Characteristic equation

About: Characteristic equation is a research topic. Over the lifetime, 4017 publications have been published within this topic receiving 67304 citations.

Theory of Functional Differential Equations

Abstract: 1 Linear differential difference equations.- 1.1 Differential and difference equations.- 1.2 Retarded differential difference equations.- 1.3 Exponential estimates of x(?, f).- 1.4 The characteristic equation.- 1.5 The fundamental solution.- 1.6 The variation-of-constants formula.- 1.7 Neutral differential difference equations.- 1.8 Supplementary remarks.- 2 Retarded functional differential equations : basic theory.- 2.1 Definition.- 2.2 Existence, uniqueness, and continuous dependence.- 2.3 Continuation of solutions.- 2.4 Differentiability of solutions.- 2.5 Backward continuation.- 2.6 Caratheodory conditions.- 2.7 Supplementary remarks.- 3 Properties of the solution map.- 3.1 Finite- or infinite-dimensional problem?.- 3.2 Equivalence classes of solutions.- 3.3 Exponential decrease for linear systems.- 3.4 Unique backward extensions.- 3.5 Range in ?n.- 3.6 Compactness and representation.- 3.7 Supplementary remarks.- 4 Autonomous and periodic processes.- 4.1 Processes.- 4.2 Invariance.- 4.3 Discrete systems-maximal compact invariant sets.- 4.4 Fixed points of discrete dissipative processes.- 4.5 Stability and maximal invariant sets in processes.- 4.6 Periodic trajectories of ?-periodic processes.- 4.7 Convergent systems.- 4.8 Supplementary remarks.- 5 Stability theory.- 5.1 Definitions.- 5.2 The method of Liapunov functional.- 5.3 Liapunov functional for autonomous systems.- 5.4 Razumikhin-type theorems.- 5.5 Supplementary remarks.- 6 General linear systems.- 6.1 Global existence and exponential estimates.- 6.2 Variation-of-constants formula.- 6.3 The formal adjoint equation.- 6.4 The true adjoint.- 6.5 Boundary-value problems.- 6.6 Stability and boundedness.- 6.7 Supplementary remarks.- 7 Linear autonomous equations.- 7.1 The semigroup and infinitesimal generator.- 7.2 Spectrum of the generator-decomposition of C.- 7.3 Decomposing C with the formal adjoint equation.- 7.4 Estimates on the complementary subspace.- 7.5 An example.- 7.6 The decomposition in the variation-of-constants formula.- 7.7 Supplementary remarks.- 8 Linear periodic systems.- 8.1 General theory.- 8.2 Decomposition.- 8.3 Supplementary remarks.- 9 Perturbed linear systems.- 9.1 Forced linear systems.- 9.2 Bounded, almost-periodic, and periodic solutions stable and unstable manifolds.- 9.3 Periodic solutions-critical cases.- 9.4 Averaging.- 9.5 Asymptotic behavior.- 9.6 Boundary-value problems.- 9.7 Supplementary remarks.- 10 Behavior near equilibrium and periodic orbits for autonomous equations.- 10.1 The saddle-point property near equilibrium.- 10.2 Nondegenerate periodic orbits.- 10.3 Hyperbolic periodic orbits.- 10.4 Supplementary remarks.- 11 Periodic solutions of autonomous equations.- 11.1 Hopf bifurcation.- 11.2 A periodicity theorem.- 11.3 Range of the period.- 11.4 The equation $$\dot x(t) = - \alpha x(t - 1)[1 + x(t)]$$.- 11.5 The equation $$\dot x(t) = - \alpha x(t - 1)[1 - {x^2}(t)]$$.- 11.6 The equation $$\ddot x(t) + f(x(t))\dot x(t) + g(x(t - r)) = 0$$.- 11.7 Supplementary remarks.- 12 Equations of neutral type.- 12.1 Definition of a neutral equation.- 12.2 Fundamental properties.- 12.3 Linear autonomous D operators.- 12.4 Stable D operators.- 12.5 Strongly stable D operators.- 12.6 Properties of equations with stable D operators.- 12.7 Stability theory.- 12.8 General linear equations.- 12.9 Stability of autonomous perturbed linear systems.- 12.10 Linear autonomous and periodic equations.- 12.11 Nonhomogeneous linear equations.- 12.12 Supplementary remarks.- 13 Global theory.- 13.1 Generic properties of retarded equations.- 13.2 The set of global solutions.- 13.3 Equations on manifolds : definitions.- 13.4 Retraded equations on compact manifolds.- 13.5 Further properties of the attractor.- 13.6 Supplementary remarks.- Appendix Stability of characteristic equations.

