integer $ n^{**} $ and a sequence $ \bar{\sigma}_{n} $ such that for each $ \bar{\sigma}_{n}(n>n^{**}) $, symmetry-breaking stationary solutions bifurcate from the annular stationary solution in the longitude direction. [ABSTRACT FROM AUTHOR]

A paradoxical nonlinear phenomenon called post-inhibitory facilitation (PIF) has been observed in the superior olive neuron for precise sound localization, which means that an inhibitory synaptic current appearing earlier than an excitatory one does not suppress but facilitates a spike for a short interval between the two currents. Although the PIF is reproduced for type III excitability/non-bifurcation and mediated by a fast low-threshold potassium current (IKLT), contradictorily, explained with a negative threshold related to type II excitability/Hopf bifurcation, even if in the absence of the IKLT. In the present paper, explanations to the seemingly contradictory nonlinear and ionic current mechanisms for the PIF are presented in a modified model containing the IKLT. Firstly, the modified model exhibits Hopf bifurcation/type II for weak or zero IKLT, in addition to non-bifurcation/type III for strong IKLT which resembles the original models. Secondly, the PIF can be evoked for types III and II with a positive threshold, which presents novel conditions for the PIF. Thirdly, with the excitatory current considered, some traditional positive thresholds become novel negative thresholds, which presents successful explanations to the novel PIF modulated by the IKLT and the inhibitory and excitatory synapses. The novel PIF appears for a strong and relatively slow inhibitory synaptic current which can induce membrane potential reduced to run across the negative threshold, while not for weak or fast inhibitory current or not for novel positive thresholds related to fast excitatory synapses. Finally, the PIF with short time interval appears for type III with fast dynamics, which is mainly mediated by the fast IKLT, resembling the experiment. The PIF with wide interval appears for type II with slow dynamics, which is mediated by the weak IKLT and/or the slow leak current (IL), presenting a novel current to mediate the PIF. The bifurcation/excitability, novel threshold surface, IKLT and IL, and inhibitory and excitatory synapses present a comprehensive viewpoint on the PIF and the potential modulations to obtain the PIF with short interval for the precise sound localization. [ABSTRACT FROM AUTHOR]

This paper addresses the challenging issues of transmission quality degradation and difficulty in obtaining dynamic response characteristics caused by the nonlinear behavior of non-circular planetary gear systems (NPG). A dynamic model for NPG was developed, encompassing axial elastic displacement, backlash, tooth surface friction, time-varying meshing stiffness, and viscoelastic damping. Fourier fitting matrices for time-varying mesh stiffness and polynomial models for dynamic backlash in non-circular gears were acquired to enhance model precision. Various analysis techniques including phase trajectory diagrams, bifurcation diagrams, time history diagrams, Poincaré mapping diagrams, and phase amplitude frequency characteristic curves were used to evaluate the nonlinear behavior of NPGs. Research results indicate that increasing the damping ratio benefits frequency response bandwidth, reduces phase lag, and improves system stability. The friction coefficient on the surface of non-circular gears also plays a role in ensuring the stability and phase consistency of NGP, although excessive coefficients can induce chaos. The output solar gear is more sensitive to internal excitation, with higher internal excitation leading to stronger system chaos. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a class of discrete predator–prey model with alternative prey. We prove the boundedness of the solution, the existence and local/global stability of equilibrium points of the model, and verify the existence of flip bifurcation and Neimark-Sacker bifurcation. In addition, we use the maximum Lyapunov exponent and isoperimetric diagrams to verify the existence of periodic structures namely Arnold tongue and the shrimp-shaped structures in bi-parameter spaces of a class of predator–prey model. [ABSTRACT FROM AUTHOR]

