Set Of Nonlinear Differential Equations Research Articles

Magnetoelastic axisymmetric multi-modal resonance and Hopf bifurcation of a rotating circular plate under aerodynamic load

In this article, an investigation of magnetoelastic axisymmetric multi-mode interaction and Hopf bifurcations of a circular plate rotating in air and uniform transverse magnetic fields is presented. The expressions of electromagnetic forces and an empirical aerodynamic model are applied in the derivation of the dynamical equations, through which a set of nonlinear differential equations for axisymmetric forced oscillation of the clamped circular plate are deduced. The method of multiple scales combined with the polar coordinate transformation is employed to solve the differential equations and achieve the phase–amplitude modulation equations for the interaction among the first three modes under primary resonance. Then, the frequency response equation for the single-mode vibration, the steady-state response equations for three-mode resonance and the corresponding Jacobian matrix are obtained by means of the modulation equations. Numerical examples are presented to show the dependence of amplitude solutions as a function of different parameters in the cases of single mode and three-mode response. Furthermore, a Hopf bifurcation can be found in three-mode equilibrium by choosing appropriate parameters, where a limit cycle occurs and then evolves into chaos after undergoing a series of period-doubling bifurcations.

Anomalous diffusion in rotating Casson fluid through a porous medium

This paper investigates the space-fractional anomalous diffusion in unsteady Casson fluid through a porous medium, based on an uncoupled continuous time random walk. The influences of binary chemical reaction and activation energy between two horizontal rotating parallel plates are taken into account. The governing equations of motion are reduced to a set of nonlinear differential equations by time derivatives discretization and generalized transformation, which are solved by bvp4c and implicit finite difference method (IFDM). Stability and convergence of IFDM are proved and some numerical comparisons to the previous study are presented with excellent agreement. The effects of involved physical parameters such as fractional derivative parameter, rotation parameter and time parameter are presented and analyzed through graphs. Results indicate that the increase of fractional derivative parameter triggers concentration increase near the lower plate, while it causes a reduction near the upper plate. It is worth mentioning that the decrease of heat transfer rate on the plate is observed with the higher time parameter.

MODELLING DSB REPAIR KINETICS FOR DNA DAMAGE INDUCED BY PROTON AND CARBON IONS.

Proton and carbon therapy are the main choices of particle therapy for cancer treatment. Particle dose distribution is superior to conventional photon therapy dose distribution due to Bragg peak. However, the basic biology of cellular damage and cell death is not well understood. The aim of this work is to present a mechanistic model of double strand break (DSB) repair that predicts the repair kinetics of damage induced by particles employed in cancer therapy. Monte Carlo Track Damage Simulation (MCDS) was employed to model DNA damage. The frequency of DSB and SSB was computed for proton and carbon ions. DSBs were subjected to repair model to calculate the repair kinetics. Two distinct DSB repair models dependent on the cell cycle were proposed. The DSB repair model contains non-homologous end joining (NHEJ), homologous recombination (HR) and back up non-homologous end joining (B-NHEJ) repair processes. The DSB complexity results in the switch in the repair pathway from NHEJ to a slower process that starts with DSB end resection. DSB end resection in early S and G1 phases of the cell cycle enhances the B-NHEJ repair pathway, while in late S and G2 phases of the cell cycle promotes HR repair pathway. The repair model was transformed to a set of nonlinear differential equations. The model calculates the overall repair kinetics and protein temporal repair activity at the site of damage. The damage and repair model provides a detailed mechanistic understanding of all processes that are involved in the damage induction and repair. The number of DSB and their complexity increase as the particle energy decreases due to the proximity of particle interactions in water. The repair kinetics show a biphasic behaviour that is due to the NHEJ fast repair of simple type DSB and HR slow repair of complex type DSB.

