Vol. 12, No. 4 (2023): Regular Issue

This paper investigates the presence of mild solutions and controllability for higher order Caputo fractional damped differential systems with impulses. The general method of Laplace transform, Grammian matrix, and sequential approximation techniques are used to establish the controllability requirement for the linear and nonlinear fractional damped dynamical systems of higher order. At the end, an example is provided to validate the theoretical results.

Despite stupendous advancement of medical science, yet mankind gets perplexed when it comes to cancer. As yet cancer poses substantial threat to life as it is lethal in some cases due to its complexity and heterogeneity. With the objective of increasing the potency of cancer treatment, scientists are now focussing on combination therapy such as chemovirotherapy. In the current study, an updated and realistic mathematical model embracing different facets like uninfected tumour cells, infected tumour cells, free virus particles, chemotherapeutic agent, tumour specific immune cells and virus specific immune cells is advocated. In addition to mutual interactions between cells, diffusion phenomenon plays a vibrant role on account of their mobility. All these biological and physical processes are embodied in the novel mathematical model. Stability analysis corresponding to the temporal system along with its sub-models undergoing comparative study is performed. Suitable numerical methods are adopted for the model outcome followed by their exhaustive delineations. Spatial distributions are visualized using graphical manifestations. Sensitized parametric variation is illustrated pictorially. The study concludes that with proper management of model parameters so that cancerous tumours can be eradicated from the body using chemovirotherapy effectively.

In this paper, the purpose of the work is to modified of an Eco-epidemiocal model and analysis with the optimal control strategies for the infected prey. Furthermore, the dynamical behaviors, i.e. the positivity, boundedness, stability and Hopf bifurcation of the proposed model are investigated. The optimal control theory is applied to control the effect of disease on prey population and plays a vital role to eliminate disease from prey population.The results obtained suggested the optimal control is ensuring the prey-predator populations coexists in a defined habitat.

It is known, there is a close connection between discrete Lotka-Volterra systems and tournaments. In the paper, we give the concept of a homogeneous tournament, which allows us to obtain new results on the location of fixed points of the Lotka-Volterra systems on the simplex.

In this article, we formulate a predator-prey interaction model among phytoplankton, zooplankton and fish species. It is assume that phytoplankton species is infected by a disease and due to infection, phytoplankton population is divide into two subpopulation such as susceptible phytoplankton and infected phytoplankton. It is consider that zooplankton consume susceptible as well as infected phytoplankton and fish consume only zooplankton. Here, it is assume that the plankton population releases some toxin which make some death of zooplankton. We also study the existence of Hopf bifurcation for the present model with respect to the disease infection rate. It is observe that increase rate of infection of phytoplankton may cause the extinction of zooplankton as well as fish species. It is find that the increase of rate of consumption of susceptible phytoplankton by zooplankton may make the system stable. The increase rate of release of toxin by phytoplankton may make the system unstable. The proposed system may continue stable steady state behaviour due to the increase of conversion rate of zooplankton in fish species. Chaotic dynamics is observe, due to the addition of diffusion term in the present model.

The aim of this paper is to present an analytical analysis of nonlinear random vibration behaviors of micro-beams resting on a visco-elastic foundation. The modifìed couple stress theory and the Euler-Bernoulli beam theory (EBT) with the von-Kármán's geometrical nonlinearity are employed for this analysis. The equation of motion of the micro-beam is established based on the Hamilton's principle. The input excitation is assumed to be a Gaussian process with zero mean. The mean-square of the micro-beam's displacement is found by the regaluted equivalent linearization method. Comparison of the obtained solutions with the published solutions and the classical solutions shows the accuracy. The influences of the material length scale parameter (MLSP), the spectral density of the input excitation and the coefficients of the visco-elastic foundation on the nonlinear random vibration response of micro-beams are investigated in detail.

