Numerical Diagrams Research Articles

Dynamic modeling of lytic virus transmission among phytoplankton driven by nitrogen and phosphorus

Lytic viruses infect phytoplankton and cause their mass mortalities. A dynamic model is proposed to explore lytic virus transmission among phytoplankton. The ecological reproductive index for the survival ability of phytoplankton and the basic reproduction number for the spreading ability of lytic viruses are established by analyzing dynamical properties of the model. We estimate parameter values of the model from Maat and Brussaard’s experimental data and theoretically examine Maat and Brussaard’s hypotheses that lytic virus transmission is driven by nitrogen and phosphorus. Our studies show that lytic viruses cannot spread among phytoplankton in low nitrogen or low phosphorus aquatic environments. The numerical diagrams also suggest that the increase in nitrogen and phosphorus during the spread of lytic viruses does not always cause phytoplankton blooms.

Phytoplankton-chytrid-zooplankton dynamics via a reaction-diffusion-advection mycoloop model.

Mycoloop is an important aquatic food web composed of phytoplankton, chytrids (one dominant group of parasites in aquatic ecosystems), and zooplankton. Chytrids infect phytoplankton and fragment them for easy consumption by zooplankton. The free-living chytrid zoospores are also a food resource for zooplankton. A dynamic reaction-diffusion-advection mycoloop model is proposed to describe the Phytoplankton-chytrid-zooplankton interactions in a poorly mixed aquatic environment. We analyze the dynamics of the mycoloop model to obtain dissipativity, steady state solutions, and persistence. We rigorously derive several critical thresholds for phytoplankton or zooplankton invasion and chytrid transmission among phytoplankton. Numerical diagrams show that varying ecological factors affect the formation and breakup of the mycoloop, and zooplankton can inhibit chytrid transmission among phytoplankton. Furthermore, this study suggests that mycoloop may either control or cause phytoplankton blooms.

Local and global dynamics of a functionally graded dielectric elastomer plate

We investigate the nonlinear vibrations of a functionally graded dielectric elastomer plate subjected to electromechanical loads. We focus on local and global dynamics in the system. We employ the Gent strain energy function to model the dielectric elastomer. The functionally graded parameters are the shear modulus, mass density, and permittivity of the elastomer, which are formulated by a common through-thickness power-law scheme. We derive the equation of motion using the Euler-Lagrange equations and solve it numerically with the Runge-Kutta method and a continuation-based method. We investigate the influence of the functionally graded parameters on equilibrium points, natural frequencies, and static/dynamic instability. We also establish a Hamiltonian energy method to detect safe regions of operating gradient parameters. Furthermore, we explore the effect of the functionally graded parameters on chaos and resonance by plotting several numerical diagrams, including time histories, phase portraits, Poincaré maps, largest Lyapunov exponent criteria, bifurcation diagram of Poincaré maps, and frequency-stretch curves. The results provide a benchmark for developing functionally graded soft smart materials.

Pareto Optimal Design of a Fuzzy Adaptive Hierarchical Sliding-mode Controller for an X-Z Inverted Pendulum System

ABSTRACT The control problem of a three-degree-of-freedom (3-DOF) X-Z inverted pendulum as an unstable, multi-input-multi-output, under-actuated and high-order nonlinear system is facing many challenges. In this paper, a Fuzzy Adaptive Hierarchical Sliding-Mode Controller (FAHSMC) is optimally designed for the X-Z inverted pendulum system utilizing the Multi-Objective Genetic Algorithm (MOGA). To this end, at first, the state variables of this system are converted to the new state variables via five steps of a forward transformation process. In this transformation, the dynamical equations of the X-Z inverted pendulum system are reorganized to an appropriate form for control implementations. Then, the novel FAHSMC is designed for stabilization and tracking control of the system, and the stability analysis of the controller is proved via the Lyapunov stability theory. After that, the state and control variables of the X-Z inverted pendulum are computed through six steps of a backward transformation process. Ultimately, the MOGA is exerted for Pareto optimal design of the proposed control approach applied on the regarded X-Z inverted pendulum system. The tracking error of the cart position in the X-direction, the tracking error of the cart position in the Z-direction and the angle error of the pendulum are selected as three inconsistent objective functions for multi-objective optimization operations. Numerical results and diagrams illustrate that the proposed approaches can accurately converge the system states to the desirable trajectories in a short time and yield superior Pareto optimal fronts in comparison with the Hierarchical Sliding-Mode Controller (HSMC) and the Adaptive Hierarchical Sliding-Mode Controller (AHSMC).