SCATTERING OF A THIN LAYER OVER A NONLINEAR RADIALLY EXTENDING SURFACE WITH MAGNETO HYDRODYNAMIC AND THERMAL DISSIPATION

The recent research is allied with the analysis of a thin layer, spreading over the nonlinear surface of a radially extended sheet. The temperature field has been taken with the accumulation of dissipation term. The similarity variables have been used to transform the basic flow equations into a set of nonlinear differential equations. The thickness of the spreading phenomenon has been taken as a variable. The approximate outcomes of the problem have been achieved using the optimal approach of the homotopy analysis method (HAM). The convergence of the HAM has been computed with numerical method. The impact of the variable thickness parameter [Formula: see text], generalized magnetic parameter [Formula: see text], Eckert number [Formula: see text] and [Formula: see text] on the spreading pattern and temperature field has been calculated and discussed. The attention has been paid to the important physical quantities of interest like the skin friction and Nusselt number under the effect of various embedded parameters.

Temporal Market Evolution of Interactive Games

Massive multiplayer online games (MMOG) are the most commercially successful computer game genre during the last two decades. Many MMOGs base their business model on monthly subscription fees. Hence, future prediction of the number of players subscribing to a given MMOG is a crucial parameter for estimating the success and valuation of the MMOG. This paper develops analytical models that estimate the number of users playing a given MMOG. The models include three categories of games: (1) simple games, in which the popularity of the game is independent of other games offered in the market; (2) competitive games, in which the player may choose between the same or similar games from different suppliers; (3) complementary games, in which one of the games is complementary to another game increasing (or reducing) its popularity. Analytic solutions are found for some simple games. The paper draws some general conclusions concerning the temporal evolution of games, for example, that there are games in which all potential players will join the game at some time and that there are other games in which some potential players never will join the game. Another general observation is that there are games where there must exist initial players to trigger the game to start growing. Some of the models with analytical tractable solutions are too simple to predict the exact temporal evolution of real games. Therefore, the paper shows how the more advanced models may be analyzed using system dynamic simulation models which accurately convert the set of differential equations describing the various games into simple simulation models. In fact, the system dynamic model solves the set of differential equations for the various games without resorting to intractable numerical solution methods for large sets of nonlinear differential equations. The paper also demonstrates that the real evolution of games such as World of Warcraft and RuneScape can be reproduced using system dynamics. This implies that system dynamics may be used to analyze different strategies before the game is marketed or for enhancing the market penetration of an ongoing game.

Fault Diagnosis in Nonlinear Hybrid Systems

Abstract The problem of fault diagnosis in hybrid systems is investigated. It is assumed that the hybrid systems under consideration consist of a finite automaton, a set of nonlinear difference equations and the so-called mode activator that coordinates the action of the other two parts. To solve the fault diagnosis problem, hybrid residual generators based on both diagnostic observers and parity relations are used. It is shown that the hybrid nature of the system imposes some restrictions on the possibility of creating such generators. Sufficient solvability conditions of the fault diagnosis problem are found. Examples illustrate details of the solution.

The Influence of Stiffness of Elastic Restraint in a Damaged Regular System on Its Resonant Vibration

The paper summarizes the findings of a study of the influence of stiffness parameters of an elastic restraint in a damaged regular system consisting of two duplicate elements and modeling a two-blade assembly, on the generation of vibration of the system under fundamental resonance. A discrete dual-mass model whose vibration is described by a nonlinear set of differential equations of second order has been chosen as a design model of the structurally regular system under consideration. Numerical investigations have been performed using the Runge-Kutta method to solve a nonlinear set of differential equations and the fast Fourier transform method to process the solutions obtained. The dependences of the resonant frequency ratios and resonant amplitude ratios of the first-harmonic vibration on the damage parameter have been derived for the discrete model of the regular system in inphase and antiphase modes, with different parameters of the elastic restraint for the damaged subsystem and for the intact one. It is demonstrated that with a preset value of the damage parameter the frequencies of the inphase and antiphase modes grow with increasing coefficient of the elastic restraint of the subsystems. This influence is the most significant in the case of the antiphase mode of vibration of the subsystems. The elastic restraint coefficient has also a substantial effect on the level of resonant amplitudes of the first-harmonic vibration for the regular-system model under study. It has been found out that as the elastic restraint coefficient grows the resonant amplitude of the intact subsystem increases under the inphase-mode vibration and decreases under the antiphase one. The values of the resonant amplitudes of vibration of the damaged subsystem are influenced by the coefficient to a lesser extent. The maximum value of the resonant amplitudes of vibration of the damaged subsystem is almost independent of the elastic restraint coefficient.

