Characterization of isochronous foci for planar analytic differential systems

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

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Isochronous Foci for Analytic Differential Systems

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007400 ◽

2003 ◽

Vol 13 (06) ◽

pp. 1617-1623 ◽

Cited By ~ 10

Author(s):

Jaume Giné

Keyword(s):

Differential Equations ◽

Analytic Functions ◽

Autonomous Systems ◽

Sufficient Conditions ◽

Two Dimensional ◽

Differential Systems ◽

Weak Focus ◽

Systems Of Differential Equations ◽

Strong Focus

Consider the two-dimensional autonomous systems of differential equations of the form [Formula: see text] where P(x, y) and Q(x, y) are analytic functions. The origin is a strong focus of this system if λ ≠ 0 and is either a weak focus or its center if λ = 0. In this work we provide some sufficient conditions to have an isochronous focus at the origin.

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SUFFICIENT CONDITIONS FOR A CENTER AT A COMPLETELY DEGENERATE CRITICAL POINT

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127402005315 ◽

2002 ◽

Vol 12 (07) ◽

pp. 1659-1666 ◽

Cited By ~ 23

Author(s):

J. GINÉ

Keyword(s):

Differential Equations ◽

Critical Point ◽

Autonomous Systems ◽

Sufficient Conditions ◽

Two Dimensional ◽

Homogeneous Polynomials ◽

Systems Of Differential Equations ◽

Degenerate Critical Point

Consider the two-dimensional autonomous systems of differential equations of the form [Formula: see text] where P3(x, y) and Q3(x, y) are homogeneous polynomials of degree 3, and P4(x, y) and Q4(x, y) are homogeneous polynomials of degree 4. The origin is a completely degenerate critical point of this system. In this work we give sufficient conditions in order to have a center at the origin.

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On the Modulii of Analytic Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-062-4 ◽

1970 ◽

Vol 13 (3) ◽

pp. 325-327 ◽

Cited By ~ 1

Author(s):

Malcolm J. Sherman

Keyword(s):

Analytic Functions ◽

Point Of View ◽

Sufficient Condition ◽

Two Dimensional ◽

Necessary And Sufficient Condition ◽

Concrete Form ◽

Norm Equality ◽

Image Position ◽

Necessary And Sufficient

The problem to be considered in this note, in its most concrete form, is the determination of all quartets f1, f2, g1, g2 of functions analytic on some domain and satisfying*where p > 0. When p = 2 the question can be reformulated in terms of finding a necessary and sufficient condition for (two-dimensional) Hilbert space valued analytic functions to have equal pointwise norms, and the answer (Theorem 1) justifies this point of view. If p ≠ 2, the problem is solved by reducing to the case p = 2, and the reformulation in terms of the norm equality of lp valued analytic functions gives no clue to the answer.

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Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2021-0010 ◽

2021 ◽

Vol 24 (1) ◽

pp. 225-253

Author(s):

Oana Brandibur ◽

Eva Kaslik

Keyword(s):

Asymptotic Stability ◽

Differential Equations ◽

Fractional Order ◽

Autonomous Systems ◽

Sufficient Conditions ◽

Two Dimensional ◽

Fractional Order Differential Equations ◽

Stability And Instability ◽

Instability Regions ◽

Necessary And Sufficient

Abstract Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterised, in terms of the main diagonal elements of the systems’ matrix, as well as its determinant.

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Radial entire solutions to even order semilinear elliptic equations in the plane

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500022101 ◽

1987 ◽

Vol 105 (1) ◽

pp. 275-287 ◽

Cited By ~ 3

Author(s):

Takaŝi Kusano ◽

Manabu Naito ◽

Charles A. Swanson

Keyword(s):

Elliptic Equations ◽

Sufficient Conditions ◽

Semilinear Elliptic Equations ◽

Two Dimensional ◽

Entire Solutions ◽

Semilinear Elliptic ◽

Negative Constant ◽

Even Order ◽

Image Position ◽

Necessary And Sufficient

SynopsisSemilinear elliptic equations of the formare considered, where Δm is the m-th iterate of the two-dimensional Laplacian Δ, p(t) is continuous in [0, ∞), and f(u is continuous and positive either in (0, ∞) or in ℝ.Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C2m(ℝ2) which depend only on |x| and satisfy (*) pointwise in ℝ2.First, necessary and sufficient conditions are established for equation (*), with p(t) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of |x|2m−2 log |x as |x| → ∞. Secondly, it is shown that, in the case p(t < 0, in [ 0, ∞) and f(u) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of |x|2m−2 log |x| as |x| → ∞. Our results seem to be new even when specialised to the prototypeswhere γ is a constant.

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Second-order impulsive differential systems with mixed and several delays

Advances in Difference Equations ◽

10.1186/s13662-021-03474-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Shyam Sundar Santra ◽

Apurba Ghosh ◽

Omar Bazighifan ◽

Khaled Mohamed Khedher ◽

Taher A. Nofal

Keyword(s):

Differential Equations ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

Oscillation Theory ◽

Second Order ◽

Neutral Delay ◽

Differential Systems ◽

Impulsive Differential Systems ◽

Several Delays ◽

Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

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On cryptographic properties of (n + 1)-bit S-boxes constructed by known n-bit S-boxes

Journal of Mathematical Cryptology ◽

10.1515/jmc-2020-0004 ◽

2020 ◽

Vol 15 (1) ◽

pp. 258-265

Author(s):

Yu Zhou ◽

Daoguang Mu ◽

Xinfeng Dong

Keyword(s):

Sufficient Conditions ◽

Sum Of Squares ◽

Basic Component ◽

Necessary And Sufficient Conditions ◽

Autocorrelation Functions ◽

Walsh Spectrum ◽

Cryptographic Algorithms ◽

Necessary And Sufficient ◽

Relationship Of

AbstractS-box is the basic component of symmetric cryptographic algorithms, and its cryptographic properties play a key role in security of the algorithms. In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (n + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (n + 1)-bit S-boxes, and one necessary and sufficient conditions of (n + 1)-bit S-boxes satisfying m-order resilient. Meanwhile, we also give one characterization of (n + 1)-bit S-boxes satisfying t-order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an n-bit S-box S0 and the (n + 1)-bit S-box S (which is constructed by S0).

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QKtype spaces of analytic functions

Journal of Function Spaces and Applications ◽

10.1155/2006/910813 ◽

2006 ◽

Vol 4 (1) ◽

pp. 73-84 ◽

Cited By ~ 18

Author(s):

Hasi Wulan ◽

Jizhen Zhou

Keyword(s):

Unit Disk ◽

Analytic Functions ◽

Sufficient Conditions ◽

Nondecreasing Function ◽

Necessary And Sufficient Conditions ◽

Type Space ◽

Necessary And Sufficient ◽

Spaces Of Analytic Functions

For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

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Separability criteria based on the Bloch representation of density matrices

Quantum Information and Computation ◽

10.26421/qic7.7-5 ◽

2007 ◽

Vol 7 (7) ◽

pp. 624-638

Author(s):

J. de Vicente

Keyword(s):

Sufficient Conditions ◽

Quantum Systems ◽

Necessary Condition ◽

Sufficient Condition ◽

Pure Product ◽

Separability Problem ◽

Arbitrary Dimensions ◽

Necessary And Sufficient ◽

Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

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