Homogenization of Differential Operators and Integral Functionals

Abstract: 1 Homogenization of Second Order Elliptic Operators with Periodic Coefficients.- 1.1 Preliminaries.- 1.2 Setting of the Homogenization Problem.- 1.3 Problems of Justification Further Examples.- 1.4 The Method of Asymptotic Expansions.- 1.5 Explicit Formulas for the Homogenized Matrix in the Two-Dimensional Case.- 1.6 Estimates and Approximations for the Homogenized Matrix.- 1.7 The Rayleigh-Maxwell Formulas.- Comments.- 2 An Introduction to the Problems of Diffusion.- 2.1 Homogenization of Parabolic Operators.- 2.2 Homogenization and the Central Limit Theorem.- 2.3 Stabilization of Solutions of Parabolic Equations.- 2.4 Diffusion in a Solenoidal Flow.- 2.5 Diffusion in an Arbitrary Periodic Flow.- 2.6 Spectral Approach to the Asymptotic Problems of Diffusion.- 2.7 Diffusion with Absorption.- Comments.- 3 Elementary Soft and Stiff Problems.- 3.1 Homogenization of Soft Inclusions.- 3.2 Homogenization of Stiff Inclusions.- 3.3 Virtual Mass.- 3.4 The Method of Asymptotic Expansions.- 3.5 On a Dense Cubic Packing of Balls.- 3.6 The Dirichlet Problem in a Perforated Domain.- Comments.- 4 Homogenization of Maxwell Equations.- 4.1 Preliminary Results.- 4.2 A Lemma on Compensated Compactness.- 4.3 Homogenization.- 4.4 The Problem of an Artificial Dielectric.- Comments.- 5 G-Convergence of Differential Operators.- 5.1 Basic Properties of G-Convergence.- 5.2 A Sufficient Condition of G-Convergence.- 5.3 G-Convergence of Abstract Operators.- 5.4 Compactness Theorem and Its Implications.- 5.5 G-Convergence and Duality.- 5.6 Stratified Media.- 5.7 G-Convergence of Divergent Elliptic Operators of Higher Order.- Comments.- 6 Estimates for the Homogenized Matrix.- 6.1 The Hashin-Shtrikman Bounds.- 6.2 Attainability of Bounds. The Hashin Structure.- 6.3 Extremum Principles.- 6.4 The Variational Method.- 6.5 G-Limit Media Attainment of the Bounds on Stratified Composites.- 6.6 The Method of Quasi-Convexity.- 6.7 The Method of Null Lagrangians.- 6.8 The Method of Integral Representation.- Comments.- 7 Homogenization of Elliptic Operators with Random Coefficients.- 7.1 Probabilistic Description of Non-Homogeneous Media.- 7.2 Homogenization.- 7.3 Explicit Formulas in Two-Dimensional Problems.- 7.4 Homogenization of Almost-Periodic Operators.- 7.5 The General Theorem of Individual Homogenization.- Comments.- 8 Homogenization in Perforated Random Domains.- 8.1 Homogenization.- 8.2 Remarks on Positive Definiteness of the Homogenized Matrix.- 8.3 Central Limit Theorem.- 8.4 Disperse Media.- 8.5 Criterion of Pointwise Stabilization A Refinement of the Central Limit Theorem.- 8.6 Stiff Problem for a Random Spherical Structure.- 8.7 Random Spherical Structure with Small Concentration.- Comments.- 9 Homogenization and Percolation.- 9.1 Existence of the Effective Conductivity.- 9.2 Random Structure of Chess-Board Type.- 9.3 The Method of Percolation Channels.- 9.4 Conductivity Threshold for a Random Cubic Structure in ?3.- 9.5 Resistance Threshold for a Random Cubic Structure in ?3.- 9.6 Central Limit Theorem for Random Motion in an Infinite Two-Dimensional Cluster.- Comments.- 10 Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients.- 10.1 Preliminary Remarks.- 10.2 Auxiliary Equation A*p = 0 on a Probability Space.- 10.3 Homogenization and the Central Limit Theorem.- 10.4 Criterion of Pointwise Stabilization.- Comments.- 11 Spectral Problems in Homogenization Theory.- 11.1 Spectral Properties of Abstract Operators Forming a Sequence.- 11.2 On the Spectrum of G-Convergent Operators.- 11.3 The Sturm-Liouville Problem.- 11.4 Spectral Properties of Stratified Media.- 11.5 Density of States for Random Elliptic Operators.