Qualitative analysis by modern methods of a stability problem in power-system analysis

First-order ordinary differential equations have some rather special properties, which result for the most part because they can only contain a limited number of terms. In fact, all linear first-order ordinary differential equations can easily be solved. This is in contrast to higher-order ordinary differential equations that become much more difficult to solve when, for example, they contain variable coefficients. Furthermore, some methods exist to solve a pretty large class of nonlinear first-order ordinary differential equations, which is generally not the case for higherorder nonlinear equations.

In this paper, a mathematical model of a prey-predator system with infectious disease in the prey population is proposed and studied. It is assumed that there is a constant refuge in prey as a defensive property against predation and harvesting from the predator. The proposed mathematical model is consisting of three first-order nonlinear ordinary differential equations, which describe the interaction among the healthy prey, infected prey, and predator. The existence, uniqueness, and boundedness of the system’ solution are investigated. The system's equilibrium points are calculated with studying their local and global stability. The persistence conditions of the proposed system are established. Finally the obtained analytical results are justified by a numerical simulation.

Tuberculosis is a disease caused by the bacteria Mycobacterium tuberculosis. This disease can spread bacteria from one individual to another. In this article, we analyzed the spread of Tuberculosis using the SEIR model. The mathematical model is presented in a system of first-order nonlinear ordinary differential equations. This mathematical model also observes the rate of case detection and treatment. This article also discusses the analysis of the equilibrium point, the stability of the equilibrium points of the model that has been formed, and the basic reproduction number (R0). This model shows a backward bifurcation, that is the appearance of an endemic equilibrium point when R0<1, which means that the disease will not necessarily disappear even though R0<1. The numerical solution for this model is obtained using the fifth order Runge-Kutta method.

ABSTRACTThe second method of Liapunov is a useful technique for investigating the stability of linear and non-linear ordinary differential equations. It is well known that the second method of Liapunov, when applied to linear differential equations with real constant coefficients, gives rise to sets of necessary and sufficient stability conditions which are alternatives to the well-known Routh-Hurwitz conditions. In this paper a direct proof of the Routh-Hurwitz conditions themselves is given using Liapunov's second method. The new proof is ‘elementary’ in that it depends on the fundamental concept of stability associated with Liapunov's second method, and not on theorems in the complex integral calculus which are required in the usual proofs. A useful by-product of this new proof is a method of determining the coefficients of a linear differential equation with real constant coefficients in terms of its Hurwitz determinants.

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

Damped and driven oscillators are generally modeled with a nonautonomous second-order nonlinear ordinary differential equation including a sinusoidal driving forcing term, such as the forced Duffing equation and the forced Holmes–Rand equation. These equations have been extensively studied during the last century and the last two decades. In the early 1990s, Abarbanel, Rabinovich and Sushchik proposed replacing the sinusoidal forcing term with a “force controlled by the movements of the oscillator itself”, i.e. by the product of two variables: the first being the solution of the oscillator itself, while the second is the solution of a first-order nonlinear ordinary differential equation. They referred to the resulting autonomous dynamical system of two coupled nonlinear ordinary differential equations as a “controlled nonlinear oscillator”. To that end, they introduced a change of variables and parameters to transform the “controlled nonlinear oscillator” that corresponds to a particular case of the forced Duffing equation into the Lorenz system. The aim of this work is to show that their idea can be further generalized and applied to many other dynamical systems, including the forced Holmes–Rand equation, Chua’s cubic circuit, Chen’s system and the forced Helmholtz oscillator. It is proved that a certain class of three-dimensional dynamical systems can be rewritten into the form of “generalized controlled nonlinear oscillators”, which can then be transformed into various Lorenz-like systems. Such a transformation could be very useful for the study of intermittent chaos.

Post-buckling behaviour of slender structures with a bi-linear bending moment–curvature relationship

Einstein's equations for a rotating pressure-free space-time are reduced to a system of four first-order non-linear ordinary differential equations in one self-similar dimensionless variable. Numerical results are given for the vacuum solution. A compatible thin disk can be specified by a surface density σ and an angular velocity ω. Self-similarity as a statement of the absence of scales implies that σ and ω can be written asσ=αc2/4πGr, ω=βc/r, and demands that α and β be pure numbers.