Rotor systems are widely used in industrial power generation and propulsion. Once the nonlinear contact stiffness and oil film force are taken into account, the dynamics and stability of rotor systems become quite complex, often accompanied by super-harmonic and chaotic motions. Furthermore, conventional methods face limitations in real-time detection of the bifurcations and complex nonlinear motions. This research investigates the bifurcations and stability induced by nonlinear factors in a rotor-bearing system from an energy perspective. Dynamic equations of a rotor-bearing system considering cubic term stiffness are established, the steady responses are obtained by the fourth-order Runge–Kutta method. The relationship between the bifurcations and energy transfers is analyzed numerically, the proposed stability criterion is validated by comparing the Lyapunov exponents. The bistable phenomenon of period-3 m (m = 0 , 1) motion is discussed in terms of numerical results and experiments which corresponds to the asymmetric jumps of the generalized energy. It is found that bifurcations and unstable motions of the nonlinear system can be captured accurately by detecting the energy transfers, the proposed generalized energy curve exhibits more detailed information than the conventional speed-up curve. These findings provide a new perspective on studying bifurcations and stability of rotating systems, which can be further applied in the condition monitoring, stability prediction as well as the design of nonlinear energy sink, representing significant progress in converting theory into engineering applications for rotor systems. [ABSTRACT FROM AUTHOR]

Analyzing the nonlinear characteristics of dual-rotor systems under thermo-mechanical (TM) coupling situations is critical, as operational conditions should be accurately determined, to avoid potential thermally-induced failures. This paper proposes the coupled TM model of a dual-rotor system, which considers multiple nonlinearities and heat generation of four bearings that couple the mechanical and thermal fields. Heat dissipation controlled by lubricant flow rates is introduced into the model to simulate different TM coupling degrees. Nonlinear phenomena and stability evolution are analyzed by the modified incremental harmonic balance method (IHB) at primary resonance regions. An increase in TM coupling degrees can lead to more bifurcation points, resonance regions with lower frequencies, larger vibration responses, and unstable regions. It can also transform resonance hysteresis phenomena into more complex nonlinear phenomena and some saddle-node bifurcation points into Neimark–Sacker bifurcation points. The reason for these transformations is that the effective radial clearance (RC) of bearings changes with rotation speed and thermal expansion. Temperature nonlinearities are induced by the radial bearing loads and the lubricant viscosity, which are investigated by various generalized nonlinear thermal forces. These findings can help further understand nonlinear coupled TM problems of complex dual-rotor systems. [ABSTRACT FROM AUTHOR]

Rumor propagation in social networks disrupts people's lives, thus affecting the stability and development of society. To explore the rumor propagation mechanism and control its spread, a novel IRLTS (Ignorant–Rumor-Spreader–Lurker–Truth-Spreader–Stifler) rumor propagation model is proposed, which considers the influence of lurking psychology and the truth of rumors. First, the basic reproduction number and the equilibria of the proposed model are computed. Second, the stability of the equilibria is demonstrated, and the trans-critical bifurcation at the equilibria is analyzed. Finally, we corroborate the theoretical results and the influence of key parameters through numerical simulations, perform sensitivity analysis of the basic reproduction number, and evaluate the proposed model through comparative experiments. Furthermore, the proposed model is applied to an actual rumor dataset to verify its validity. We employ the least squares method to estimate the proposed model's parameters, which are subsequently utilized to forecast rumor propagation trends. The R-squared value for the proposed model on the actual dataset is 0.9567. The outcomes indicate the model is valid and capable of predicting rumor spreading trends, thereby providing theoretical support for formulating strategies to control rumor propagation. [ABSTRACT FROM AUTHOR]

Snap-through is a buckling instability that allows slender objects, including those in plant and biological systems, to generate rapid motion that would be impossible if they were to use their internal forces exclusively. In microfluidic devices, such as micromechanical switches and pumps, this phenomenon has practical applications for manipulating fluids at small scales. The onset of this elastic instability often drives the surrounding fluid into motion – a process known as snap-induced flow. To analyse the complex dynamics resulting from the interaction between a sheet and a fluid, we develop a prototypical model of a thin sheet that is compressed between the two sides of a closed channel filled with an inviscid fluid. At first, the sheet bends towards the upstream direction and the system is at rest. However, once the pressure difference in the channel exceeds a critical value, the sheet snaps to the opposite side and drives the fluid dynamics. We formulate an analytical model that combines the elasticity of thin sheets with the hydrodynamics of inviscid fluids to explore how external pressure differences, material properties and geometric factors influence the system's behaviour. To analyse the early stages of the evolution, we perform a linear stability analysis and obtain the growth rate and the critical pressure difference for the onset of the instability. A weakly nonlinear analysis suggests that the system can exhibit a pressure spike in the vicinity of the inverted configuration. [ABSTRACT FROM AUTHOR]