- 11.6 Asymptotics of the Density of States.- Comments.- 12 Homogenization in Linear Elasticity.- 12.1 Some General Facts from the Theory of Elasticity.- 12.2 G-Convergence of Elasticity Tensors.- 12.3 Homogenization of Periodic and Random Tensors.- 12.4 Fourth Order Operators.- 12.5 Linear Problems of Incompressible Elasticity.- 12.6 Explicit Formulas for Two-Dimensional Incompressible Composites.- 12.7 Some Questions of Analysis on a Probability Space.- 13 Estimates for the Homogenized Elasticity Tensor.- 13.1 Basic Estimates.- 13.2 The Variational Method.- 13.3 Two-Phase Media Attainability of Bounds on Stratified Composites.- 13.4 On the Hashin Structure.- 13.5 Disperse Media with Inclusions of Small Concentration.- 13.6 Fourth Order Operators Systems of Stokes Type.- Comments.- 14 Elements of the Duality Theory.- 14.1 Convex Functions.- 14.2 Integral Functionals.- 14.3 On Two Types of Boundary Value Problems.- 14.4 Dual Boundary Value Problems.- 14.5 Extremal Relations.- 14.6 Examples of Regular Lagrangians.- Comments.- 15 Homogenization of Nonlinear Variational Problems.- 15.1 Random Lagrangians.- 15.2 Two Principal Lemmas.- 15.3 Homogenization Theorems.- 15.4 Applications to Boundary Value Problems in Perforated Domains.- 15.5 Chess Lagrangians Dychne's Formula.- Comments.- 16 Passing to the Limit in Nonlinear Variational Problems.- 16.1 Definition of ?-Convergence of Lagrangians Formulation of the Compactness Theorems.- 16.2 Convergence of Energies and Minimizers.- 16.3 Proof of the Compactness Theorems.- 16.4 Two Examples: Ulam's Problem Homogenization Problem.- 16.5 Compactness of Lagrangians in Plasticity Problems Application to Ll-Closedness.- 16.6 Remarks on Non-Convex Functionals.- Comments.- 17 Basic Properties of Abstract ?-Convergence.- 17.1 ?-Convergence of Functions on a Metric Space.- 17.2 ?-Convergence of Functions Defined in a Banach Space.- 17.3 ?-Convergence of Integral Functionals.- Comments.- 18 Limit Load.- 18.1 The Notion of Limit Load.- 18.2 Dual Definition of Limit Load.- 18.3 Equivalence Principle.- 18.4 Convergence of Limit Loads in Homogenization Problems.- 18.5 Surface Loads.- 18.6 Representation of the Functional $$\bar F$$ on BV0.- 18.7 ?-Convergence in BV0.- Comments.- Appendix A. Proof of the Nash-Aronson Estimate.- Appendix C. A Property of Bounded Lipschitz Domains.- References.

Fokker-Planck Equation for an Inverse-Square Force

Abstract: The contribution to the Fokker-Planck equation for the distribution function for gases, due to particle-particle interactions in which the fundamental two-body force obeys an inverse square law, is investigated. The coefficients in the equation, $〈\ensuremath{\Delta}\mathrm{v}〉$ (the average change in velocity in a short time) and $〈\ensuremath{\Delta}\mathrm{v}\ensuremath{\Delta}\mathrm{v}〉$, are obtained in terms of two fundamental integrals which are dependent on the distribution function itself. The transformation of the equation to polar coordinates in a case of axial symmetry is carried out. By expanding the distribution function in Legendre functions of the angle, the equation is cast into the form of an infinite set of one-dimensional coupled nonlinear integro-differential equations. If the distribution function is approximated by a finite series, the resultant Fokker-Planck equations may be treated numerically using a computing machine. Keeping only one or two terms in the series corresponds to the approximations of Chandrasekhar, and Cohen, Spitzer and McRoutly, respectively.

Stability analysis of linear fractional differential system with multiple time delays

Abstract: In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R+)n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.