On the exponential behaviour of non-autonomous difference equations

2013 ◽  
Vol 56 (3) ◽  
pp. 643-656
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Asymptotic Behaviour ◽  
Lyapunov Exponents ◽  
Difference Equations ◽  
Small Perturbations ◽  
Exponential Behaviour ◽  
Exponential Rates

AbstractGiven a sequence of matrices(Am)m∈ℕwhose Lyapunov exponents are limits, we show that this asymptotic behaviour is reproduced by the sequencesxm+1=Amxm+ fm(xm)for any sufficiently small perturbationsfm. We also consider the general case of exponential rates ecρmfor an arbitrary increasing sequenceρm. Our approach is based on Lyapunov's theory of regularity.

Exponential trends of Ornstein–Uhlenbeck first-passage-time densities

1985 ◽  
Vol 22 (02) ◽  
pp. 360-369 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Asymptotic Behaviour ◽  
Excellent Agreement ◽  
First Passage Time ◽  
Passage Time ◽  
Computational Results ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Exponential Behaviour

The asymptotic behaviour of the first-passage-time p.d.f. through a constant boundary for an Ornstein–Uhlenbeck process is investigated for large boundaries. It is shown that an exponential p.d.f. arises, whose mean is the average first-passage time from 0 to the boundary. The proof relies on a new recursive expression of the moments of the first-passage-time p.d.f. The excellent agreement of theoretical and computational results is pointed out. It is also remarked that in many cases the exponential behaviour actually occurs even for small values of time and boundary.

NONUNIFORM EXPONENTIAL BEHAVIOUR AND TOPOLOGICAL EQUIVALENCE

2015 ◽  
Vol 58 (2) ◽  
pp. 279-291
Author(s):  
LUIS BARREIRA ◽  
LIVIU HORIA POPESCU ◽  
CLAUDIA VALLS
Keyword(s):  
Asymptotic Behaviour ◽  
Ergodic Theory ◽  
Canonical Form ◽  
Exponential Dichotomy ◽  
Topological Equivalence ◽  
Exponential Dichotomies ◽  
Exponential Behaviour ◽  
Nonuniform Exponential Dichotomies ◽  
Nonuniform Exponential Dichotomy ◽  
Topological Conjugacies

AbstractWe show that any evolution family with a strong nonuniform exponential dichotomy can always be transformed by a topological equivalence to a canonical form that contracts and/or expands the same in all directions. We emphasize that strong nonuniform exponential dichotomies are ubiquitous in the context of ergodic theory. The main novelty of our work is that we are able to control the asymptotic behaviour of the topological conjugacies at the origin and at infinity.

scholarly journals Growth theorems for homogeneous second-order difference equations

2002 ◽  
Vol 43 (4) ◽  
pp. 559-566 ◽  
Author(s):  
Stevo Stević
Keyword(s):  
Asymptotic Behaviour ◽  
Difference Equations ◽  
Constant Coefficient ◽  
Second Order ◽  
Order Difference ◽  
Vibrating String

AbstractIn this paper we investigate the boundedness and asymptotic behaviour of the solutions of a class of homogeneous second-order difference equations with a single non-constant coefficient. These equations model, for example, the amplitude of oscillation of the weights on a discretely weighted vibrating string. We present several growth theorems. Two examples are also given.

scholarly journals Asymptotic behaviour of solutions of nonlinear delay difference equations in Banach spaces

2005 ◽  
Vol 2005 (17) ◽  
pp. 2769-2774
Author(s):  
Anna Kisiolek ◽  
Ireneusz Kubiaczyk
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Banach Spaces ◽  
Difference Equations ◽  
Measure Of Noncompactness ◽  
Nonlinear Difference Equations ◽  
Measure Of Weak Noncompactness ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

We consider the second-order nonlinear difference equations of the formΔ(rn−1Δxn−1)+pnf(xn−k)=hn. We show that there exists a solution(xn), which possesses the asymptotic behaviour‖xn−a∑j=0n−1(1/rj)+b‖=o(1),a,b∈ℝ. In this paper, we extend the results of Agarwal (1992), Dawidowski et al. (2001), Drozdowicz and Popenda (1987), M. Migda (2001), and M. Migda and J. Migda (1988). We suppose thatfhas values in Banach space and satisfies some conditions with respect to the measure of noncompactness and measure of weak noncompactness.

Sign in / Sign up

Export Citation Format

Share Document