The aeroelasticity of a panel in the presence of a shock is a fundamental issue of great significance in the development of hypersonic vehicles. In practical engineering, cavity pressure emerges as a crucial factor that influences the nonlinear dynamical characteristics of the panel. This study focuses on the aeroelastic bifurcation of a flexible panel subjected to both cavity pressure and oblique shock. To this end, a computational method is devised, coupling a high-fidelity reduced-order model for unsteady aerodynamic loads with nonlinear structural equations. The solution is meticulously tracked by continuous calculations. The obtained results indicate that cavity pressure plays a pivotal role in determining the bifurcation and stability characteristics of the system. First, the system exhibits hysteresis behaviour in response to the ascending and descending dynamic pressures. The evolution of hysteresis behaviour originates from the phenomenon of cusp catastrophe. Second, variations in cavity pressure induce three types of bifurcation phenomena, exhibiting characteristics akin to supercritical Hopf bifurcation, subcritical Hopf bifurcation and saddle-node bifurcation of cycles. The system's response at the critical points of these bifurcations manifests as long-period asymptotic flutter or explosive flutter. Lastly, the evolution of the dynamical system among these three types of bifurcations is an important factor contributing to the discrepancies observed in certain research results. This study enhances the understanding of the nonlinear dynamical behaviour of panel aeroelasticity in complex practical environments and provides new explanations for the discrepancies observed in certain research results. [ABSTRACT FROM AUTHOR]

During the 2018 K $\unicode{x012B}$ lauea lower East Rift Zone eruption, lava from 24 fissures inundated more than 8000 acres of land, destroying more than 700 structures over three months. Eruptive activity eventually focused at a single vent characterized by a continuously fed lava pond that was drained by a narrow spillway into a much wider, slower channelized flow. The spillway exhibited intervals of 'pulsing' behaviour in which the lava depth and velocity were observed to oscillate on time scales of several minutes. At the time, this was attributed to variations in vesiculation originating at depth. Here, we construct a toy fluid dynamical model of the pond–spillway system, and present an alternative hypothesis in which pulsing is generated at the surface, within this system. We posit that the appearance of pulsing is due to a supercritical Hopf bifurcation driven by an increase in the Reynolds number. Asymptotics for the limit cycle near the bifurcation point are derived with averaging methods and compare favourably with the cycle periodicity. Because oscillations in the pond were not observable directly due to the elevation of the cone rim and an obscuring volcanic plume, we model the observations using a spatially averaged Saint-Venant model of the spillway forced by the pond oscillator. The predicted spillway cycle periodicity and waveforms compare favourably with observations made during the eruption. The unusually well-documented nature of this eruption enables estimation of the viscosity of the erupting lava. [ABSTRACT FROM AUTHOR]

In this paper, the predator–prey model with strong Allee and fear effects is considered. The existence of the equilibria and their stability are established. Especially it is found that there is an interesting degenerate point, which is a cusp point with codimension 2 or higher codimension, or an attracting (repelling)-type saddle-node, subject to different conditions. Then the Hopf bifurcation and its direction, the saddle-node bifurcation and the Bogdanov–Tankens bifurcation are further explored. Afterwards, with the help of the energy estimates and the Leray–Schauder degree, the nonexistence and existence of the nonconstant steady states of the model are presented. From the obtained results, we find that strong Allee effect will cause the per capita growth rate of prey species from negative to positive; both the fear and Allee effects could affect the existence of equilibria and bifurcations; meanwhile, the diffusion rates will affect the existence of the nonconstant steady states. [ABSTRACT FROM AUTHOR]

