Vol. 2, No. 4 (2013): Regular Issue

A stochastic variant of the Brusselator model is investigated. The model accounts for a long range coupling among constituents, as well as for the finite capacity of the embedding medium. The mean field limit of the model is studied and the condition for Turing and wave instability obtained. A degenerate, cusp like transition that separates the domains of Turing and wave order can take place. The point of transition is worked out analytically. Interestingly, the region of Turing instability, as delimited by such transition point, can set in also if the inhibitor diffuses slower then the activator. This is a consequence of the generalized diffusion scheme here analyzed and which originates from having imposed an effect of spatial competition. Beyond the deterministic, mean field picture, we elaborate on the role of stochastic corrections. Granularity, endogenous to the system, can eventually materialize in waves or Turing like patterns, that we here categorized in distinct classes.

The concept of dispersion relations is widely used in physics and ap- plied mathematics in investigating wave type solutions of differential equations. On the other hand, Lie group analysis provides another useful method for constructing exact solutions of linear and nonlinear differential equations via the concept of invariant solutions. We show in the present paper that for certain types of differential equations there is a remarkable connection between these two concepts. Namely, the idea of dispersion relations leads to a special type of invariant solutions.

Synchronization of chaotic oscillators has an important application in cryptography. When two identical oscillators are coupled, they can be completely synchronized and the chaotic output of the transmitter oscillator can be used to mask a message. Although the oscillator parameters are usually used as secret keys, the sensitivity of such cryptosystems to parameter changes has never been systematically analyzed. To cryptanalyze a communication system based on synchro- nization of chaotic oscillators, we use a synchronization attack that allows estimating all unknown parameters by minimizing the synchronization error. Using this attack we cryptanalyze popular communication systems based on the Rössler and Chua chaotic electronic circuits. We suggest to include this attack as a standard security test for crypt-analysis of chaotic communication systems.

The method of the so-called “Clean Numerical Simulation” (CNS) is applied to gain reliable chaotic solutions of Saltzman’s dynamic system, a simplified model for convection flows of fluid. Based on the high-order Taylor series method with data in multiple precision library and a validation check of global reliability of result, the CNS provides us a practical way to gain reliable, accurate enough solutions of chaotic dynamic systems in a finite but long enough time interval. The parallel computation is used to greatly increase the computational efficiency. The numerical noises of the CNS can be controlled to be so small that even the influence of the micro-level inherent uncertainty of initial conditions can be investigated in details. It is found that the micro-level inherent physical uncertainty (i.e. the unavoidable statistical fluctuation of temperature and velocity of fluid) of initial conditions of chaotic Saltzman’s system transfers into macroscopic randomness. This suggests that chaos might be a bridge between micro-level inherent physical uncertainty and macroscopic randomness. The current work illustrates that the above conclusion holds not only for Lorenz equation with three ODEs but also for Saltzman’s equation with up-to nine ODEs, and thus has general meanings.

This paper presents the initial steps towards the development of a compartmental model for drug diffusion in the human body, using fractional calculus. The model presented here preserves the mass balance, therefore it maintains the link between physiological and mathematical concepts. The final purpose of this model is to predict drug pharma-cokinetics and pharmacodynamics during general anesthesia. However, in this case the model is derived for a general class of drugs, therefore it can be employed in many biomedical applications.

We consider second order functional differential equations with generalized piecewise constant argument. Conditions for existence, uniqueness and stability of Bohr almost periodic solutions are defined. Appropriate examples which illustrate the results are provided.

We establish solvability in the sense of sequences in the appropriate H2 spaces for certain linear nonhomogeneous elliptic problems involving Schrödinger type operators without Fredholm property using the technique developed in our preceding work [23]. We show the existence of bounded solutions for certain nonlinear Lippmann-Schwinger equations.

A new technological technique based on the “moist” carbon dioxide generation in the wellbore zone of the oil formations or at the wellhead is proposed. In this new technology a carbon dioxide is generated as a result of the chemical reaction between an aqueous acid solution and a natural calcium carbonate rock (limestone) as a “gas yielding” component. The mechanism of the process allows controlling a generation rate and a volume of the gas, the thermobaric conditions of the oil formation, mineral and acidity levels of water and a phase state of carbon dioxide. An industrial application of this technology in a large scale has a significant ecological importance as this technology utilizes the wastes of processing plants of the natural minerals, which is consid- ered as a major source of the pollution of atmosphere by greenhouse gas emissions.