Chaos control applied to heart rhythm dynamics

The dynamics of cardiovascular rhythms have been widely studied due to the key aspects of the heart in the physiology of living beings. Cardiac rhythms can be either periodic or chaotic, being respectively related to normal and pathological physiological functioning. In this regard, chaos control methods may be useful to promote the stabilization of unstable periodic orbits using small perturbations. In this article, the extended time-delayed feedback control method is applied to a natural cardiac pacemaker described by a mathematical model. The model consists of a modified Van der Pol equation that reproduces the behavior of this pacemaker. Results show the ability of the chaos control strategy to control the system response performing either the stabilization of unstable periodic orbits or the suppression of chaotic response, avoiding behaviors associated with critical cardiac pathologies.     

An analysis of a vacuum diode model

One of the vacuum diode models is written as a singular boundary value problem for two ODEs of the second order with one parameter. The parameter is unknown and must be found along with the initial conditions at the origin to satisfy the boundary conditions at the end of the interval. The problem presents considerable difficulties in its theoretical and, especially, numerical study. Here we give a complete analytical and numerical solution of this problem.     

Qualitative analysis of a mathematical model for tissue inflammation dynamics

The non-linear differential eguation $$\dot X = \frac{{rX}}{{I + X}} - \frac{{XY}}{{X + Y}},\dot Y = \alpha (I + \theta X - Y)$$ is studied. It is the main aim of this paper to show the existence of bifurcation of saddle connection type; and to show the creation of limit cycles under certain conditions of the parameters, together with their biological significance.     

Chaotic dynamics of a simple oscillator — a pictorial introduction

That seemingly simple mechanical oscillators show already many general phenomena including local and global bifurcations as well as chaotic behavior makes their study so important in the theory of nonlinear dynamics. A considerable body of knowledge has been developed in recent years for analyzing and evaluating this variety of behavior. By studying a Duffing oscillator some well approved techniques are demonstrated and discussed. The presentation is highly illustrated to improve understanding.     

An analytical model for conflict dynamics

A coherent dynamic conflict model is developed from basic principles. The governing equations have a striking resemblance to the continuity equation in fluid dynamics with an additional term for the response to pressure by the opponent. The salient feature of the model is a moving confrontation line which is an excellent indicator for the evolution of conflict. The developed model also permits investigation of the necessary minimum involvement of a third party actor such as an international organization to establish a status quo between the actors. The model is demonstrated on the Russian–Chechen conflict and the Bosnian war.     

An improved model of age-structured population dynamics

The classical model of age-dependent population dynamics is improved. Instead of the traditional renewal equation, a new approach is developed to describe the reproduction process of the population. The composition of a population is redefined to contain the pre-birth individuals, and the disadvantages of the classical model avoided. Moreover, the improved model turns out to be an initial value problem, which is mathematically more convenient to deal with. Existence and uniqueness results for the nonlinear nonautonomous system of model equations are obtained. It is shown that the classical model and its time delay generalization are two degenerate cases of the improved model.     

Graphene-based mass sensors: Chaotic dynamics analysis using the nonlocal strain gradient model

Nonlinear oscillations of graphene resonators are unavoidable due to enhancing the mass sensitivity of graphene-based mass sensors and the nonlinear behavior of the systems provides the route to chaos. In this paper, the nonlinear and chaotic behavior of the graphene-based mass sensor is investigated. The nano-mechanical sensor includes an electrostatically actuated fully clamped single-graphene sheet as a nano-resonator with an attached concentrated mass. By neglecting the rotary inertia, the equation of motion of the nano- resonator and the attached mass is derived using the nonlocal strain gradient theory of elasticity. The nano-resonator is modeled as a Kirchhoff nano-plate with the von Kármán-type geometric nonlinearity. Applying the Galerkin decomposition method to the partial differential equation of motion leads to the ordinary differential equation. Based on the Melnikov's integral method two analytical criteria are derived which provide necessary conditions that determine the chaotic region of the system. The chaotic dynamics of the system are also scrutinized and verified through plotting the Lyapunov exponent diagram, phase plane trajectories and Poincaré maps.     

Adelic model of harmonic oscillator

Adelic quantum mechanics is formulated. The corresponding model of the harmonic oscillator is considered. The adelic harmonic oscillator exhibits many interesting features. One of them is a softening of the uncertainty relation.This work was partially supported by the Russian Foundation for Fundamental Research, Grant No. 93-011-147.Institute of Physics, P.O. Box 57, 11001 Belgrade, Yugoslavia. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 3, pp. 349–359, December, 1994.     

Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics

A generator of the complex algebra within the framework of general formulation obeys the quadratic equation of the type e² = a₁e − a₀. In this paper we construct the general complex algebras of the n-th order where the generators obey n-order polynomial equation of the type eⁿ = a_{n - 1}e^{n - 1} − a_{n - 2}e^{n - 2} + ... + (−)^{n + 1}a₀, with real coefficients a_k, k = 0, 1, ... n − 1. This algebra induces a generalized trigonometry ((n + 1)-gonometry), subyacent to the n-th order oscillator model and to the n-th order Hamilton equations.     

