Exponential behavior in mean for cocycles

2015 ◽  
Vol 15 (04) ◽  
pp. 1550029
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Probability Measure ◽  
Discrete Time ◽  
Exponential Dichotomy ◽  
Exponential Behavior ◽  
Classical Notion ◽  
Linear Perturbations

For cocycles with discrete time, we consider the notion of an exponential dichotomy in mean. This corresponds to replace the classical notion of an exponential dichotomy by the much weaker requirement that the same happens in mean with respect to some probability measure. We show that the exponential behavior in mean is robust, in the sense that it persists under sufficiently small linear perturbations.

Admissibility for exponential dichotomies in average

2015 ◽  
Vol 15 (03) ◽  
pp. 1550014 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Probability Measure ◽  
Linear Operator ◽  
Exponential Dichotomy ◽  
Simple Manner ◽  
Exponential Dichotomies ◽  
Linear Perturbations ◽  
Bounded Sequences

We characterize completely the notion of an exponential dichotomy in average in terms of an admissibility property. The notion corresponds to a generalization of that of an exponential dichotomy to measurable cocycles acting on L1 functions with respect to a given probability measure. The admissibility property is described in terms of the injectivity and surjectivity of a certain linear operator in the space of bounded sequences of L1 functions. The characterization is then used to establish in a simple manner the robustness of the notion, in the sense that it persists under sufficiently small linear perturbations. We note that we consider both ℤ-cocycles and ℕ-cocycles.

Exponential dichotomies with respect to a sequence of norms and admissibility

2014 ◽  
Vol 25 (03) ◽  
pp. 1450024 ◽  
Author(s):  
Luis Barreira ◽  
Davor Dragičević ◽  
Claudia Valls
Keyword(s):  
Exponential Dichotomy ◽  
Linear Operators ◽  
Conditional Stability ◽  
Exponential Behavior ◽  
Exponential Dichotomies ◽  
Linear Perturbations ◽  
Nonuniform Exponential Dichotomy

For a nonautonomous dynamics defined by a sequence of linear operators, we introduce the notion of an exponential dichotomy with respect to a sequence of norms and we characterize it completely in terms of the admissibility in lp spaces, both for the space of perturbations and the space of solutions. This allows unifying the notions of uniform and nonuniform exponential behavior. Moreover, we consider the general case of a noninvertible dynamics. As a nontrivial application we show that the conditional stability of a nonuniform exponential dichotomy persists under sufficiently small linear perturbations.

scholarly journals Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 47 ◽  
Author(s):  
Davor Dragičević ◽  
Ciprian Preda
Keyword(s):  
Hilbert Space ◽  
Exponential Stability ◽  
Discrete Time ◽  
Lyapunov Functions ◽  
Complete Characterization ◽  
Skew Product ◽  
Linear Perturbations

For linear skew-product three-parameter semiflows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufficiently small linear perturbations.

scholarly journals Nonuniform Exponential Trichotomy for Linear Discrete-Time Systems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xiao-qiu Song ◽  
Tian Yue ◽  
Dong-qing Li
Keyword(s):  
Exponential Stability ◽  
Banach Spaces ◽  
Discrete Time ◽  
Difference Equations ◽  
Exponential Dichotomy ◽  
Linear Difference Equations ◽  
Discrete Time Systems ◽  
Exponential Trichotomy ◽  
Time Systems

The aim of this paper is to give several characterizations for nonuniform exponential trichotomy properties of linear difference equations in Banach spaces. Well-known results for exponential stability and exponential dichotomy are extended to the case of nonuniform exponential trichotomy.

scholarly journals A FRAMEWORK FOR INTERNEURAL DYNAMICS IN A CEREBRAL CORTEX CENTER

2013 ◽  
Vol 23 (09) ◽  
pp. 1350151
Author(s):  
ABRAHAM BOYARSKY ◽  
ZHENYANG LI ◽  
PAWEŁ GÓRA
Keyword(s):  
Cerebral Cortex ◽  
Steady State ◽  
Probability Measure ◽  
Discrete Time ◽  
Sensory Input ◽  
Invariant Probability Measure ◽  
Chaotic State ◽  
Time Map

We model the innervation dynamics of interneurons in a cerebral cortex center A between the time of initial sensory input and acquisition of a sustained steady state. The model assumes that interneurons in A are heavily interconnected allowing synchronization. This invites modeling the dynamics by means of a discrete time map. The model takes into account the influence of excitatory and inhibitory cells and reflects the architecture of synapses along the axons. The acquisition of a sustained chaotic state is characterized by means of a natural invariant probability measure. The time to attain this probability measure can be estimated.

Exponential behavior in Banach spaces: robustness of trichotomies in discrete time

2014 ◽  
Vol 68 (2) ◽  
pp. 207-221
Author(s):  
L. Barreira ◽  
L. H. Popescu ◽  
C. Valls
Keyword(s):  
Banach Spaces ◽  
Discrete Time ◽  
Exponential Behavior

Lyapunov theorems for exponential dichotomies in Hilbert spaces

2016 ◽  
Vol 27 (04) ◽  
pp. 1650033 ◽  
Author(s):  
Davor Dragičević ◽  
Ciprian Preda
Keyword(s):  
Hilbert Spaces ◽  
Exponential Dichotomy ◽  
Complete Characterization ◽  
Linear Operators ◽  
Nonlinear Perturbations ◽  
Exponential Behavior ◽  
Exponential Dichotomies ◽  
Linear And Nonlinear ◽  
Unstable Direction

For a nonautonomous dynamics defined by a sequence of linear operators, we obtain a complete characterization of the notion of a uniform exponential dichotomy in terms of the existence of appropriate Lyapunov sequences. In sharp contrast to previous results, we consider the case of noninvertible dynamics, thus requiring only the invertibility of operators along the unstable direction. Furthermore, we deal with operators acting on an arbitrary Hilbert space. As a nontrivial application of our work, we study the persistence of uniform exponential behavior under small linear and nonlinear perturbations.

scholarly journals Uniform Dichotomy Concepts for Discrete-Time Skew Evolution Cocycles in Banach Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2177
Author(s):  
Ariana Găină ◽  
Mihail Megan ◽  
Carmen Florinela Popa
Keyword(s):  
Dynamical Systems ◽  
Banach Spaces ◽  
Discrete Time ◽  
Exponential Dichotomy ◽  
Uniform Dichotomy

In the present paper, we consider the problem of dichotomic behaviors of dynamical systems described by discrete-time skew evolution cocycles in Banach spaces. We study two concepts of uniform dichotomy: uniform exponential dichotomy and uniform polynomial dichotomy. Some characterizations of these notions and connections between these concepts are given.

scholarly journals Characterizations of Generalized Exponential Trichotomies for Linear Discrete-time Systems

2019 ◽  
Vol 27 (2) ◽  
pp. 153-166 ◽  
Author(s):  
Ioan-Lucian Popa ◽  
Traian Ceauşu ◽  
Ovidiu Bagdasar ◽  
Ravi P. Agarwal
Keyword(s):  
Discrete Time ◽  
Linear Time ◽  
Time Varying ◽  
Adjoint Systems ◽  
Discrete Time Systems ◽  
Classical Notion ◽  
Exponential Trichotomy ◽  
Time Varying Systems ◽  
Time Systems

AbstractThe concept of generalized exponential trichotomy for linear time-varying systems is investigated in relationship with the classical notion of uniform exponential trichotomy. Some key properties of generalized exponential trichotomy are explored through supplementary projections. These results are also extended to the case of projection sequences, while certain applications for adjoint systems are suggested.

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