Stability of Continuous Dynamic Systems With Parametric Excitation

1969 ◽  
Vol 36 (2) ◽  
pp. 212-216 ◽  
Author(s):  
J. R. Dickerson ◽  
T. K. Caughey

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

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk

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.


2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng

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.


2004 ◽  
Vol 14 (09) ◽  
pp. 1361-1399 ◽  
Author(s):  
V. GIOVANGIGLI ◽  
B. GRAILLE

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.


1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei

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.


2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Youjun Liu ◽  
Jianwen Zhang ◽  
Jurang Yan

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.


2004 ◽  
Vol 04 (03) ◽  
pp. 385-404 ◽  
Author(s):  
J. A. LANGA ◽  
B. SCHMALFUSS

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.


Author(s):  
Rick I. Zadoks ◽  
Charles M. Krousgrill

Abstract As a first approximation, a steel-belted radial tire can be modeled as a one dimensional rotating ring connected elastically to a moving hub. This ring can be modeled mathematically using a set of three nonlinear partial differential equations, where the three degrees of freedom are a radial displacement, a tangential displacement and a section rotation. In this study, only quadratic geometric nonlinearities are considered. The system is excited by a temporally harmonic point load f^(t) and a temporally harmonic hub motion z^(t) that have the same harmonic frequency. The point load f^(t) appears in the equations of motion as a single in-homogeneous term, while the hub motion z^(t) appears in inhomogeneous and parametric excitation terms. To simplifying the ensuing analysis, the rotation rate of the hub is assumed to be constant. The partial differential equations of motion are reduced to a set of four second-order ordinary differential equations by using two linear normal modes to approximate the spatial distribution of the displacements. A region of the parameter space, as defined by ranges of values of the excitation amplitude z and the excitation frequency ω (or detuning parameter σ), is identified, from a Strutt diagram, where the parametric excitation is expected to be dominant. In this region σ is varied to locate a secondary Hopf bifurcation that leads to a set of complex steady-state quasi-periodic solutions. These solutions contain two families of frequency components where the fundamental frequencies of these families are non-commensurate, and they are characterized by Poincaré sections with closed or nearly closed “orbits” as opposed to the distinct points displayed by periodic responses and the strange attractor sections displayed by chaotic solutions.


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