Discrete dynamical systems feature a recurrent structure and rich qualitative behavior, including... more Discrete dynamical systems feature a recurrent structure and rich qualitative behavior, including chaos, making them suitable for algorithmic art. After defining an orbit of a system, we discuss techniques for aesthetically enhancing their renderings. We use additional functions to control drawing attributes such as the point size, color, opacity and more. Introduction A discrete dynamical system (DDS), or an iterative map, is a function with a fixed rule that determines the future states by iterating on previous states. This fact alone makes DDS suitable for algorithmic art. Some well-known examples of the use of DDS in algorithmic art include the work of Joel [3], the Bridges papers by Krawczyk (e.g. [5, 6]), and some of the references in both papers. These works mainly focus on the generation of drawings of strange attractors by exploiting the orbits of iterative maps. In addition to the orbits, in this paper, we will use iterative maps to generate many other attributes of algori...
The fully distributed and asynchronous interference reduction algorithm GADIA developed by B. Bab... more The fully distributed and asynchronous interference reduction algorithm GADIA developed by B. Babadi and V. Tarokh [1] is a very powerful and effective algorithm. It has been the source of various other asynchronous distributed interference reduction algorithms. Due to the time varying channel gains, the convergence speed in all these asynchronous algorithms plays an important role. In this paper, we turn the asynchronous GADIA algorithm into a pairwise synchronous dynamic system in order to improve the convergence speed, and propose a two-node synchronous system. We analyze the proposed system from a mathematical perspective using the results in [5]. Wireless systems simulations show that the proposed pairwise synchronous algorithm (PSA) remarkably increases the convergence speed at the expense of some extra signal exchanges between nodes and at the expense of a probability of a slight performance deterioration as compared to the fully asynchronous GADIA algorithm.
This paper deals with the local asymptotic stability of non-hyperbolic fixed points of one-dimens... more This paper deals with the local asymptotic stability of non-hyperbolic fixed points of one-dimensional maps. There are, basically, two stability conditions introduced in this study. One of them is for the stability of fixed points of non-oscillatory maps. The second one is a sufficient condition for the stability for oscillatory maps. Some properties and applications are also presented.
2011 International Conference on Computational Science and Its Applications, 2011
Time scale analysis is the generalization of discrete and continuous analysis. In this research p... more Time scale analysis is the generalization of discrete and continuous analysis. In this research paper, we extend our previous study and develop some important concepts; tangent planes and partial derivatives for multi-variable functions on time scales with Mathematica.
Host parasite models are similar to host parasitoid models except that the parasite does not nece... more Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model (Leslie and Gower in Biometrika 47(3/4):219-234, 1960) played a historical role in ecology. We consider the stability of Misra and Mitra's model (Misra and Mitra in Comput. Math. Appl. 52:525-538, 2006). We study this system analytically and improve the results of Misra and Mitra (Comput. Math. Appl. 52:525-538, 2006). MSC: 39A11; 92D25
The concept of analyticity for complex functions on time scale complex plane was introduced by Bo... more The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy–Riemann equations for delta case.In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy–Riemann equations for nabla case.
Predator-prey models are similar to host-parasite and host-parasitoid models. We investigate the ... more Predator-prey models are similar to host-parasite and host-parasitoid models. We investigate the stability and invariant manifolds of a discrete predator-prey model by using center manifold theory which is not addressed in Çelik and Duman (Chaos Solitons
Mathematica is extremely popular with a wide range of researchers from all sorts of disciplines. ... more Mathematica is extremely popular with a wide range of researchers from all sorts of disciplines. It is a symbolic, numerical and graphical manipulation package.
We investigate the stability of the equilibria and the invariant manifolds of the host-parastoid ... more We investigate the stability of the equilibria and the invariant manifolds of the host-parastoid model due to Beddington, Free, and Lawton [2] subject to the Allee effect.
Host parasite models are similar to host parasitoid models except that the parasite does not nece... more Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model[6] played a historical role in ecology. We consider the stability of Misra and Mitra's model . We study this system analytically and improve the results of Misra and Mitra .
... Page 3. ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my adviserAssoc. Pro... more ... Page 3. ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my adviserAssoc. Prof. Dr. ¨Unal ... I would like to thank Prof. Dr. O˘guz YILMAZ and Assist. Prof. Dr. Serap TOPAL for being in my thesis committee, it was a great honor. ...