An analytical model for close combat dynamics

A coherent dynamic combat model is developed from basic principles. The governing set of equations has a striking resemblance to the continuity equation in fluid dynamics with an additional term for the losses of combat units. The salient features of the model are a moving battle front, the replenishment of losses, and the withdrawal of combat units while others are still engaged. A basic example shows that the often used force ratio of three can produce a frontline movement up to 90% of the speed of the attacker. Another example simulates a well documented battle from the American Civil War. It is shown that terrain influences and the absence of reconnaissance had a large adverse effect on the outcome of the battle for the Confederate forces.     

Defence operational analysis using system dynamics

This paper reflects on experiences gained in applying system dynamics as a modelling methodology to create a feedback perspective of army defence situations. The ideas of the approach are outlined, and a case study model presented to demonstrate the type of insight which can be generated by the method. This example is further used to make a comparison between system dynamics and other methods of defence operational analysis.     

N-symmetric Chebyshev polynomials in a composite model of a generalized oscillator

We continue to study a composite model of a generalized oscillator generated by an N-periodic Jacobi matrix. The foundation of the model is a system of orthogonal polynomials connected to this matrix for N = 3, 4, 5. We show that such polynomials do not exist for N ?? 6.     

Mathematical analysis of a model for the transmission dynamics of bovine tuberculosis

A deterministic model for studying the transmission dynamics of bovine tuberculosis in a single cattle herd is presented and qualitatively analyzed. A notable feature of the model is that it allows for the importation of asymptomatically infected cattle (into the herd) because re‐stocking from outside sources. Rigorous analysis of the model shows that the model has a globally‐asymptotically stable disease‐free equilibrium whenever a certain epidemiological threshold, known as the reproduction number, is less than unity. In the absence of importation of asymptomatically infected cattle, the model has a unique endemic equilibrium whenever the reproduction number exceeds unity (this equilibrium is globally asymptotically stable for a special case). It is further shown that, for the case where asymptomatically infected cattle are imported into the herd, the model has a unique endemic equilibrium. This equilibrium is also shown to be globally asymptotically stable for a special case. Copyright © 2011 John Wiley & Sons, Ltd.     

A system dynamics model for intentional transmission of HIV/AIDS using cross impact analysis

The system dynamics approach is a holistic way of solving problems in real-time scenarios. This is a powerful methodology and computer simulation modeling technique for framing, understanding, and discussing complex issues and problems. System dynamics modeling and simulation is often the background of a systemic thinking approach and has become a management and organizational development paradigm. This paper proposes a system dynamics approach for modeling the phenomenon of intentional transmission of HIV/AIDS by non-disclosure. The model is built using the Cross Impact Analysis (CIA) method of relating entities and attributes relevant to the risky conduct of HIV+ individuals in any given community. The proposed model uses a hypothetical cross impact matrix that relates pairs of attributes. The factors that impact non-disclosure are identified by simulating the model through dynamic difference equations. After the simulation results are reviewed, two policies are introduced and tested in order to observe the progress in the system state.     

A temperature-calibrated continuum model for vibrational analysis of the fullerene family using molecular dynamics simulations

In the present study, a model was proposed to determine the elastic properties of the family of fullerenes at different temperatures (between 300 and 2000 K) using a combination of molecular dynamics simulation and continuum shell theory. The fullerenes molecules examined here are eight spherical fullerenes, including C₆₀, C₈₀, C₁₈₀, C₂₄₀, C₂₆₀, C₃₂₀, C₅₀₀, and C₇₂₀. First, the breathing mode frequency and the radius of gyration of the molecules were obtained at different temperatures by molecular dynamics simulations using AIREBO potential. Then, these data were used in a continuum model to obtain the elastic coefficients of these closed clusters of carbon in terms of temperature changes. As another result of this paper is finding a nearly linear relationship between the changes in radius and breathing mode frequency of molecules versus temperature variations. Validation of the results was accomplished by comparing the results with the available laboratory as well as quantum mechanics results.     

A mathematical analysis for a diffusive epidemic model with criss-cross dynamics

Abstract. In this paper, an initial boundary value problem with homogeneous Neumann bound-ary condition is studied for a reaction diffusion system which models the spread of infectious dis-eases within two population groups by means of serf and criss-cross infection mechanism, Exis-tence, uniqueness and houndedness of the nonnegative global solution     

An analysis of a mathematical model of trophoblast invasion

We present a mathematical model that describes the initial stages of placental development during which trophoblast cells begin to invade the uterine tissue. We then carry out a mathematical analysis of a simpler submodel that describes the final stages of normal embryo implantation and suggests that as the timescale of interest increases, the dominant migratory mechanism of the trophoblasts switches from chemotaxis to nonlinear random motion.