We studied the nonlinear dynamics of a shunted inductive Josephson junction coupled to a diode and a negative conductance. Taking into account the non-harmonicity of the junction, based on Kirchhoff's laws, we have developed the mathematical model which governs the dynamics of the circuit. The fixed points of the system are determined, and their stabilities are analyzed using the Routh–Hurwitz criterion. The bifurcation and transition to chaos of the model are studied using the the fourth-order Runge–Kutta method; the system displays a rich dynamics. The range of values of each parameter leading to periodic and chaotic electrical oscillations is obtained through the analysis of the effect of these parameters on each type of dynamics. Finally, the implementation by microcontroller is carried out in order to experimentally verify the different dynamics obtained numerically. [ABSTRACT FROM AUTHOR]

The traditional nonlinear aerostatic instability of long-span bridges is based on a two-layer iteration method that accurately predicts the structural equilibrium path before the critical buckling point. Due to strong nonlinearity after buckling, this traditional method cannot easily calculate the structural equilibrium and possible bifurcation using either Newton–Raphson or arc-length methods. In this study, a reduced order modeling (ROM) method for long-span bridge aerostatic deformation is proposed to approximate the bridge aerostatic equilibrium path after the critical point. The structural deformation mode shapes are extracted through singular value decomposition performed on the deformation matrix, and the nonlinear structural stiffness matrix is determined through the indirect displacement-based method on the finite-element method (FEM) platform. The ROM method is validated through comparison against the aerostatic deformation by the traditional two-layer iteration method based on FEM. By extending to higher wind speed, the ROM method can approximate the bridge deformation after initial buckling, and pitchfork bifurcation is observed after the structure undergoes rapid deformation growth. The stability of the equilibrium paths is examined through the Jacobian of restoring force vector, and the "snap-through" phenomenon exists for the equilibrium path before the bifurcation point. [ABSTRACT FROM AUTHOR]

This study on Lossy Nonlinear Electrical Transmission Line Models (LNETLM) highlights several novel contributions: (1) Introducing the modified G ′ G 2 -expansion method applied to LNETLM with a beta derivative, providing precise soliton solutions previously undocumented in the field; (2) Through extensive computer simulations, uncovering diverse wave phenomena such as bright, solitary, and parabolic solitons, alongside oscillatory singular waves, advancing the characterization of wave dynamics within LNETLM; (3) Contrasting conformable, M-truncated derivatives with the beta derivative, revealing the unique advantages of the beta derivative approach in modeling electrical transmission line systems; (4) Transforming the LNETLM equation into a Hamiltonian system, analyzing phase portraits, Chaos, and bifurcation phenomena, enhancing understanding of system dynamics and stability; (5) Expanding the knowledge frontier in lossy electrical transmission line models, with implications for theoretical understanding and practical applications in signal transmission and communication systems. In conclusion, this research underscores the potential of pulse-like solitons for enhancing data transmission rates in telecommunication systems. The insights gained pave the way for advancements in telecommunication technology, promising improved efficiency and performance. [ABSTRACT FROM AUTHOR]

The classical prey-predator model assumes that prey species has no life history. But many species whose individuals have a life history with two classes, namely immature and mature. Since, immature prey can be easily captured by the predators because of their short life experiences and less activity, while predators make some extra effort to catch mature prey. So, in this paper, a mathematical model is proposed to study the complex behaviour of a delayed prey-predator system with stage in prey population. We have included a maturation delay which is the time required for immature prey to join mature class and a cooperative hunting delay which is the time taken by predator to cooperate and attack on prey. Stability analysis has been addressed with different pairs of delays. Hopf bifurcation analysis with its directions is also performed. Global stability of predator free equilibrium point has been examined analytically and shown graphically. Both maturity and cooperative hunting delay significantly influence the system dynamics and produce multi-times stability switches. The presence of cooperative hunting delay of predators reduces the maturation time of prey due to which prey becomes mature very fast. Increase in cooperative hunting rate of predators suppress mature prey populations. Numerical simulations have been performed in various ways to support our theoretical findings. [ABSTRACT FROM AUTHOR]