Sensitive dependence to initial conditions for one dimensional maps

Abstract

Abstract. This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.There has been recent interest in the relationship between the "chaotic" asymptotic behavior of complicated solutions to ordinary differential equations and physically unstable phenomena such as those encountered in fluid flow [21]. This has led to numerical studies of a variety of systems of differential and difference equations which appear to have large sets of initial conditions yielding complicated asymptotic behavior. The mathematical theory of Axiom A "strange attractors" provides a satisfying description of some systems which do have complicated asymptotic behavior, but there is little overlap between the numerical studies referred to above and the class of systems with Axiom A attractors. Perhaps the "Lorenz system" [16], is the only example of an explicit system of equations used in a physical problem for which there is a convincing argument that a large set of its solutions have complicated asymptotic behavior. Nonetheless, the numerical studies of such examples as the "Henon" map [9], the mechanical systems studied by Holmes and Moon [11], the "strange attractor" of Spiegel [26], and the density dependent population models of [8] all provide evidence for the prevalence of complicated solutions in systems near those with homoclinic tangencies. From a practical point of view the distinction between trajectories tending to complicated periodic orbits with very long periods and trajectories with aperiodic asymptotic behavior may be slight, but we would like to understand the extent to which numerical computations of strange attractors reflect the mathematical properties of the systems being studied. This paper aims to study some of these mathematical questions in the simplest situation in which they arise : the iteration of a single real valued function of"quadratic" type. In this context we give a reasonably complete topological picture of what is involved in chaotic behavior and sensitive dependence to initial conditions while failing to answer the outstanding question about their prevalence.

Atomic relaxation and dynamical generation of ordered and disordered chemical vapour infiltration (CVI) SiC polytypes

“…The mathematical characterization of periodicity, aperiodicity and their frequencies is based on the so-called kneading theory [29,30,23]. Methods used to study the existence of chaos for the two-parameter family of gap maps [25][26][27] are easy to adapt to the sawtooth map case, confirming the fundamental fact of the set-up of chaotic structures in a deterministic growth model (sec appendix).…”

“…Careful X-ray investigations of 2H-SiC [17] and 6H-SiC [18] have led to estimates of non-ideal bond lengths and angles in the bulk structure. 29 Si and 13 C NMR studies [19] reported the existence of different kinds of sites in common polytypes (3C, 4H, 6H, 15R), and have been related to the orientations of their layer and the neighbouring ones [19,20]. Ab initio pseudopotential calculations including relaxation effects have been made for 3C, 2H, 4H, 6H and 15R structures [21]: they led to an estimation of the bond lengths in reasonable agreement with the X-ray data, mainly for the 6H polytype.…”

Sensitive dependence on parameters

“…If is a Riemannian manifold, then this is the (normed) Riemann measure, etc. Definition 2.1 (Guckenheimer,[3]). The system ( , ) has sensitive dependence on initial conditions if there is a set ⊂ of positive measure and > 0 such that for any ∈ and neighborhood of there is ∈ and ≥ 0 with ( ( ), ( )) > .…”

“…This is due to the fact that many systems have sensitive dependence on initial conditions. This notion has been introduced in 1979 by John Guckenheimer [3] (without a precise definition, it has been used slightly earlier by David Ruelle [7]). We will define it in the next section.…”

Dynamical compactness and sensitivity

“…[16], [3], [15], [2], [4], [5], [1], [21], [30]). For the recent development of sensitivity in topological dynamics see for example the survey [31] by Li and Ye. According to the works by Guckenheimer [17], Auslander and Yorke [7] a dynamical system (X, T ) is sensitive if there exists δ > 0 such that for every x ∈ X and every neighborhood U x of x, there exist y ∈ U x and n ∈ N with d(T n x, T n y) > δ. Such a δ is also called a sensitive constant of the system (X, T ).…”