This paper essays a new solution to the landmark navigation problem of planar robots in the presence of randomly fixed obstacles through a new dynamic updating rule involving the orientation and steering angle parameters of a robot. The dynamic updating rule utilizes a first-order nonlinear ordinary differential equation for the changing of landmarks so that whenever a landmark is updated, the path followed by the robot remains continuous and smooth. This waypoints guidance is via specific landmarks selected from a new set of rules governing the robot’s field of view. The governing control laws guarantee asymptotic stability of the 2D point robot system. As an application, the landmark motion planning and control of a car-like mobile robot navigating in the presence of fixed elliptic-shaped obstacles are considered. The proposed control laws take into account the geometrical constraints imposed on steering angle and guarantee eventual uniform stability of the car-like system. Computer simulations, using Matlab software, are presented to illustrate the effectiveness of the proposed technique and its stabilizing algorithm.

Explicit and exact travelling wave solutions for Konopelchenko–Dubrovsky equation

The improved Fan sub-equation method and its application to the SK equation

Lyapunov stability and numerical analysis of excursive instability for forced two-phase boiling flow in a horizontal channel

The dynamic behavior of a physical system can concisely be described by the least action principle. In the centrum of its mathematical presentations is a specific function of the coordinates and velocities, i.e., the Lagrangian. If the integral of a Lagrangian is stationary, then the system is moving along an extremal path through the phase space. All Lie symmetries of a Lagrangian correspond to a conserved quantity, and the conservation principle is explained by variation symmetry. Briefly, that is the meaning of Nother’s theorem. After showing that Goodwin’s cyclical growth model has a Lagrangian, we introduce the generalized Nother’s theorem and apply it to Goodwin’s 2D model in order to get its Hamiltonian. We prove that the cyclical motion in his model derives from its dynamic Lie symmetries. These cyclical trajectories are extremal trajectories in the phase space, and along these trajectories the first integral of the model’s Lagrangian is stationary, which by the principle of least action also means that they satisfy the first-order necessary conditions. The optimality still needs to satisfy the sufficient condition. Since the Legendre’s second-order sufficient condition is not applicable here, we show it satisfies another systems of sufficient conditions, the local convex surface of Lagrangian with the minimum non-trivial fixed-point and contour lines. Our conclusion is that all systems’ solutions described by the other first-order nonlinear ordinary differential equations are optimal if they have a Lagrangian that satisfies the sufficient and necessary conditions.

Gene translation is the process in which intracellular macro-molecules, called ribosomes, decode genetic information in the mRNA chain into the corresponding proteins. Gene translation includes several steps. During the elongation step, ribosomes move along the mRNA in a sequential manner and link amino-acids together in the corresponding order to produce the proteins. The homogeneous ribosome flow model (HRFM) is a deterministic computational model for translation-elongation under the assumption of constant elongation rates along the mRNA chain. The HRFM is described by a set of n first-order nonlinear ordinary differential equations, where n represents the number of sites along the mRNA chain. The HRFM also includes two positive parameters: ribosomal initiation rate and the (constant) elongation rate. In this paper, we show that the steady-state translation rate in the HRFM is a concave function of its parameters. This means that the problem of determining the parameter values that maximize the translation rate is relatively simple. Our results may contribute to a better understanding of the mechanisms and evolution of translation-elongation. We demonstrate this by using the theoretical results to estimate the initiation rate in M. musculus embryonic stem cell. The underlying assumption is that evolution optimized the translation mechanism. For the infinite-dimensional HRFM, we derive a closed-form solution to the problem of determining the initiation and transition rates that maximize the protein translation rate. We show that these expressions provide good approximations for the optimal values in the n-dimensional HRFM already for relatively small values of n. These results may have applications for synthetic biology where an important problem is to re-engineer genomic systems in order to maximize the protein production rate.

This paper deals with the mathematical analysis of a model of urban dynamics, which was proposed by Miyata and Yamaguchi in the context of a region of Japan. The model is a complex system of first-order nonlinear ordinary differential equations. The study undertaken by Miyata and Yamaguchi is essentially computational, while an extensive study of the asymptotic behavior of the solutions is performed in this paper related to a detailed analysis of the qualitative properties of the system.