Plasma confinement in fusion devices like Tokamaks depends on the existence of closed magnetic field lines with toroidal geometry. The magnetic field line structure in toroidal plasma devices is a Hamiltonian system, where the role of time is played by an ignorable coordinate. Nonsymmetrical perturbations lead to a nonintegrable hamiltonian system that can exhibit area-filling chaotic orbits. If exits are suitably positioned on a chaotic magnetic field line region, the Hamiltonian system becomes open and one is interested to know the corresponding escape basins, i.e., the sets of initial conditions for which the corresponding field lines escape through a given exit. From general mathematical arguments, it can be shown that these escape basins are fractal. In this paper, we investigate quantitatively fractal escape basins in the magnetic field line structure in Tokamaks described by an area-preserving map proposed by Balescu et al, using the uncertainty dimension to characterize the fractal structure of the magnetic field lines. We also use the concept of basin entropy in order to quantify the final state uncertainty, a relevant issue that arises when fractal basins are involved.

A basic simple cancer model is studied to see the effectiveness of a therapeutic approach to control or reduce the cancer cells through a specific virus injecting into the patient's tumour site. Existence, uniqueness, positivity, persistency, boundedness of the system's solutions are established. The system's local and global behaviour about all possible equilibria are studied. Bifurcation analysis of the system is discussed as well. The basic reproduction number $\mathcal{R}_0$ is calculated using Next Generation Matrix method, which plays a significant role in the disease endemicity. We have analyzed the centre manifold near the tumour-free equilibrium point $E_0$, $\mathcal{R}_0=1$. For the proposed cancer model system, it is shown that there exists an optimal control of the virus replication rate. The characterization of the optimal control is explained as well. The optimality of the viral cytotoxicity is also studied. Numerical simulations are presented to validate the analytical findings. Finally, we conclude some epidemiological remarks made through analytical and numerical observations.

An innumerable number of phenomena that take place in nature can be represented as dynamical systems, which in many cases are not linear. One of the common tasks performed in the study of these systems is to analyse through time and frequency the relationships among their components. In this work, we present, discuss, and extend for the first time in the study of nonlinear dynamical systems, a mathematical and computational tool, the wavelet local multiple correlation (WLMC) to analyse quantitatively and visually the behaviour among components of nonlinear dynamical system. The Lorenz system is used as a case study. The WLMC analysis presented shows that the WLMC is able to capture the most relevant periodic and chaotic dynamics of the Lorenz system as well as the "dominant" components of this dynamical system. These results confirm that the WLMC is an adequate mathematical tool to analyse nonlinear and chaotic dynamical systems with multiple components.

In this study, we examined the nonlinear longitudinal wave equation in a magneto-electro-elastic circular rod of fourth order. Firstly, bifurcation behavior was examined by the help of a planar dynamical system to analyze the bifurcation structures of traveling wave solution forms of the considered equation. All possible phase portraits are exposed with some parameter assignment. Secondly, generalized Kudryashov and auxiliary equation methods were used to investigate the traveling wave solutions for this equation. The solutions of the longitudinal wave equation types as solitary waves, periodic traveling waves, periodic cusp wave solutions are discussed. The existence of these solutions was also demonstrated by bifurcation analysis. The physical properties of these longitudinal waves were trying to be discovered by drawing dynamic analysis of these systematically produced solutions.

This article is a generalized study of the dynamics of a delayed Solow model with a structured population. Within this framework, we investigate the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as a $\tau$-delay passes through critical values. Then using the standard form theory and center form reduction, we drive an explicit algorithm that allows determining the direction of Hopf bifurcation and the stability of the resulting periodic solutions. We give some numerical examples to motivate the proposal and illustrate our results.

This research paper studies a new chaotic system modified from Lorenz system. Firstly, the stability of the fixed points are studied. then, the dynamics of the novel system are investigated using bifurcations diagrams, Lyapunov exponents, 0-1 test, as well as Poincare sections. Furthermore, we use the three previously mentioned methods and see how our system exhibit a chaotic behavior from doubling of period. In addition to that, comparison of our proposed system with 30 other systems are implemented in order to see their advantage in secure communication using Kaplan-York dimension. In the second part of this paper, we implement an identical synchronization in less than $0.4$ seconds via an adaptive control law. Moreover, any results we get analytically will be checked by Matlab simulation.