Impact of Internal Heat Source on Mixed Convective Transverse Transport of Viscoplastic Material under Viscosity Variation

This communication addresses the impact of heat source/sink along with mixed convection on oblique flow of Casson fluid having variable viscosity. Similarity analysis has been utilized to model governing equations, which are simplified to set of nonlinear differential equations. Computational procedure of shooting algorithm along with 4th order Range-Kutta-Fehlberg scheme is opted to attain the velocity and temperature distributions. Impact of imperative parameters on Casson fluid flow, temperature, significant physical quantities such as skin friction, local heat flux and streamlines are displayed via graphs.

Spherically symmetric charged black hole in conformal teleparallel equivalent of general relativity

We studied 4-dimensional non-charged and charged spherically symmetric spacetimes in conformal teleparallel equivalent of general relativity. For this aim, we apply the field equations of non-charged and charged to diagonal and non-diagonal vierbeins and derive their sets of non-linear differential equations. We investigate in details that the Schwarzschild, for the non-charged case, and the Reissner-Nordström, for the charged case, are the only black hole solutions for the spherically symmetric case in the frame of conformal teleparallel equivalent of general relativity theory. Our conclusion indicates that the scalar field in the conformal teleparallel equivalent of general relativity theory has no effect for the spherically symmetric manifold.

An optimal analysis of radiated nanomaterial flow with viscous dissipation and heat source

The present work explores the analytical study of nanofluid flow above a stretching medium with the heat source and viscous dissipation. Additional radiative effects are also incorporated. The main physical problem is offered and changed into an arrangement of combined nonlinear differential equations with appropriate transformations. Optimal homotopy analysis method is used to attain the analytical solutions of the set of nonlinear differential equations. Important predictions of the flow phenomena are explored and deliberated by means of graphs and numerical tables. Moreover, the accurateness of the existing findings is verified by equating them with the previously available work.

Stability analysis on the flow and heat transfer of nanofluid past a stretching/shrinking cylinder with suction effect

Abstract The steady boundary layer flow over a stretching/shrinking cylinder with suction effect is numerically studied. Using a similarity transformations, the governing partial differential equations are transformed into a set of nonlinear differential equations and have been solved numerically using a bvp4c code in Matlab software. The nanofluid model used is taking into account the effects of Brownian motion and thermophoresis. The influences of the governing parameters namely the curvature parameter γ, mass suction parameter S, Brownian motion parameter Nb and thermophoresis parameter Nt on the flow, heat and mass transfers characteristics are presented graphically. The numerical results obtained for the skin friction coefficient, local Nusselt number and local Sherwood number are thoroughly determined and presented graphically for several values of the governing parameters. From our investigation, it is found that the non-unique (dual) solutions exist for a certain range of mass suction parameter. It is observed that as curvature parameter increases, the skin friction coefficient and heat transfer rate decrease, meanwhile the mass transfer rates increase. Moreover, the stability analysis showed that the first solution is linearly stable, while the second solution is linearly unstable.

Novel relationships between some coordinate systems and their effects on mechanics of an intrinsically curved filament

The fixed frame, Frenet-Serret frame and generalized Frenet-Serret frame are commonly used coordinate systems in the study of a filament or a moving rigid body. In terms of Eulerian angles, we derive some relations in these frames and apply these relations to find some significant results. Especially, we find the angle between the normal of centerline of a filament and the line of nodes which is the crossover between the horizontal plane of fixed frame and normal plane of centerline. We prove that the general solution of a set of nonlinear differential equations represents a circular helix or the corresponding filament has a unique helical ground-state configuration. We show that the effective description of a planar filament depends on the value of its torsional modulus. Finally, we find the expression of energy for a three-dimensional intrinsically curved filament when its cross-section area vanishes, and show that under an applied force the finite intrinsic curvature alone can induce a discontinuous transition in extension.