M-current induced Bogdanov\u2013Takens bifurcation and switching of neuron excitability class

In this work, we consider a general conductance-based neuron model with the inclusion of the acetycholine sensitive, M-current. We study bifurcations in the parameter space consisting of the applied current I_{app}, the maximal conductance of the M-current g_{M} and the conductance of the leak current g_{L}. We give precise conditions for the model that ensure the existence of a Bogdanov–Takens (BT) point and show that such a point can occur by varying I_{app} and g_{M}. We discuss the case when the BT point becomes a Bogdanov–Takens–cusp (BTC) point and show that such a point can occur in the three-dimensional parameter space. The results of the bifurcation analysis are applied to different neuronal models and are verified and supplemented by numerical bifurcation diagrams generated using the package MATCONT. We conclude that there is a transition in the neuronal excitability type organised by the BT point and the neuron switches from Class-I to Class-II as conductance of the M-current increases.

On the non-linear dynamics of torus-shaped and cylindrical shell structures

In this study, the non-linear dynamic analysis of torus-shaped and cylindrical shell-like structures has been studied. The applied material is assumed as the functionally graded material (FGM). The structures are considered to be used for important machines such as wind turbines. The effects of some environmental factors on the analysis like temperature and humidity have been considered. The strain field has been calculated in general form and in continue the dynamic governing equations of torus structure have been derived based on the first-order shear deformation theory. The rotation around two independent axes in the torus coordinate system is considered and time-dependent equations are solved using SAPM semi-analytical method. The stresses and deformations generated in the torus and cylindrical shaped structures are plotted. The rotation of structures has been attended due to some transportation purposes. The effect of internal pressures as well as rotational speed at torus and cylindrical structures has been investigated in several numerical diagrams. The results are presented in the form of graphs that consider the rotational effects, loading, thermal and humid (hygro-thermal) environments, and size of the structures. This research can provide scientific perspectives to researchers who will examine the dynamic analysis of torus and cylindrical shaped structures.

Magic Squares and Other Numerical Diagrams on the Chittagong Plaster Replicas in the David Eugene Smith Collection

Magic Squares and Other Numerical Diagrams on the Chittagong Plaster Replicas in the David Eugene Smith Collection

Influence of the higher-order effects on the solutions of the complex cubic–quintic Ginzburg–Landau equation

ABSTRACT The influence of intrapulse Raman scattering (IRS), self-steepening (SS) and third-order of dispersion (TOD) on the solutions of the complex cubic–quintic Ginzburg–Landau equation is studied by means of bifurcation analysis of the dynamical model. The numerical bifurcation diagram of the amplitude of the solution with respect to the nonlinear gain has revealed a cascade of bifurcations that leads to chaotic behaviour. We have found that the IRS leads to the larger shifts in positions and widths of the zones with different types of solutions in comparison to the influence of SS and TOD. Using numerical bifurcation diagrams of the amplitude of the solution with respect to the parameter describing IRS the following types of transformations have been identified: (a) from chaotic into two-periodic solution; (b) from two-periodic into a limit cycle; and (c) from a limit cycle into a stationary solution.

A dynamical model and bifurcation analysis for glucagon and glucose regulatory system

A new mathematical delayed model with two discrete time delays is presented for description of the behavior of the blood glucose regulation system. This study includes glucagon as a third effective variable. Sufficient conditions are obtained for local stability and the existence of Hopf bifurcation caused by variation of delays as the bifurcation parameters. Also we find some explicit formulas to determine the direction of the Hopf bifurcation and the stability of the bifurcated periodic solutions via the theory of normal form and center manifolds. Moreover, some numerical diagrams are illustrated for better understanding of the theoretical results.