Discrete dynamical systems feature a recurrent structure and rich qualitative behavior, including... more Discrete dynamical systems feature a recurrent structure and rich qualitative behavior, including chaos, making them suitable for algorithmic art. After defining an orbit of a system, we discuss techniques for aesthetically enhancing their renderings. We use additional functions to control drawing attributes such as the point size, color, opacity and more. Introduction A discrete dynamical system (DDS), or an iterative map, is a function with a fixed rule that determines the future states by iterating on previous states. This fact alone makes DDS suitable for algorithmic art. Some well-known examples of the use of DDS in algorithmic art include the work of Joel [3], the Bridges papers by Krawczyk (e.g. [5, 6]), and some of the references in both papers. These works mainly focus on the generation of drawings of strange attractors by exploiting the orbits of iterative maps. In addition to the orbits, in this paper, we will use iterative maps to generate many other attributes of algori...
The fully distributed and asynchronous interference reduction algorithm GADIA developed by B. Bab... more The fully distributed and asynchronous interference reduction algorithm GADIA developed by B. Babadi and V. Tarokh [1] is a very powerful and effective algorithm. It has been the source of various other asynchronous distributed interference reduction algorithms. Due to the time varying channel gains, the convergence speed in all these asynchronous algorithms plays an important role. In this paper, we turn the asynchronous GADIA algorithm into a pairwise synchronous dynamic system in order to improve the convergence speed, and propose a two-node synchronous system. We analyze the proposed system from a mathematical perspective using the results in [5]. Wireless systems simulations show that the proposed pairwise synchronous algorithm (PSA) remarkably increases the convergence speed at the expense of some extra signal exchanges between nodes and at the expense of a probability of a slight performance deterioration as compared to the fully asynchronous GADIA algorithm.
This paper deals with the local asymptotic stability of non-hyperbolic fixed points of one-dimens... more This paper deals with the local asymptotic stability of non-hyperbolic fixed points of one-dimensional maps. There are, basically, two stability conditions introduced in this study. One of them is for the stability of fixed points of non-oscillatory maps. The second one is a sufficient condition for the stability for oscillatory maps. Some properties and applications are also presented.
2011 International Conference on Computational Science and Its Applications, 2011
Time scale analysis is the generalization of discrete and continuous analysis. In this research p... more Time scale analysis is the generalization of discrete and continuous analysis. In this research paper, we extend our previous study and develop some important concepts; tangent planes and partial derivatives for multi-variable functions on time scales with Mathematica.
Host parasite models are similar to host parasitoid models except that the parasite does not nece... more Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model (Leslie and Gower in Biometrika 47(3/4):219-234, 1960) played a historical role in ecology. We consider the stability of Misra and Mitra's model (Misra and Mitra in Comput. Math. Appl. 52:525-538, 2006). We study this system analytically and improve the results of Misra and Mitra (Comput. Math. Appl. 52:525-538, 2006). MSC: 39A11; 92D25
The concept of analyticity for complex functions on time scale complex plane was introduced by Bo... more The concept of analyticity for complex functions on time scale complex plane was introduced by Bohner and Guseinov in 2005. They developed completely delta differentiability, delta analytic functions on products of two time scales, and Cauchy–Riemann equations for delta case.In this research paper we study on continuous, discrete and semi-discrete analytic functions and developed completely nabla differentiability, nabla analytic functions on products of two time scales, and Cauchy–Riemann equations for nabla case.
Predator-prey models are similar to host-parasite and host-parasitoid models. We investigate the ... more Predator-prey models are similar to host-parasite and host-parasitoid models. We investigate the stability and invariant manifolds of a discrete predator-prey model by using center manifold theory which is not addressed in Çelik and Duman (Chaos Solitons
Mathematica is extremely popular with a wide range of researchers from all sorts of disciplines. ... more Mathematica is extremely popular with a wide range of researchers from all sorts of disciplines. It is a symbolic, numerical and graphical manipulation package.
We investigate the stability of the equilibria and the invariant manifolds of the host-parastoid ... more We investigate the stability of the equilibria and the invariant manifolds of the host-parastoid model due to Beddington, Free, and Lawton [2] subject to the Allee effect.
Host parasite models are similar to host parasitoid models except that the parasite does not nece... more Host parasite models are similar to host parasitoid models except that the parasite does not necessarily kill the host. Leslie/Gower model[6] played a historical role in ecology. We consider the stability of Misra and Mitra's model . We study this system analytically and improve the results of Misra and Mitra .
... Page 3. ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my adviserAssoc. Pro... more ... Page 3. ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my adviserAssoc. Prof. Dr. ¨Unal ... I would like to thank Prof. Dr. O˘guz YILMAZ and Assist. Prof. Dr. Serap TOPAL for being in my thesis committee, it was a great honor. ...
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