The method of judging satisfactory consistency of linguistic judgment matrix based on adjacency matrix and 3-loop matrix

Language phrases are an effective way to express uncertain pieces of information, and easily conforms to the language habits of decision makers to describe the evaluation of things. The consistency judgment of a linguistic judgment matrices is the key to analytic hierarchy process (AHP). If a linguistic judgment matrix has a satisfactory consistency, then the rank of the decision schemes can be determined. In this study, the comparison relation between the decision schemes is first represented by a directed graph. The preference relation matrix of the linguistic judgment matrix is the adjacency matrix of the directed graph. We can use the $n - 1$ st power of the preference relation to judge the linguistic judgment matrix whether has a satisfactory consistency. The method is utilized if there is one and only one element in the $n - 1$ st power of the preference relation, and the element 1 is not on the main diagonal. Then the linguistic judgment matrix has a satisfactory consistency. If there are illogical judgments, the decision schemes that form a 3-loop can be identified and expressed through the second-order sub-matrix of the preference relation matrix. The feasibility of this theory can be verified through examples. The corresponding schemes for illogical judgments are represented in spatial coordinate system

A note on the periodic orbits of a self excited rigid body

The aim of the present paper is to study the periodic orbits of a perturbed self excited rigid body with a fixed point. For studying these periodic orbits we shall use averaging theory of first order.The first and third authors were partially supported by MICINN/FEDER grant number MTM2011-22587. The second author was partially supported by AYA 2010 Ministerio de Ciencia e Innovación grant number 22039-C02-01 and ACOMP Cosellería de Educación de la Generalitat Valenciana grant number 2012/128

A note on the Definition of a–limit Set

Many phenomena coming from the biology, economy, engineering are modeled using discrete dynamical systems. The concept of backward orbit is an essential concept for understanding the dynamics of the system. In the literature various definitions of the concept of the alpha–limit point (respectively set) have been historically used. The aim of this paper is to analyze the forcing relationships between them via the proof of the valid relationships and the construction of counterexamples for the converse situation in order to clarify the scenario for the computation of these objects. Moreover, we present a discrete dynamical system (X, f ) with the following paradoxical behavior: for every point x ∈ X, its alpha–limit set is equal to the whole space X; there is a complete negative trajectory of x whose alpha–limit set is equal to a fixed point; there is a complete negative trajectory of x whose alpha–limit set is equal to X

Design and Numerical Solutions of a Novel Third-Order Nonlinear Emden–Fowler Delay Differential Model

In this study, the design of a novel model based on nonlinear third-order Emden–Fowler delay differential (EF-DD) equations is presented along with two types using the sense of delay differential and standard form of the second-order EF equation. The singularity at ξ = 0 at single or multiple points of each type of the designed EF-DD model are discussed. The detail of shape factors and delayed points is provided for both types of the designed third-order EF-DD model. For the verification and validation of the model, two numerical examples are presented of each case and numerical results have been performed using the artificial neural network along with the hybrid of global and local capabilities. The comparison of the obtained numerical results with the exact solutions shows the perfection and correctness of the designed third-order EF-DD model

Limit cycles of a generalised Mathieu differential system

We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre ̇x = y, ̇y = −x, when it is perturbed in the form x˙=y-ɛ(1+coslθ)P(x,y),    y˙=-x-ɛ(1+cosmθ)Q(x,y),\dot x = y - \varepsilon \left( {1 + {{\cos }^l}\theta } \right)P\left( {x,y} \right),\,\,\,\,\dot y = - x - \varepsilon \left( {1 + {{\cos }^m}\theta } \right)Q\left( {x,y} \right), where ε > 0 is a small parameter, l and m are positive integers, P(x, y) and Q(x, y) are arbitrary polynomials of degree n, and θ = arctan(y/x). As we shall see the differential system (1) is a generalisation of the Mathieu differential equation. The tool for studying such limit cycles is the averaging theory

On the libration collinear points in the restricted three – body problem

In the restricted problem of three bodies when the primaries are triaxial rigid bodies, the necessary and sufficient conditions to find the locations of the three libration collinear points are stated. In addition, the Linear stability of these points is studied for the case of the Euler angles of rotational motion being θi = 0, ψi + φi = π/2, i = 1, 2 accordingly. We underline that the model studied in this paper has special importance in space dynamics when the third body moves in gravitational fields of planetary systems and particularly in a Jupiter model or a problem including an irregular asteroid

On local fractional Volterra integral equations in fractal heat transfer

In the article, the fractal heat-transfer models are described by the local fractional integral equations. The local fractional linear and nonlinear Volterra integral equations are employed to present the heat transfer problems in fractal media. The local fractional integral equations are derived from the Fourier law in fractal media