scholarly journals Stability and Multiscroll Attractors of Control Systems via the Abscissa

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Edgar-Cristian Díaz-González ◽  
Baltazar Aguirre-Hernández ◽  
Jorge Antonio López-Rentería ◽  
Eric Campos-Cantón ◽  
Carlos Arturo Loredo-Villalobos
Keyword(s):  
Control Systems ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Dissipative Systems ◽  
Linear Type ◽  
Hurwitz Matrix ◽  
Hurwitz Polynomial ◽  
Hurwitz Polynomials ◽  
Starting Point ◽  
Linear Differential

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss–Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.

scholarly journals LOWER BOUNDS FOR THE MAXIMUM NUMBER OF LIMIT CYCLES OF DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH A STRAIGHT LINE OF SEPARATION

2013 ◽  
Vol 23 (04) ◽  
pp. 1350066 ◽  
Author(s):  
J. LLIBRE ◽  
M. A. TEIXEIRA ◽  
J. TORREGROSA
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Unique Equilibrium ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

In this paper, we provide a lower bound for the maximum number of limit cycles of planar discontinuous piecewise linear differential systems defined in two half-planes separated by a straight line. Here, we only consider nonsliding limit cycles. For those systems, the interior of any limit cycle only contains a unique equilibrium point or a unique sliding segment. Moreover, the linear differential systems that we consider in every half-plane can have either a focus (F), or a node (N), or a saddle (S), these equilibrium points can be real or virtual. Then, we can consider six kinds of planar discontinuous piecewise linear differential systems: FF, FN, FS, NN, NS, SS. We provide for each of these types of discontinuous differential systems examples with two limit cycles.

scholarly journals Stable Factorization of Strictly Hurwitz Polynomials

2010 ◽  
Vol 5 (5) ◽  
pp. 701
Author(s):  
Ömer Eğecioğlu ◽  
B. Siddik Yarman
Keyword(s):  
Positive Function ◽  
Quadratic Equations ◽  
Root Finding ◽  
Real Frequency ◽  
Hurwitz Polynomial ◽  
Hurwitz Polynomials ◽  
Starting Point ◽  
Factorization Technique ◽  
Multiple Variables ◽  
Selection Of

We propose a stable factorization procedure to generate a strictly Hurwitz polynomial from a given strictly positive even polynomial. This problem typically arises in applications involving real frequency techniques. The proposed method does not require any root finding algorithm. Rather, the factorization process is directly carried out to find the solution of a set of quadratic equations in multiple variables employing Newton’s method. The selection of the starting point for the iterations is not arbitrary, and involves interrelations among the coefficients of the set of solution polynomials differing only in the signs of their roots. It is hoped that this factorization technique will provide a motivation to perform the factorization of two-variable positive function to generate scattering Hurwitz polynomials in two variables for which root finding methods are not applicable.

scholarly journals Linear Differential Equations on Some Classes of Weighted Function Spaces

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1113
Author(s):  
Ahmed El-Sayed Ahmed ◽  
Amnah E. Shammaky
Keyword(s):  
Specific Growth ◽  
Function Spaces ◽  
Entire Solutions ◽  
Linear Type ◽  
Weighted Function ◽  
Weighted Function Spaces ◽  
Type Classes ◽  
Type Differential Equation ◽  
Linear Differential ◽  
Weighted Type

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation h(k)+Kk−1(υ)h(k−1)+…+K1(υ)h′+K0(υ)h=0, will be discussed in this current research, whereas the considered coefficients K0(υ),…,Kk−1(υ) are holomorphic in the disc ΓR={υ∈C:|υ|<R},0<R≤∞. In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature.

scholarly journals On explicit estimates for linear forms in the values of a class of E-functions

1982 ◽  
Vol 25 (1) ◽  
pp. 37-69 ◽  
Author(s):  
Xu Guangshan ◽  
Wang Lianxiang
Keyword(s):  
Differential Equations ◽  
Lower Bounds ◽  
Linear Differential Equations ◽  
Linear Forms ◽  
Linear Differential

We apply methods of Mahler to obtain explicit lower bounds for certain combinations of E-functions satisfying systems of linear differential equations as studied by Makarov. Our results sharpen and generalise earlier results of Mahler, Shidlovskii, and Väänänen.

scholarly journals Determination of the Starting Point in Time Series for Trend Detection Based on Overlapping Trend

2016 ◽  
Vol 16 (6) ◽  
pp. 98-110
Author(s):  
Gao Xuedong ◽  
Gu Kan
Keyword(s):  
Time Series ◽  
Piecewise Linear ◽  
Negative Influence ◽  
Trend Detection ◽  
Computational Accuracy ◽  
Starting Point ◽  
Time Series Studies ◽  
Definition Of ◽  
Stage Characteristic

Abstract The traditional time series studies consider the time series as a whole while carrying on the trend detection; therefore not enough attention is paid to the stage characteristic. On the other hand, the piecewise linear fitting type methods for trend detection are lacking consideration of the possibility that the same node belongs to multiple trends. The above two methods are affected by the start position of the sequence. In this paper, the concept of overlapping trend is proposed, and the definition of milestone nodes is given on its base; these way not only the recognition of overlapping trend is realized, but also the negative influence of the starting point of sequence is effectively reduced. The experimental results show that the computational accuracy is not affected by the improved algorithm and the time cost is greatly reduced when dealing with the processing tasks on dynamic growing data sequence.

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