Narrow Results

This paper explores the chaotic properties of an advection system expressed in difference equations form. In the beginning the Aref's blinking vortex system is examined. Then several new lines are explored related to the sink problem (one central sink, two symmetric sinks, eccentric sink and others). Chaotic forms with or without space contraction are presented, analyzed and simulated. Several chaotic objects are formulated especially when special rotation angles or a complex sinus rotation angle are introduced in the rotation-translation difference equations. Very interesting chaotic forms arise when elliptic rotation-translation equations are applied. The simulated chaotic images and attractors express several vortex-like forms resulting in various situations and especially in fluid dynamics.

In this paper we model and simulate the so-called Von Karman Streets, a characteristic form of turbulent (vortex) flow that appears in small or large systems. Very interesting examples are the large turbulent formations in the sea or in the clouds that have been viewed and photographed from the space (from satellites). The modeling approach is based on the Rotation-Reflection-Translation theory developed in the recent book [1]. The related theory was already applied in various fields and especially in flows.

The development of the last year disaster in the Stock Markets all over the world gave rise to reconsidering the previous models used. It is clear that, even in an organized international or national context, large fluctuations and sudden losses may occur. This paper explores a two populations' model. The populations are conflicting into the same environment (a Stock Market) by following the main rules present, that is mutual interaction between adopters, potential adopters, word-of-mouth communication and of course by taking into consideration the innovation diffusion process. The proposed model has special futures expressed by third order terms providing characteristic stationary points.

The previous years' disaster in the stock markets all over the world and the resulting economic crisis lead to serious criticisms of the various models used. It was evident that large fluctuations and sudden losses may occur even in the case of a well organized and supervised context as it looks to be the European Union. In order to explain the economic systems, we explore models of interacting and conflicting populations. The populations are conflicting into the same environment (a Stock Market or a Group of Countries as the EU). Three models where introduced 1) the Lotka-Volterra 2) the Lanchester or the Richardson model and 3) a new model for two conflicting populations. These models assume immediate interaction between the two conflicting populations. This is usually not the case in a stock market or between countries as delays in the information process arise. The main rules present include mutual interaction between adopters, potential adopters, word-of-mouth communication and of course by taking into consideration the innovation diffusion process. In a previous paper (Skiadas, 2010 [9]) we had proposed and analyzed a model including mutual interaction with delays due to the innovation diffusion process. The model characteristics where expressed by third order terms providing four characteristic symmetric stationary points. In this paper we summarize the previous results and we analyze the case of a non-symmetric case where the leading part receives the information immediately while the second part receives the information following a delay mechanism due to the innovation diffusion process (the spread of information) which can be expressed by a third order term. In the later case the non-symmetric process leads to gains of the leading part while the second part oscillates between gains and losses during time.

In this paper, a modified chaotic shift keying method is proposed to transmit digital bits securely over a communication channel. The scheme is based upon encrypting the digital bits 0 and 1 into infinite levels by applying the keystream such that there is no recognisable pattern in the encoded transmitted signal. The encoded transmitting signal generated is shown to resist popular attack method therefore realizing a secure and trustworthy digital communication system.