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11
Chaotic Maps As Models of Packet Traffic
, 1994
"... this paper. In 2 we summarize the considerable literature on the subject along with an introduction to our approach. 3 presents results indicating the traffic characteristics that can be generated with simple piecewise linear and nonlinear maps. 4 shows how a queueing system can be represented by ..."
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Cited by 33 (0 self)
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this paper. In 2 we summarize the considerable literature on the subject along with an introduction to our approach. 3 presents results indicating the traffic characteristics that can be generated with simple piecewise linear and nonlinear maps. 4 shows how a queueing system can be represented by a 2D deterministic transformation, and outlines a potential performance analysis approach. 5 concludes this paper with a description of future directions for this work.
Stability of the Spectrum for Transfer Operators
 ANN. SCUOLA NORM. SUP. PISA CL SCI
, 1998
"... We prove stability of the isolated eigenvalues of transfer operators satisfying a LasotaYorke type inequality under a broad class of random and nonrandom perturbations including Ulamtype discretizations. The results are formulated in an abstract framework. ..."
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Cited by 31 (5 self)
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We prove stability of the isolated eigenvalues of transfer operators satisfying a LasotaYorke type inequality under a broad class of random and nonrandom perturbations including Ulamtype discretizations. The results are formulated in an abstract framework.
Random Fibonacci sequences and the number 1.13198824...
 MATHEMATICS OF COMPUTATION
, 1999
"... For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n>2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 =1.61803398.... But for a simple modification with both additions and subtractions — the random Fibonacci sequences define ..."
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Cited by 24 (2 self)
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For the familiar Fibonacci sequence (defined by f1 = f2 = 1, and fn = fn−1 + fn−2 for n>2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 =1.61803398.... But for a simple modification with both additions and subtractions — the random Fibonacci sequences defined by t1 = t2 =1, and for n>2, tn = ±tn−1 ± tn−2, where each ± sign is independent and either + or − with probability 1/2 — it is not even obvious if tn  should increase with n. Our main result is that n tn  →1.13198824... as n →∞ with
Rigorous numerical investigation of the statistical properties of piecewise expanding maps  A feasibility study
, 2000
"... I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of ..."
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Cited by 13 (1 self)
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I explore the concrete applicability of recent theoretical results to the rigorous computation of relevant statistical properties of a simple class of dynamical systems: piecewise expanding maps 1 Introduction The aim of the present paper is to investigate the possibility of answering questions of the type: ffl Given a piecewise expanding map is it possible to decide if it is ergodic or mixing? ffl Is it possible to determine with arbitrary precision its absolutely continuous invariant measure? ffl If the map is mixing, is it possible to compute the exact rate of decay of correlations for a given function? Of course, the literature contains many papers in which some of these question are discussed either theoretically (especially, but not exclusively, as far as the invariant density is concerned) or numerically (e.g. [3], [4, 5, 6, 7], [8, 9], [14], [15], [18, 19], [21, 22], [23, 24, 25], [27, 28, 29, 30, 31, 32, 33], [34, 35], [38], [39, 40], [48], [49], [52], [55], [62], [66]). N...
Lyapunov Exponents From Random Fibonacci Sequences To The Lorenz Equations
 Department of Computer Science, Cornell University
, 1998
"... this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because dete ..."
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Cited by 10 (1 self)
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this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because deterministic matrix products that govern sensitive dependence on initial conditions are barely understood; it is as if the strong law of large numbers were well understood without a satisfactory theory of convergence of infinite series. The elements of the theory of random matrix products are carefully explained in the beautiful monograph by Bougerol [16]. The basic result about Lyapunov exponents, lim
Invariant Measures and Their Properties. A Functional Analytic Point of View
, 2002
"... In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the ..."
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Cited by 8 (1 self)
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In this series of lectures I try to illustrate systematically what I call the \functional analytic approach" to the study of the statistical properties of Dynamical Systems. The ideas are presented via a series of examples of increasing complexity, hoping to give in this way a feeling of the breadth of the method.
Approximating Physical Invariant Measures of Mixing Dynamical Systems
 Nonlinear Analysis, Theory, Methods, & Applications
, 1998
"... Invariant measures of higher dimensional transformations are hard to calculate. We present new results on the estimation of absolutely continous invariant measures of mixing transformations, including a new method of proof of Ulam's conjecture. The method involves constructing finite matrix app ..."
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Cited by 8 (4 self)
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Invariant measures of higher dimensional transformations are hard to calculate. We present new results on the estimation of absolutely continous invariant measures of mixing transformations, including a new method of proof of Ulam's conjecture. The method involves constructing finite matrix approximations to the PerronFrobenius operator from increasingly finer partitions of the state space X. We show that at a finite stage, our approximations are close to a special operator which would yield a correct answer. The exponential mixing property guarantees that the system is sufficiently insensitive to any approximation errors, showing our computed invariant density is close to the true invariant density. Our method has the advantages of having very relaxed conditions on the partitions, being applicable to higher dimensional systems, and potentially applicable to a wide class of maps. 1 Introduction In dynamical systems, one is often interested in the statistical properties of orbits of s...
Modeling Packet Traffic with Chaotic Maps
, 1994
"... We investigate the application of deterministic chaotic maps to model traffic sources in packet based networks, motivated in part by recent measurement studies which indicate the presence of significant statistical features in packet traffic more characteristic of fractal processes than conventional ..."
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Cited by 5 (1 self)
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We investigate the application of deterministic chaotic maps to model traffic sources in packet based networks, motivated in part by recent measurement studies which indicate the presence of significant statistical features in packet traffic more characteristic of fractal processes than conventional stochastic processes. We outline one approach whereby traffic sources can be modeled by chaotic maps, and illustrate the traffic characteristics that can be generated by analyzing three maps. We show that low order nonlinear maps can capture several of the fractal properties observed in actual data. Finally, we outline a potential performance analysis approach based on chaotic maps that can be used to assess the traffic significance of fractal properties. It is our conclusion that while there are considerable analytical difficulties, chaotic maps may allow accurate, yet concise, models of packet traffic, with some potential for transient and steady state analysis.
Finite Precision Representation of the Conley Decomposition:
 J. Dynam. Differential Equations
, 1998
"... We present a theoretical basis for a novel way of studying and representing the long time behaviour of finite dimensional maps. It is based on a finite representation of fflpseudo orbits of a map by the sample paths of a suitable Markov chain based on a finite partition of phase space. The use ..."
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Cited by 2 (0 self)
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We present a theoretical basis for a novel way of studying and representing the long time behaviour of finite dimensional maps. It is based on a finite representation of fflpseudo orbits of a map by the sample paths of a suitable Markov chain based on a finite partition of phase space. The use of stationary states of the chain and the corresponding partition elements in approximating the attractors of maps and differential equations was demonstrated in [7] [3] and proved for a class of stable attracting sets in [8]. Here we explore the relationship between the communication classes of the Markov chain and basic sets of the Conley Decomposition of a dynamical system. We give sufficient conditions for the existence of a chain transitive set and show that basic sets are isolated from each other by neighborhoods associated with closed communication classes of the chain. A partition element approximation of an isolating block is introduced that is easy to express in terms of s...