arXiv subjects
Publications and source records attributed to Marek Berezowski.
The paper is focused on the discussion of the phenomenon of transitional chaos in dynamic autonomous and non-autonomous systems. This phenomenon involves the disappearance of chaotic oscillations in specific time periods and the system becoming predictable. Variable dynamics of the system may be used to control the process.
In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical relationship was also derived and it was shown that all prime numbers must satisfy it. If a given number does not satisfy the above dependence, then it is a composite number. One linear equation and 8 numbers are enough to carry out all of the above-mentioned analyses. In addition, the astonishing cyclical nature of the occurrence of composite numbers in the set of natural numbers has been demonstrated.
The work concerns the spectral, entropy and bifurcation analysis of the dynamics of a reverse-flow system. The existence of chaotic oscillations was demonstrated in a wide range of changes in the parameters of the model. The model of such a system can be compared to a periodically stimulated pendulum. The periodic apparatus here is a tubular reactor, which can generate its own oscillations. Reverse flow is cyclical stimulation. As a result, both of these devices can start to oscillate chaotically, and they actually do. The purpose of using reverse flow is to increase the conversion degree. This is because after reverse, the cold stream of raw material falls into the heated reactor inlet.
The dynamics of the tubular chemical reactor with mass recycle were examined. In such a system, temperature and concentrations may oscillate chaotically. This means that state variable values are then unpredictable. In this paper it has been shown that despite the chaos, the behaviour of such a reactor can be predictable. It has been shown that this phenomenon can occur in two cases. The first case concerns intermittent chaos. It has been shown that intermittent outbursts can occur at regular intervals. The second case concerns transient chaos, i.e. a situation when the chaos occurs only for a certain period of time, e.g. only during start-up. This phenomenon makes it impossible to predict what will occur in the reactor in the nearest time, but, makes it possible to precisely determine the values of the variables even in the distant future. Both of these phenomena were tested by numerical simulation of the mathematical model of the reactor.
In the paper the distribution of prime numbers and digits of $π$ were presented as chaotic.
It is possible that self-induced oscillations appear in reactors, and that their range does not reach the steady state, although such state exists. To prove this, a cascade of tank reactors coupled with mass recycle loop was tested numerically. The above-mentioned phenomenon is characterized by the location of the steady point out of the limit cycle in the phase portrait. This incident may be beneficial to the process, as low steady state does not have to exclude an independent increase of the conversion degree, despite being the only state and not generating oscillations.
The analysis performed as well as extensive numerical simulations have revealed the possibility of the generation of homoclinic orbits as a result of homoclinic bifurcation in a porous pellet. A method has been proposed for the development of a special type of diagrams - the socalled bifurcation diagrams. These diagrams comprise the locus of homoclinic orbits together with the lines of limit points bounding the region of multiple steady states as well as the locus of the points of Hopf bifurcation. Thus, they define a set of parameters for which homoclinic bifurcation can take place. They also make it possible to determine conditions under which homoclinic orbits are generated. Two kinds of homoclinic orbits have been observed, namely semistable and unstable orbits. It is found that the character of the homoclinic orbit depends on the stability features of the limit cycle which is linked with the saddle point. Very interesting dynamic phenomena are associated with the two kinds of homoclinic orbits; these phenomena have been illustrated in the solution diagrams and phase diagrams.
The paper discusses a simple method of using the parametric continuation method to designate complex diagrams of steady states. The main advantage of the discussed approach is the fact that it does not require the installation of huge professional IT systems. The deliberations are illustrated by examples of a reactor models described by the algebraic and differential equations. Both models render solutions in the form of the so called multiple steady states.
The paper deals with the mathematical - numerical analysis of the Mandelbrot equation extended by the dynamic continuous term. The possibilities of generation of fractal patterns with the mathematical form, defined in such a manner, were investigated. The decay of fractal structure with time was demonstrated.
The scope of the paper is the theoretical analysis of the time rate in which a dynamical system reaches a stable stationary state or stable oscillations. The method used for the analysis is based on the so-called iterative time profiles, demonstrating a chaotic and fractal nature of some of the profiles. The results were presented in the form of two and three-dimensional graphs.
The scope of the paper is a theoretical analysis of the dynamical system, the model of which was reduced to Weierstrasse function. A fractal structure of the trajectory was proved and the entropy of the system information designated.
The scope of the paper is the analysis of the impact of flow reversal on the dynamics of cascades of reactors. Periodic and chaotic oscillations occur in the analyzed system. There is a dependence between the oscillation period of the state variable of the system without flow reversal and the recurrence period of windows of chaos in the steady-state diagram of the system with flow reversal.
The present study deals with a theoretical analysis of the effect of delay time of energy transport upon the generation of complex dynamics in continuous physical system. The importance of this time for the presence of quasi - periodicity and chaos in a reactor is demonstrated. The considerations are preceded by the analysis of one - dimensional mathematical model.
The paper deals with the theoretical analysis of a logistic system composed of at least two elements with distributed parameters. It has been shown that such a system may generate specific oscillations in spite of the fact that the solutions of the mathematical method are characterized by no dynamic bifurcations. It has also been shown that the time series of the state variables of such a system may behave in a semi-chaotic way. This means that they have then predictable and unpredictable fragments. The analysis has been illustrated by two examples, viz. of a simple logistic model and of a reactor with feedback.
Three kinds of fractal solutions of model of reactors are presented. The first kind concerns the structure of Feigenbaum_s diagram on the limit of chaos. The second kind and the third one concern the effect of initial conditions on the dynamic solutions of models. In the course of computations two types of recirculation were considered, viz. the recirculation of mass (return of a part of products stream) and recirculation of heat (heat exchange in the external heat exchanger).
The mathematical - numerical analysis of a discrete dynamical model with two independent delays was performed. Such model may describe a continuous system with delays that have real rational number values. Applicable characteristic equations were derived for both a single and double cycle. The results of the analysis were illustrated by numerical examples in the form of boundary bifurcation curves and Feigenbaums diagrams.
The scope of the paper is the analysis of the possibility of increasing conversion in different types of systems based on continuous stirred tank reactors. A numerical model of the cascade with constant direction of material flow is tested, a model of changes of direction flow and a relaxation model. The analysis is conducted by means of parametric continuation method and numerical simulation, with the designation of bifurcation diagrams and time series. An essential impact of the relaxation method on the increase of the conversion degree is indicated.
The adaptive approach of strongly non-linear fast-changing signals identification is discussed. The approach is devised by adaptive sampling based on chaotic mapping in yourself of a signal. Presented sampling way may be utilized online in the automatic control of chemical reactor (throughout identification of concentrations and temperature oscillations in real-time), in medicine (throughout identification of ECG and EEG signals in real-time), etc. In this paper, we presented it to identify the Weierstrass function and ECG signal.