scholarly journals Some relations between Bohl exponents and the exponential dichotomy spectrum

2019 ◽  
Vol 25 (4) ◽  
pp. 573-582 ◽  
Author(s):  
Nicolás Pinto ◽  
Gonzalo Robledo
Keyword(s):  
Exponential Dichotomy ◽  
Dichotomy Spectrum

DICHOTOMY SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS

2002 ◽  
Vol 02 (02) ◽  
pp. 175-201 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
STEFAN SIEGMUND
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Exponential Dichotomy ◽  
Spectral Theorem ◽  
Random Dynamical System ◽  
Dichotomy Spectrum ◽  
Random Intervals ◽  
Two Parameter

We investigate a concept of dichotomy spectrum for nonautonomous linear stochastic differential equations, which is defined with sample-wise exponential dichotomy of the two-parameter flow generated by the equation. We use random norm and cohomology to capture the nature of the stochastic nonuniformity. The main result is our spectral theorem stating that the dichotomy spectrum consists of compact random intervals with corresponding spectral manifolds, which are Oseledets spaces if the equation generates a random dynamical system. The dichotomy spectrum is nonrandom and equals the Lyapunov spectrum if the stochastic differential equation is Lyapunov regular.

scholarly journals STABILITY OF LYAPUNOV EXPONENTS, WEAK INTEGRAL SEPARATION AND NONUNIFORM DICHOTOMY SPECTRUM

2021 ◽  
Author(s):  
Hailong Zhu ◽  
Zhaoxiang LI
Keyword(s):  
Lyapunov Exponents ◽  
Exponential Dichotomy ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dichotomy Spectrum ◽  
Integral Separation ◽  
Linear Differential ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Nonuniform Exponential Dichotomy

In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system is proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Reduction principle at work

2021 ◽  
Vol 8 (1) ◽  
pp. 46-74
Author(s):  
Christian Pötzsche ◽  
Evamaria Russ
Keyword(s):  
Center Manifold ◽  
Reduction Principle ◽  
Critical Stability ◽  
Center Manifold Reduction ◽  
Infinite Dimensional ◽  
Linearized Stability ◽  
Dichotomy Spectrum ◽  
Nonautonomous Case ◽  
Necessary And Sufficient ◽  
Manifold Reduction

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

scholarly journals On Exponential Dichotomy of Variational Difference Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Bogdan Sasu
Keyword(s):  
Difference Equations ◽  
Exponential Dichotomy ◽  
New Method ◽  
Skew Product ◽  
Skew Product Flows

We give very general characterizations for uniform exponential dichotomy of variational difference equations. We propose a new method in the study of exponential dichotomy based on the convergence of some associated series of nonlinear trajectories. The obtained results are applied to difference equations and also to linear skew-product flows.

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