Dynamic interaction between adjacent buildings through nonlinear soil during earthquakes

This paper evaluates the effect of Structure-Soil-Structure Interaction (SSSI) between two buildings given different parameters of the buildings, inter-building spacing, and soil type. A two-dimensional simple discrete nonlinear model is proposed that is described by a set of nonlinear differential equations of motion. A nonlinear phenomenological Bouc-Wen model, for the soil directly underneath the foundations, linear rotational interaction spring between buildings and linear behaviour of buildings are assumed. The seismic ground motion employed is spectrally matched with EC8 elastic spectra. The results showed that there are both unfavourable and beneficial configurations of the two buildings that produce important differences between nonlinear SSSI and nonlinear SSI (the uncoupled building case). Importantly it is demonstrated that the adverse effects of SSSI can be more pronounced when the nonlinear soil behaviour is assumed.

Dynamical phase diagrams of a love capacity constrained prey\u2013predator model

One interesting question in love relationships is: finally, what and when is the end of this love relationship? Using a prey–predator Verhulst–Lotka–Volterra (VLV) model we imply cooperation and competition tendency between people in order to describe a “love dilemma game”. We select the most simple but immediately most complex case for studying the set of nonlinear differential equations, i.e. that implying three persons, being at the same time prey and predator. We describe four different scenarios in such a love game containing either a one-way love or a love triangle. Our results show that it is hard to love more than one person simultaneously. Moreover, to love several people simultaneously is an unstable state. We find some condition in which persons tend to have a friendly relationship and love someone in spite of their antagonistic interaction. We demonstrate the dynamics by displaying flow diagrams.

Об устойчивости репрессилятора с запаздывающей экспрессией генов

The repressor is the first genetic regulatory network in synthetic biology, which was artificially constructed at Harvard University in 2000. It is a closed network of three genetic elements lacI , λ cI and tetR , which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In this paper, the nonlinear dynamics of a repressilator, which has time delays in all parts of the regulatory network, has been studied for the first time. Delay can be both natural, i.e. arises during the transcription / translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using synthetic biology technologies. It is assumed that the regulation is carried out by proteins being in a dimeric form. In the paper, the nonlinear dynamics has been considered within the framework of the deterministic description. By applying the method of many time scales, the set of nonlinear differential equations with delay on a slow manifold has been obtained. It is shown that there exists a single equilibrium state which loses its stability in an oscillatory manner at certain values of the control parameters. For a symmetric repressilator, in which all three genes are identical, an analytical solution for the neutral Andronov-Hopf bifurcation curve has been obtained. For the general case of an asymmetric repressilator, neutral curves are found numerically. The place of the model proposed in the present work among other theoretical models of the repressilator is discussed.

Fault Diagnosis in Hybrid Systems by Nonparametric Method

The problem of fault diagnosis in hybrid systems is investigated. The hybrid systems under consideration consist of finite automaton, the set of nonlinear difference equations and so-called mode activator that coordinates the action of these two parts. The hybrid residual generator based on the nonparametric method is found that solves the fault diagnosis problem under certain (sufficient) solvability conditions. Examples illustrate details of the solution.

Identifying Elastic and Viscoelastic Material Parameters by Means of a Tikhonov Regularization

For studying the interaction of displacements, stresses, and acting forces for elastic and viscoelastic materials, it is of utmost importance to have a decent mathematical model available. Usually such a model consists of a coupled set of nonlinear differential equations together with appropriate boundary conditions. However, since the different material classes vary significantly with respect to their physical and mechanical behavior, the parameters which appear in these equations are unknown and therefore have to be determined before the equations can be used for further investigations or simulations. It is this very step which is addressed in this article where we consider elastic as well as viscoelastic material behavior. The idea is to compute the parameters as solutions of a minimization problem for Tikhonov functionals. Tikhonov regularization is a well-established solution technique for tackling inverse problems. On the one hand, it assures a computation that is stable with respect to noisy input data, and on the other hand, it involves desired a priori information on the solution. In this article we develop problem adapted Tikhonov functionals and prove that a Tikhonov regularization improves the accuracy especially when the underlying system is ill-conditioned.

