Deterministic Nonperiodic Flow

Abstract

a distinction between a deterministic system and a predictable system. Lorenz's original mathematical model (Lorenz 1963 -note its title!), in common with more recent simple models used to illustrate chaotic behaviour, is perfectly deterministic, i.e. its evolution is perfectly determined by its initial conditions. Nevertheless it also exhibits unpredictability, because the smallest error in the initial conditions eventually leads to complete loss of forecast skill. Therefore we cannot conclude that a physical system is not deterministic just because we find it to be unpredictable.Perhaps the most we can conclude about physical systems which exhibit chaotic behaviour is that there is a severe and fundamental limit on our ability to investigate whether and to what extent such systems are deterministic. We certainly cannot conclude, on the basis of these arguments, that "determinism is an untenable proposition".

Decay of correlations and laws of rare events for transitive random maps

“…Lorenz-like maps. Lorenz maps are the one-dimensional maps associated to the geometric Lorenz models, which were constructed as an attempt to understand the numerically observed behaviour of the Lorenz attractor introduced by Lorenz in [24]. The Lorenz equationsẋ = a(y − x),ẏ = (r − z)x − y,ż = xy − bz, (3.1) with the parameters a = 10, r = 28/3 and b = 8/3 were intended as an extremely simplified model for thermal fluid convection, in order to understand the atmospherical circulation.…”

A Simple Hybrid Synchronization for a Class of Chaotic Financial Systems

“…Since the 1960s, with the discovery of chaotic systems, chaos has set off a nonlinear dynamic research boom [1,2]. The chaos phenomena in economics were first discovered in 1985 [3].…”

Analysis and prognostication of changes of concentrations of dioxide of sulphur in the atmosphere of industrial city (on the example of the Gdansk region)

“…У вивченні антропогенного впливу на екологічний стан атмосфери промислового міста мето-ди теорії хаосу дозволяють, наприклад, на основі до-слідження часових рядів концентрацій забруднюючих атмосферу промислового міста речовин давати цілком адекватний аналіз і навіть короткостроковий прогноз подальшої часової еволюції концентрацій забруднюю-чих речовин [3][4][5][6][7][8][9][10][11][12][13][14]. Такий підхід з успіхом використову-вався при аналізі флуктуацій метеорологічних та гідро-логічних характеристик [15][16][17].…”