Stability of Continuous Dynamic Systems With Parametric Excitation

1969 ◽  
Vol 36 (2) ◽  
pp. 212-216 ◽  
Author(s):  
J. R. Dickerson ◽  
T. K. Caughey
Keyword(s):  
Partial Differential Equations ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Dynamic Systems ◽  
Parametric Excitation ◽  
Sufficient Conditions ◽  
Partial Differential ◽  
Continuous Dynamic

A Lyapunov-type approach is used to establish sufficient conditions guaranteeing the asymptotic stability of a class of partial differential equations with parametric excitation.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

scholarly journals Oscillation criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

ASYMPTOTIC STABILITY OF EQUILIBRIUM STATES FOR AMBIPOLAR PLASMAS

2004 ◽  
Vol 14 (09) ◽  
pp. 1361-1399 ◽  
Author(s):  
V. GIOVANGIGLI ◽  
B. GRAILLE
Keyword(s):  
Partial Differential Equations ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Null Space ◽  
Current Model ◽  
Debye Length ◽  
Equilibrium States ◽  
Electron Mass ◽  
Partial Differential ◽  
Stability Of Equilibrium

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equations can be written as a symmetric form by using entropic variables. The corresponding dissipation matrices satisfy the null space invariant property and the system of partial differential equations can be written as a normal form, i.e. in the form of a symmetric hyperbolic–parabolic composite system. By properly modifying the chemistry source terms and/or the diffusion matrices, asymptotic stability of equilibrium states is established and decay estimates are obtained. We also establish the continuous dependence of global solutions with respect to vanishing electron mass.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

scholarly journals Oscillation Properties for Systems of Higher-Order Partial Differential Equations with Distributed Deviating Arguments

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Youjun Liu ◽  
Jianwen Zhang ◽  
Jurang Yan
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Partial Differential ◽  
Deviating Arguments ◽  
Distributed Deviating Arguments ◽  
Functional Differential ◽  
Oscillation Properties

New sufficient conditions are obtained for oscillation for the solutions of systems of a class of higher-order quasilinear partial functional differential equations with distributed deviating arguments. The obtained results are illustrated by example.

FINITE DIMENSIONALITY OF ATTRACTORS FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS GIVEN BY PARTIAL DIFFERENTIAL EQUATIONS

2004 ◽  
Vol 04 (03) ◽  
pp. 385-404 ◽  
Author(s):  
J. A. LANGA ◽  
B. SCHMALFUSS
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Fractal Dimensionality ◽  
Natural Generalization ◽  
Pullback Attractor ◽  
Compact Sets ◽  
Qualitative Properties ◽  
Partial Differential

In the last decade, the concept of pullback attractor has become one of the usual tools to describe some qualitative properties of non-autonomous partial differential equations. A pullback attractor is a family of compact sets, invariant for the corresponding process related to the equation, and attracting from the past, and it assumes a natural generalization of the now classical concept of global attractor for autonomous partial differential equations. In this work we give sufficient conditions in order to prove the finite Hausdorff and fractal dimensionality of pullback attractors for non-autonomous infinite dimensional dynamical systems, and we apply our results to a generalized non-autonomous partial differential equation of Navier–Stokes type.

scholarly journals Interval oscillation criteria for impulsive conformable partial differential equations

2019 ◽  
Vol 13 (1) ◽  
pp. 325-345
Author(s):  
G.E. Chatzarakis ◽  
K. Logaarasi ◽  
T. Raja ◽  
V. Sadhasivam
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Qualitative Properties ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
All Solutions ◽  
Two Factors ◽  
Interval Oscillation ◽  
Impulsive Delay

In this paper, we present some sufficient conditions for the oscillation of all solutions of forced impulsive delay conformable partial differential equations. We consider two factors, namely impulse and delay that jointly affect the interval qualitative properties of the solutions of those equations. The results obtained in this paper extend and generalize some of the known results for forced impulsive conformable partial differential equations. An example illustrating the results is also given.

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