Simulation of mixed-mode fracture (I-II) on PFRC specimens with various fibre proportions using an embedded cohesive crack model

The study of fibre-reinforced concrete (FRC) has become of increasing interest in the last decades. Although it is not a new technology, it has experienced a remarkable progress with the appearance of some recommendations in the standards. More specifically, the use of polyolefin fibres has proved to increase the tensile strength of concrete without the problems usually found with steel fibres, especially those related to corrosion. This type of fibres have been studied in depth and its fracture behaviour has been successfully simulated in the past by means of an embedded crack model using a trilinear softening function. Nevertheless, these simulations have been always focused on cases where fracture took place under pure mode I conditions, namely using the classical three-point bending test on notched specimens. In this study, such embedded crack model is used to reproduce the fracture behaviour on notched specimens subjected to a modified three-point bending test that induces fracture under a combination of modes I and II. Three PFRC mixes are analysed, all of them with the same proportions of concrete components but different proportions of polyolefin fibres. The experimental and numerical diagrams properly agree and allow identifying how the increasing proportion of fibres can be reflected in the trilinear softening function that numerically drives the damage evolution.

Bifurcation analysis in a delay model of IVGTT glucose-insulin interaction.

In this paper, a delayed differential model based on the intravenous glucose tolerance test is considered. The conditions to determine stability or instability of the model's steady state are obtained. We obtain the necessary conditions for the appearance of a bifurcation, and we investigate the direction and stability of the local bifurcation. For this purpose, the normal form theory is used. In addition, the numerical diagrams in the direction of theoretical results are drawn.

Heat equation model for rod and thin plate by partial q-difference operator

Heat equation model for homogeneous rod and thin plate based on Newton’s law of cooling is constructed by partial q-difference operator. The propagation of heat, the nature of material used and its corresponding solutions of heat equations are the focus of this paper. In particular, logarithmic solution for this heat equation model is arrived. Through numerical simulations and diagrams generated using MATLAB, solutions are validated and relevant applications are derived.

The Bifurcation of Two Invariant Closed Curves in a Discrete Model

A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

One- and multiconfigurational study of excited states of He atom in a small impenetrable cavity

The system of ten low-lying electronic states of a helium atom placed in the center of an impenetrable spherical cavity of radius R c is studied for the case of small cavities. The methods for numerical and qualitative study of the atomic systems in small cavities are analyzed. The problem is studied by the finite-difference variants of the restricted SCF, the simple restricted CI and explicitly correlated Hylleraas-type approximations (for S-states). The limited applicability of the SCF approximation for excited states is demonstrated. The importance of the relatively low-lying configurations 2p 2 and corresponding states 1S, 3P and 1D is confirmed by numerical calculations and correlation diagrams.

Weak gravitational lensing of quantum perturbed lukewarm black holes and cosmological constant effect

The aim of the paper is to study weak gravitational lensing of quantum (perturbed) and classical lukewarm black holes (QLBHs and CLBHs respectively) in the presence of cosmological parameter Λ. We apply a numerical method to evaluate the deflection angle of bending light rays, image locations θ of sample source and corresponding magnifications μ. There are no obtained real values for Einstein ring locations for CLBHs but we calculate them for QLBHs. As an experimental test of our calculations, we choose mass M of 60 types of the most massive observed galactic black holes acting as a gravitational lens and study quantum matter field effects on the angle of bending light rays in the presence of cosmological constant effects. We calculate locations of non-relativistic images and corresponding magnifications. Numerical diagrams show that the quantum matter effects cause absolute values of the quantum deflection angle to be reduced with respect to the classical ones. The sign of the quantum deflection angle is changed with respect to the classical values in the presence of the cosmological constant. This means dominance of the anti-gravity counterpart of the cosmological horizon on the angle of bending light rays with respect to absorbing effects of 60 local types of the most massive observed black holes. Variations of the image positions and magnifications are negligible when increasing dimensionless cosmological constant . The deflection angle takes positive (negative) values for CLBHs (QLBHs) and they decrease very fast (slowly) by increasing the closest distance x0 of bending light ray and/or dimensionless cosmological parameter for sample giant black holes with .