A new approach for nonlinear vibration analysis of thin and moderately thick rectangular plates under inplane compressive load

In this study, a hybrid method is proposed to investigate the nonlinear vibrations of pre- and post-buckled rectangular plates for the first time. This is an answer to an existing need to develope a fast and precise numerical model which can handle the nonlinear vibrations of buckled plates under different boundary conditions and plate shapes. The method uses the differential quadrature element, arc-length, harmonic balance and direct iterative methods. The governing differential equations of plate vibration have been extracted considering shear deformations and the initial geometric imperfection. The solution is assumed to be the sum of the static and dynamic parts which upon inserting them into the governing equations, convert them into two sets of nonlinear differential equations for static and dynamic behaviors of the plate. First, the static solution is calculated using a combination of the differential quadrature element method and an arc-length strategy. Then, putting the first step solutions into the dynamic nonlinear differential equations, the nonlinear frequencies and modal shapes of the plate are extracted using the harmonic balance and direct iterative methods. Comparing the obtained solutions with those published in the literature confirms the accuracy and the precision of the proposed method. The results show that an increase in the nonlinear vibration amplitude increases the nonlinear frequencies.

Mathematical insights on psoriasis regulation: Role of Th₁ and Th₂ cells.

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th₁), Type 2 (Th₂) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th₁-Cells viz. Interferon gamma (IFN-γ), Transforming growth factor beta (TGF-β) and Tumor necrosis factor alpha (TNF-α) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 (IL-10), Interleukin 4 (IL-4) etc., released by Th₂-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / IL-10) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

On the Relation between Chemical Oscillations and Self-Replication.

One proposed scenario for the emergence of biochemical oscillations is that they may have provided the basic mechanism behind cellular self-replication by growth and division. However, alternative scenarios not requiring any chemical oscillation have also been proposed. Each of the various protocell models proposed to support one or another scenario comes with its own set of specific assumptions, which makes it difficult to ascertain whether chemical oscillations are required or not for cellular self-replication. This article compares these two cases within a single whole-cell model framework. This model relies upon a membrane embedding a chemical reaction network (CRN) synthesizing all the cellular constituents, including the membrane, by feeding from an external nutrient. Assuming the osmolarity is kept constant, the system dynamics are governed by a set of nonlinear differential equations coupling the chemical concentrations and the surface-area-to-volume ratio. The resulting asymptotic trajectories are used to determine the cellular shape by minimizing the membrane bending energy (within an approximate predefined family of shapes). While the stationary case can be handled quite generally, the oscillatory one is investigated using a simple oscillating CRN example, which is used to identify features that are expected to hold for any network. It is found that cellular self-replication can be reached with or without chemical oscillations, and that a requirement common to both stationary and oscillatory cases is that a minimum spontaneous curvature of the membrane is required for the cell to divide once its area and volume are both doubled. The oscillatory case can result in a greater variety of cellular shape trajectories but raises additional constraints for cellular division and self-replication: (i) the ratio of doubling time to oscillation period should be an integer, and (ii) if the oscillation amplitude is sufficiently high, then the spontaneous curvature must be below a maximum value to avoid early division before the end of the cycle. Because of these additional stringent constraints, it is likely that early protocells did not rely upon chemical oscillations. Biochemical oscillations typical of modern evolved cells may have emerged later through evolution for other reasons (e.g., metabolic advantage) and must have required additional feedback mechanisms for such a self-replicating system to be robust against even slight environmental variations (e.g., temperature fluctuations).