scholarly journals The absence of residual property for strong exponents of oscillation of linear systems

2021 ◽  
Vol 31 (1) ◽  
pp. 59-69
Author(s):  
A.G. Stash
Keyword(s):  
Linear Systems ◽  
Differential System ◽  
Finite Interval ◽  
Autonomous Systems ◽  
Residual Property ◽  
Differential Systems ◽  
Sign Changes

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. It is proved that all strong exponents of oscillations (unlike Sergeev's frequencies of sign changes, zeros and roots, as well as all the weak exponents of oscillations) considered as functions on the set of solutions to linear homogeneous $n$-dimensional differential systems with continuous coefficients on the semi-line are not residual (i.e. can be changed when changing solution on a finite interval). Besides, at any beforehand given natural $n\ge2$ we give the example of $n$-dimensional differential system, for some solution of which all strong oscillation exponents differ from corresponding weak exponents. In this case, all weak and all strong exponents on the chosen solution coincide with each other, respectively. When proving the results of this work, the case of parity and odd $n$ are considered separately.

scholarly journals Approximate solution of nonlinear differential system with time variation

2021 ◽  
Vol 44 (2) ◽  
pp. 121-130
Author(s):  
Rezaul Karim ◽  
Pinakee Dey ◽  
Saikh Shahjahan Miah
Keyword(s):  
Approximate Solution ◽  
Perturbation Method ◽  
Differential System ◽  
Time Variation ◽  
Autonomous Systems ◽  
Differential Systems ◽  
Matlab Software ◽  
Kbm Method ◽  
Academy Of Sciences ◽  
Nonlinear Differential Systems

this paper develops a reliable algorithm based on the general Struble’s technique and extended KBM method for solving nonlinear differential systems. Moreover, we find a solution based on the KBM and general Struble’s technique of nonlinear autonomous systems with time variation, which is more powerful than the existing perturbation method. Finally, results are discussed, primarily to enrich the physical prospects, and shown graphically by utilizing MATHEMATICA and MATLAB software. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 121-130, 2020

scholarly journals On the equivalence of three-dimensional differential systems

2020 ◽  
Vol 18 (1) ◽  
pp. 1164-1172
Author(s):  
Jian Zhou ◽  
Shiyin Zhao
Keyword(s):  
Periodic Solutions ◽  
Differential System ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Periodic Systems ◽  
Structural Form ◽  
Differential Systems

Abstract In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t.

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

scholarly journals Nilpotent Global Centers of Linear Systems with Cubic Homogeneous Nonlinearities

2020 ◽  
Vol 30 (01) ◽  
pp. 2050010
Author(s):  
J. D. García-Saldaña ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Linear Systems ◽  
Linear Form ◽  
Differential Systems ◽  
Polynomial Differential Systems

In this paper, we characterize the global nilpotent centers of polynomial differential systems of the linear form plus cubic homogeneous terms.

scholarly journals Formal Weierstrass Nonintegrability Criterion for Some Classes of Polynomial Differential Systems in ℂ2

2020 ◽  
Vol 30 (04) ◽  
pp. 2050064
Author(s):  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Differential System ◽  
Integrable Systems ◽  
Necessary Conditions ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Formal Power

In this paper, we present a criterion for determining the formal Weierstrass nonintegrability of some polynomial differential systems in the plane [Formula: see text]. The criterion uses solutions of the form [Formula: see text] of the differential system in the plane and their associated cofactors, where [Formula: see text] is a formal power series. In particular, the criterion provides the necessary conditions in order that some polynomial differential systems in [Formula: see text] would be formal Weierstrass integrable. Inside this class there exist non-Liouvillian integrable systems. Finally we extend the theory of formal Weierstrass integrability to Puiseux Weierstrass integrability.

scholarly journals Initial Time Difference Stability of Causal Differential Systems in terms of Lyapunov Functions and Lyapunov Functionals

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Coşkun Yakar ◽  
Mustafa Bayram Gücen
Keyword(s):  
Differential System ◽  
Lyapunov Functions ◽  
Lyapunov Functional ◽  
Initial Position ◽  
Time Difference ◽  
Initial Time ◽  
Qualitative Behavior ◽  
Lyapunov Functionals ◽  
Comparison Result ◽  
Differential Systems

We investigate the qualitative behavior of a perturbed causal differential equation that differs in initial position and initial time with respect to the unperturbed causal differential equations. We compare the classical notion of stability of the causal differential systems to the notion of initial time difference stability of causal differential systems and present a comparison result in terms of Lyapunov functions. We have utilized Lyapunov functions and Lyapunov functional in the study of stability theory of causal differential systems when establishing initial time difference stability of the perturbed causal differential system with respect to the unperturbed causal differential system.

LIAO STYLE NUMBERS OF DIFFERENTIAL SYSTEMS

2004 ◽  
Vol 06 (02) ◽  
pp. 279-299 ◽  
Author(s):  
XIONGPING DAI
Keyword(s):  
Differential System ◽  
Compact Riemannian Manifold ◽  
Ergodic Measure ◽  
Ergodic System ◽  
View Point ◽  
Differential Systems ◽  
Linearly Independent ◽  
Characterization Theorems ◽  
Positive Integers ◽  
Style Number

For any C1 differential system S on a compact Riemannian manifold M of dimension d with d≥2, this paper studies the Liao style numbers, κ(S) (or respectively, κ*(S)) of S from the view-point of ergodic theory. Here κ(S) (κ*(S)) is the largest number of moving vectors (or respectively, conjugate-) of the differential system S that are mean linearly independent. For any ergodic measure ν of S, two positive integers κ*(ν) and κ(ν), called the reduced and non-reduced style number of ν respectively, are introduced. The connection between the style numbers of the system (M,S) and ones of the ergodic system (M,S; ν) are discovered by the variational principle of style number proved in the paper. Several characterization theorems with respect to the style numbers κ*(S), κ*(ν), κ(S) and κ(ν) are presented respectively.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

Center Problems and Limit Cycle Bifurcations in a Class of Quasi-Homogeneous Systems

2015 ◽  
Vol 25 (10) ◽  
pp. 1550135 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han ◽  
Yong Wang
Keyword(s):  
Limit Cycle ◽  
Differential System ◽  
Homogeneous Polynomial ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
First Order ◽  
Polynomial Differential Systems ◽  
Perturbed System ◽  
Generalized Polynomial ◽  
Necessary And Sufficient

In this paper, we first classify all centers of a class of quasi-homogeneous polynomial differential systems of degree 5. Then we extend this kind of systems to a generalized polynomial differential system and provide the necessary and sufficient conditions to have a center at the origin. Furthermore, we study the Poincaré bifurcation for its perturbed system as it has a center at the origin, find the Poincaré cyclicity up to first order of ε.

Global attraction for systems with almost C1 vector fields

2005 ◽  
Vol 135 (4) ◽  
pp. 815-832
Author(s):  
Zhanyuan Hou
Keyword(s):  
Linear Systems ◽  
Global Attractor ◽  
Differential System ◽  
Vector Fields ◽  
Single Point ◽  
Sufficient Conditions ◽  
Global Attraction ◽  
Almost Everywhere ◽  
Image Position ◽  
Special Case

Sufficient conditions are given for an autonomous differential system to have a single point global attractor (repeller) with f continuously differentiable almost everywhere. These results incorporate those of Hartman and Olech as a special case even when the condition f ∈ C1(D, ℝN) is fully met. Moreover, these criteria are simplified for a class of region-wise linear systems in ℝN.

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