Vol. 15, No. 1 (2026): Regular Issue

Mathematical models of 3D printers in Cartesian and Polar configurations driven by stepper motors, referred to henceforth as Unified Models, are formulated and derived as combinations of the basic building blocks. The building blocks are mathematical models of stepper motor, single-stage reduction gearbox, timing belt assembly, power screw assembly, turntable assembly and rack & pinion assembly. The Unified Models formulated have DC voltage pulse applied to the two phases of the Hybrid Bipolar stepper motor as the input and the displacement of the printing head or the printing table of the 3D printer as the output. The sin-cos micro-stepping voltage pulse profiles applied to the two phases of the motor are also derived to demonstrate the printer operation in micro-stepping mode. The operation uses an open loop paradigm in the absence of the encoder and phase current measurement feedback. The stepper motor velocity profile is controlled by changing the duration of the pulse or the micro-pulse. An exponential motion profile in a Speed-up Factor, which is the inverse of the pulse time duration, is also developed to facilitate the motion feasibility, as the traditional S-Curve Velocity profile used in closed loop paradigm does not work with the open loop control paradigm in stepper motors. The unified Motor-Gearbox-Mechanism models in Cartesian and Polar configurations, the Unified Models, are simulated with the designed micro-stepping voltage pulses to demonstrate various printing head and printing table motions encountered during 3D printing process. The Unified Models of Cartesian and Polar Configurations are also simulated in the closed loop paradigm to ascertain the best available controlled response. Lastly, the importance of the open and closed loop motion analysis and establishing the motion parameters for the Unified Models in the theoretical development of the non-traditional closed loop controller algorithm is discussed.

It has been investigated how proper predation and density dependent mortality affect chaotic dynamics in ecosystems using the Leslie-Gower type tritrophic food chain model. The model demonstrates how proper predation, in which predators choose lesser prey, can regulate population dynamics and avert food chain disruption. The system can also be stabilised by density-dependent mortality, which occurs when the mortality rate of prey rises as population density does. The model also shows that interactions between populations of predators and prey determine how effective these control mechanisms are. Proper predation and density-dependent mortality have a stronger effect on stability when the interaction is moderate, but their impact is lessened when the interaction is robust. The management of ecosystems and conservation efforts will be significantly impacted by these discoveries. We can contribute to the maintenance of stable population dynamics and avert population collapse by encouraging appropriate predation and density-dependent mortality in ecosystems. This emphasises how crucial it is to take into account the intricate relationships between predator and prey populations when managing and conserving ecosystems.

In this study, we examine the spatiotemporal synchronization behaviour of chaotic systems within the framework of a complex network. The network is characterized by dynamic coupling connections that change stochastically over time. Specifically, we consider a one-dimensional ring of coupled chaotic logistic maps, whose evolution is governed by the nearest-neighbour wave equation. Our investigation focuses on the effects of three key parameters: the degree of randomness $(p)$, the coupling strength $(\epsilon)$, and the system parameter $(r)$. Through both analytical and numerical approaches, we conduct a linear stability analysis of the synchronized steady states. Our analytical results are found to be in excellent agreement with the numerical simulations, providing a comprehensive understanding of the synchronization dynamics in this complex system.

The stability of dynamic systems under parametrically periodic excitations with a single frequency has been extensively investigated. However, parametrically arbitrary excitations with multiple frequencies are common in various sciences and engineering, including earthquakes and blasting. This paper proposes a new numerical method to study the stability of dynamic systems under parametric excitations with multiple frequencies. The critical step of the numerical method involves approximating the system with multiple frequencies by a system with a single principal frequency (or period) as closely as possible. Subsequently, a numerical algorithm is proposed to calculate both the state transition matrix on one principal period, which determines the dynamic stability, and the responses at any specific time. The efficiency and accuracy of the proposed numerical method are demonstrated. As an application example, the dynamic stability of a column under parametric loads with multiple frequencies is obtained through parametric studies involving the magnitude of incommensurate frequency, the number of frequencies, damping, and semi-rigid connections.

In this paper, we present a new method for nonlinear control and stabilization of chaos based on the synchronization technique. The idea behind this is to synchronize the slave chaotic system with a master stable system. The stability proof is carried out using the linearization method. Two cases of continuous and discrete systems are considered and the synchronization for stabilization is made between identical and different systems. Theoretical proofs and numerical simulations are given to demonstrate the efficiency of the proposed approach.

This paper investigates the blow-up phenomena of non-negative solutions of a two-species Keller-Segel chemotaxis model with the Lotka-Volterra competitive source terms under Neumann boundary conditions in a bounded domain $\Omega\subset\mathbb{R}^2$ with smooth boundary. We establish the results for the finite time blow-up of solutions when $\frac{\chi_1}{\alpha}=\frac{\chi_2}{\beta}$ for some positive constants $\alpha$ and $\beta$. The concavity method determines the main result in a two-dimensional space domain with a suitable auxiliary function. Also, the lower bounds for the finite time blow-up of solutions using the differential inequality techniques are estimated.

The flow of two dimensional Buogiorno nanofluid across an extending sheet having magnetic effect with a non-Darcy porous medium is explored analytically. By applying the transformation of similarity, the associated dimensional equations are simplified into dimensionless equations. With the aid of Ananthaswamy-Sivasankari approach and Modified Homotopy Analysis Technique, the expressions regarding the corresponding non-dimensional temperature, velocity, and concentration equations are attained. When the outcomes are compared to the numerical solution, a very excellent fit is found. Numerous controlling parameters including magnetic, porosity and thermophoresis are graphically depicted to reveal how they affect the flow. Furthermore, the physical parameters, especially~the reduced number of Sherwood, and the reduced number of Nusselt are visually displayed.