A stochastic lot sizing model with partial backordering and imperfect production processes

This research presents an inventory system with partial backordering and imperfect process with a stochastic numbers of products that are defective per order. This problem is known and there are some customers that do not wait for their orders to be fulfilled so a particular proportion of backordered items become lost sales. This paper considers both mentioned situations simultaneously while the number of defective items follows a uniform distribution and the proportion of backordering is constant. This condition is modelled. The cost function of this inventory model includes order cost, holding cost and two types of shortage costs, one of them is related to backordered items and the other one is related to lost sales. This paper also provides a solution method to obtain optimum values for the decision variables, order quantity and total shortage, then we derive the value of the total cost function according to the optimum values obtained for the decision variables. Finally, some numerical results and diagrams are provided to show how some parameters affect the values of the decision variables and the cost function.

Hopf-bifurcation Limit Cycles of an Extended Rosenzweig-MacArthur Model

<p>In this paper, we formulated a new topologically equivalence dynamics of an Extended Rosenzweig-MacArthur Model. Also, we investigated the local stability criteria, and determine the existence of co-dimension-1 Hopf-bifurcation limit cycles as the bifurcation-parameter changes. We discussed the dynamical complexities of this model using numerical responses, solution curves and phase-space diagrams.</p>

An EPQ model with reliability in an imperfect production system at the presence of two sale policies of prepayment and delay in payment

An economic production quantity model is developed in this paper for an imperfect production system. The conditions which are considered for this model involve inflation, deterioration, rising prices under inflation, and price-dependent demand. The demand becomes principally dependent on the time due to the dependency of the price on the time. In this model, it is assumed that in the specified conditions, production system shifts from in-control state to out of control state and produce imperfect items which must be reworked by spending a specified cost in order to be ready for sale. In order to reduce the imperfect items production, we must increase the system's reliability that requires further investment for improvement of production system. The proposed model finds out the optimal case among these costs by selecting the reliability as the decision making variable. The numerical example, diagrams and sensitivity analysis have been provided to illustrate the model.

Periodic and Chaotic Dynamic Responses of Face Gear Transmission System

The periodic and chaotic dynamic responses of face gear transmission system considering time-varying mesh stiffness and backlash nonlinearity are studied. Firstly, a nonlinear time-varying dynamic model of face gear pair is developed and the motion equations are presented, the real accurate mesh stiffness is obtained by applying Finite element approach. Then, the dynamic equations are solved using Runge-Kutta numerical integral method and bifurcation diagrams are presented and analyzed. The stability properties of steady state responses are illustrated with Floquet multipliers and Lyapunov exponents. The results show that a process of periodic-chaotic-periodic motion exists with the dimensionless pinion rotational frequency as control parameters. The analysis can be a reference to avoid the chaotic motion and unstable periodic motion through choosing suitable rotational frequency.

Dynamics and circuit implementation of three simplified chaotic systems

To improve the performance of chaotic secure communication, three simplified chaotic systems with one variable parameter were investigated. Basic properties were analyzed including symmetry, dissipation and topological structure. Complex dynamical behaviors of the systems including chaos and periodic orbits were verified by numerical simulations, Lyapunov exponents and bifurcation diagrams. Interestingly, the three systems were integrated in a common circuit, and their dynamical behaviors were easily observed by adjusting regulable resistors R28, R14 and R17, respectively, and the relations between the variable resistor and the system parameter were deduced. The circuit experiment results agree well with the simulation results. Finally, a secure communication scheme based on chaos shift keying (CSK) was presented, which lays an experiment foundation for chaotic digital secure communication.