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In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given. 1

Design of the fractional-order PIλDµ controllers based on the optimization with self-organizing migrating algorithm

Design of fractional-order controllers based on optimization methods is one of the intensively developed trends of the present time. There are several quality control criterions to evaluate the controller performance and to design the controller parameters by optimization. All of these objective functions are almost always multimodal in this case - so they have too complex geometric surface with many local extrema. In this context the choice of the optimization method is very important. In this paper we present a synthesis method for the design of fractional-order PIλDµ controllers based on an intelligent optimization method with so called self-organizing migrating algorithm utilizing the principles of artificial intelligence. Along with the mathematical description we will present also simulation results on illustrative examples to demonstrate the advantages of this method and advantages of the fractional-order PIλDµ controllers in comparison with traditional PID controllers

Identification of fractional-order dynamical systems based on nonlinear function optimization

In general, real objects are fractional-order systems and also dy- namical processes taking place in them are fractional-order processes, although in some types of systems the order is very close to an integer order. So we con- sider dynamical system whose mathematical description is a differential equa- tion in which the orders of derivatives can be real numbers. With regard to this, in the task of identification, it is necessary to consider also the fractional order of the dynamical system. In this paper we give suitable numerical solutions of differential equations of this type and subsequently an experimental method of identification in the time domain is given. We will concentrate mainly on the identification of parameters, including the orders of derivatives, for a chosen structure of the dynamical model of the system. Under mentioned assump- tions, we would obtain a system of nonlinear equations to identify the system. More suitable than to solve the system of nonlinear equations is to formulate the identification task as an optimization problem for nonlinear function mini- mization. As a criterion we have considered the sum of squares of the vertical deviations of experimental and theoretical data and the sum of squares of the corresponding orthogonal distances. The verification was performed on systems with known parameters and also on a laboratory object

Chua's Equation With Cubic Nonlinearity

In this tutorial paper we present one of the simplest autonomous differential equations capable of generating chaotic behavior. Some of the fundamental routes to chaos and bifurcation phenomena are demonstrated with examples. A brief discussion of equilibrium points and their stability is given. For the convenience of the reader, a short computer program written in QuickBASIC is included to give the reader a possibility of quick hands-on experience with the generation of chaotic phenomena without using sophisticated numerical simulators. All the necessary parameter values and initial conditions are provided in a tabular form. Eigenvalue diagrams showing regions with particular eigenvalue patterns are given. 1. Introduction Chaotic phenomena in physical systems have been observed for a long time, but their in-depth research has only been conducted in recent decades. Observations of chaotic behavior in the field of electrical circuits date back to Van der Pol [1927], who reported "irreg..

Analogue Realization of Fractional-Order Dynamical Systems

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Analogue realization of fractional-order dynamical systems

As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented