scholarly journals Partial qualitative analysis of planar 𝓐Q-Riccati equations

2020 ◽  
Vol 55 (2) ◽  
pp. 351-366
Author(s):  
Borut Zalar ◽  
◽  
Brigita Ferčec ◽  
Yilei Tang ◽  
Matej Mencinger ◽  
...  
Keyword(s):  
Differential Equation ◽  
Qualitative Analysis ◽  
Critical Points ◽  
Riccati Equations ◽  
Nonassociative Algebra ◽  
Real Interval ◽  
Complex Numbers ◽  
Limit Case ◽  
Real Algebra ◽  
Simultaneous Stability

If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of ℂ. In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).

Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Riccardo Barsotti
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Nonlinear Ordinary Differential Equation ◽  
Experimental Results ◽  
Limit Case ◽  
Reasonable Approximation ◽  
Constituent Material ◽  
Actual Solution ◽  
Wrinkled Membrane

The axisymmetric inflation problem for a wrinkled membrane is solved by means of a simple nonlinear ordinary differential equation. The solution is illustrated in full details. Both the free and constrained cases are addressed, in the limit case where the membrane is fully wrinkled. In the constrained inflation problem, no slippage is allowed between the membrane and the constraining surfaces. It is shown that an actual membrane can in no way reach the fully wrinkled configuration during free inflation, regardless of the membrane's initial configuration and constituent material. The fully wrinkled solution is compared to some finite element results obtained by means of an expressly developed iterative–incremental procedure. When the values of the inflating pressure and length of the meridian lie within a suitable applicability range, the fully wrinkled solution may represent a reasonable approximation of the actual solution. A comparison with some numerical and experimental results available in the literature is illustrated.

scholarly journals Near-idempotents, near-nilpotents and stability of critical points for Riccati equations

2018 ◽  
Vol 53 (2) ◽  
pp. 331-342 ◽  
Author(s):  
Borut Zalar ◽  
◽  
Matej Mencinger ◽  
◽  
◽  
...  
Keyword(s):  
Critical Points ◽  
Riccati Equations

scholarly journals On the Growth of Solutions of Some Second Order Linear Differential Equations with Entire Coefficients

2013 ◽  
Vol 21 (2) ◽  
pp. 35-52
Author(s):  
Benharrat Belaïdi ◽  
Habib Habib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Numbers ◽  
Order Of Growth ◽  
Hyper Order ◽  
Linear Differential ◽  
Growth Of Solutions

Abstract In this paper, we investigate the order and the hyper-order of growth of solutions of the linear differential equation where n≥2 is an integer, Aj (z) (≢0) (j = 1,2) are entire functions with max {σ A(j) : (j = 1,2} < 1, Q (z) = qmzm + ... + q1z + q0 is a nonoonstant polynomial and a1, a2 are complex numbers. Under some conditions, we prove that every solution f (z) ≢ 0 of the above equation is of infinite order and hyper-order 1.

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

scholarly journals On solutions of the Chazy equation

2021 ◽  
pp. 51-59
Author(s):  
Kiryl G. Atrokhau ◽  
Elena V. Gromak
Keyword(s):  
Differential Equation ◽  
Critical Points ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Elliptic Functions ◽  
Third Order ◽  
Chazy Equation ◽  
Painlevé Property ◽  
Necessary And Sufficient

The Chazy system determines the necessary and sufficient conditions for the absence of movable critical points of solutions of the particular third order differential equation that was considered by Chazy in one of the first papers on the classification of higher-order ordinary differential equations with respect to the Painlevé property. The solution of the complete Chazy system in the case of constant poles has been already obtained. However, the question of integrating the Chazy equation remained open until now. In this paper, we prove that in the case of constant poles, under some additional conditions, this equation is integrated in elliptic functions.

scholarly journals An Octave Package to Perform Qualitative Analysis of Nonlinear Systems Immersed in R4

2020 ◽  
Author(s):  
Eder Escobar ◽  
Richard Abramonte ◽  
Antenor Aliaga ◽  
Flabio Gutierrez
Keyword(s):  
Differential Equation ◽  
Hopf Bifurcation ◽  
Qualitative Analysis ◽  
Initial Conditions ◽  
Autonomous Systems ◽  
Electronic Circuits ◽  
Four Dimensions ◽  
System Sensitivity ◽  
Linear Differential

In this work, the AutonomousSystems4D package is presented, which allows the qualitative analysis of non-linear differential equation systems in four dimensions, as well as drawing the phase surfaces by immersing R4 in R3. The package is programmed in the computational tool Octave. As a case study applied to the new Lorenz 4D System, sensitivity was found in the initial conditions, Lyapunov exponents, Kaplan Yorke dimension, a stable and unstable critical point, limit cycle, Hopf bifurcation, and hyperattractors. The package could be adapted to perform qualitative analysis and visualize phase surfaces to autonomous systems, e.g. Sprott 4D, Rossler 4D, etc. The package can be applied to problems such as: design, analysis, implementation of electronic circuits; to message encryption.

scholarly journals Qualitative Analysis of the Dynamics of a Two-Component Chiral Cosmological Model

Universe ◽  
2020 ◽  
Vol 6 (11) ◽  
pp. 195
Author(s):  
Viktor Zhuravlev ◽  
Sergey Chervon
Keyword(s):  
Asymptotic Behavior ◽  
Cosmological Model ◽  
Qualitative Analysis ◽  
Critical Points ◽  
Scalar Fields ◽  
Two Component ◽  
Different Types ◽  
Asymptotic Dynamics

We present a qualitative analysis of chiral cosmological model (CCM) dynamics with two scalar fields in the spatially flat Friedman–Robertson–Walker Universe. The asymptotic behavior of chiral models is investigated based on the characteristics of the critical points of the selfinteraction potential and zeros of the metric components of the chiral space. The classification of critical points of CCMs is proposed. The role of zeros of the metric components of the chiral space in the asymptotic dynamics is analysed. It is shown that such zeros lead to new critical points of the corresponding dynamical systems. Examples of models with different types of zeros of metric components are represented.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

2020 ◽  
pp. 163-181
Author(s):  
Anar Adiloğlu-Nabiev
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Problem ◽  
Boundary Value ◽  
Asymptotic Formulas ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Sturm Liouville ◽  
Complex Valued

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n&gt;1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

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