The current paper recovers a bright 1--soliton solution to the dispersive concatenation model that is considered with power--law of nonlinear self--phase modulation. The semi--inverse variational principle is applied to recover the soliton solution. The parameter constraints that naturally emerge from the analysis for the existence of bright solitons are also presented in the work.

We propose a prey-predator type model with four interacting species. We carry out a mathematical analysis of the proposed model followed by a numerical analysis. We plan to examine the dynamics of different populations in an interaction where the super predator has a diverse food source. The mathematical analysis first concerns the existence, bounding and stability (local and global using the Routh-Hurwitz criterion and the Lyapunov principle) of the solutions. In addition, we look for conditions under which solutions persist or die out. Finally, numerical simulations are carried out to illustrate the theoretical results.

Reline, being a sub--class of ionic liquids consisting of hydrogen bond acceptor (HBA) and hydrogen bond donor (HBD), exhibits properties such as low melting points, high thermal stability, and tunable properties. It is one of the deep eutectic solvents with the composition Choline Chloride--Urea and is also known as an environmentally friendly solvent in various chemical processes. This attribute of the reline aligns well with sustainable practices in the chemical industry as an impressive alternative to conventional solvents. Its application extends to some of the major fields that mankind is dependent upon, such as pharmaceuticals, biotechnology, energy storage, etc. On the other hand, single--walled carbon nanotubes provide a higher thermal conductivity and specific heat capacity, making them useful in various heat exchangers, thermal storage systems, metallurgical applications, etc. Hence, in this study, the nanofluid is formed by considering reline as the base fluid with SWCNT suspensions. The simulation is performed to understand the various aspects of reline--based nanofluid flow inside elliptic ducts. The study explores the impact of the Reynolds number, the aspect ratio of the duct, and the thermophysical properties of SWCNT$-$reline nanofluid on the system's behaviour. Navier--Stokes' equation is utilised to perform the Mathematical analysis to understand the flow and heat transfer behaviour of SWCNT$-$reline nanofluid. From the results, it was clearly observed that the velocity at the narrow region decreased as the pressure rose, and Reynold's number profile indicated the presence of turbulent flow behaviour.

In this paper, we studied the dynamical behaviour of Hassell-Varley model in the presence of non-linear harvesting and Richards' growth in prey population by using imprecise biological parameters. Moreover, anti-predator behaviour and discrete time delay due to gestation or digestion of the species are considered in the model system. The positivity and boundedness of the system are studied and the criterion for the extinction of the predator-prey populations are discussed. The criterion for the co-existence of prey and predator, and their stability is studied analytically. In addition, the effect of fear factor on the dynamics of the system is studied. Moreover sufficient condition for the hopf bifurcation was noted under consideration of discrete time delay and for other different biological parameters. Numerical simulations are presented to validate the analytical results obtained and with the outcome the biological relevance of the model is discussed.

A symbolic computation approach is employed to calculate rogue wave solutions in a bilinear equation with a controllable center, focusing on higher-order rogue waves of the generalized (2+1)-dimensional Kadomtsev-Petviashvili equation. The specific forms of the sixth- and seventh-order rogue waves of the KP equation have been obtained. The basic idea is to set the introduced parameters in the symbolic computation approach to 0 in order to simplify the calculation. Taking a third-order rogue wave as an example, some figures are given to shed light on the effect of the introduced parameters on the dynamic properties of the rogue waves by choosing appropriate values of the introduced parameters.

Considering the increasing virus spread in the society, it is important to understand the connection between respiratory illnesses, the prevalence of respiratory viruses, and meteorological conditions in various nations in order to effectively prepare hospital services for admissions. The paper addresses the fractional four dimensional chaotic system predicting respiratory diseases. The system is thoroughly analyzed by using dynamical tools of phase portraits, bifurcation diagrams, Lyapunov diagrams etc. Adaptive SMC method is applied for controlling chaos in presence of uncertainties and disturbances. Theoretical studies is verified numerically using MATLAB.

This study uses actual data from Pakistan to investigate an established mathematical framework that describes the dynamics and epidemiology of measles transmission. Sanitation and immunization are taken as mitigating strategies in the model. The primary model's six components are Susceptible, Recovered, Infected, Exposed, Hospitalized, and Vaccinated that can be addressed semi-analytically using the homotopy analysis approach. To show the influence of various model parameter categories including the frequency of hospitalized persons with measles visit due to complications, rate of vaccinating susceptible class and recruitment rate into susceptible class, graphical illustrations are provided. The outcomes show that this approach is the most practical, easy to use, and efficient. A satisfactory match is obtained by comparing the findings with the numerical simulation (MATLAB). This technique will be extended to tackle epidemic models especially, SIR, SEIR, SVIR, SVEIR,SLVEIR based on malaria, chikungunya, tuberculosis, HIV, hepatitis A virus, typhoid, Ebola, Cholera etc.

The nonlinear dynamics of hydraulic actuators, coupled with their inherent model uncertainty, present considerable challenges in achieving accurate motion control in hydraulically driven systems. This research proposes a sampling-based incremental nonlinear dynamic inverse control methodology designed to address the force tracking problem associated with hydraulic actuators. The proposed control strategy employs differential pressure derivatives as feedback and functions independently of a precise mathematical model of the hydraulic actuator or the necessity for parameter calibration, thereby demonstrating robust resilience to model uncertainty. A theoretical analysis is performed to evaluate the robustness of the incremental nonlinear dynamic inverse control approach in the presence of parameter uncertainty, and stability bounds critical to maintaining control performance are established. The validity of the proposed method is substantiated through comparative simulations with a primary focus on the force tracking aspect.