Chaos Theory: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=132
en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemThu, 18 Dec 2014 17:22:01 GMTThu, 18 Dec 2014 17:22:01 GMTNew applications in the Chaos Theory categoryhttp://www.mapleprimes.com/images/mapleapps.gifChaos Theory: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=132
Hopalong Attractor
http://www.maplesoft.com/applications/view.aspx?SID=153557&ref=Feed
<p>Hopalong attractors are fractals, introduced by Barry Martin of Aston University in Birmingham, England. This application allows you to explore the Hopalong by varying the parameters, the number of iterations, the iterates' symbol size, and the background color choice. You can also change the starting values of each of the three orbits by dragging the cross symbols appearing in the plot. Full details on how this application was created using the Explore command with a user-defined module are included.</p><img src="/view.aspx?si=153557/95fa944692de1fb724cb7e758e6c56e5.gif" alt="Hopalong Attractor" align="left"/><p>Hopalong attractors are fractals, introduced by Barry Martin of Aston University in Birmingham, England. This application allows you to explore the Hopalong by varying the parameters, the number of iterations, the iterates' symbol size, and the background color choice. You can also change the starting values of each of the three orbits by dragging the cross symbols appearing in the plot. Full details on how this application was created using the Explore command with a user-defined module are included.</p>153557Mon, 28 Apr 2014 04:00:00 ZDave LinderDave LinderBifTools - Package for Bifurcation Analysis in Dynamical Systems
http://www.maplesoft.com/applications/view.aspx?SID=128951&ref=Feed
<p>BifTools is a package for symbolic and numeric bifurcation analysis of equilibrium points in dynamical systems. The package consists of five main procedures:</p>
<ul>
<li><strong>BifTools[calcOneZeroEigenvalueBifPoints]</strong> calculates the bifurcation points of an ODE system with a single zero eigenvalue of the Jacobian;</li>
</ul>
<ul>
<li><strong>BifTools [calcOneZeroEigenvalueBif]</strong> calculates the normal form of the steady states bifurcations with a single zero eigenvalue of the Jacobian;</li>
</ul>
<ul>
<li><strong>BifTools [calcHopfBifPoints]</strong> calculates the Andronov-Hopf bifurcation points of an ODE system, using the method of resultants;</li>
</ul>
<ul>
<li><strong>BifTools [calcHopfBif]</strong> calculates the normal form of the Andronov-Hopf bifurcation of the equilibrium points;</li>
</ul>
<ul>
<li><strong>BifTools [calcBTBif]</strong> calculates the normal form of the Bogdanov-Takens (double zero) bifurcation, using the projection or the direct method for center manifold reduction.</li>
</ul><img src="/view.aspx?si=128951/427373\BifTools.jpg" alt="BifTools - Package for Bifurcation Analysis in Dynamical Systems" align="left"/><p>BifTools is a package for symbolic and numeric bifurcation analysis of equilibrium points in dynamical systems. The package consists of five main procedures:</p>
<ul>
<li><strong>BifTools[calcOneZeroEigenvalueBifPoints]</strong> calculates the bifurcation points of an ODE system with a single zero eigenvalue of the Jacobian;</li>
</ul>
<ul>
<li><strong>BifTools [calcOneZeroEigenvalueBif]</strong> calculates the normal form of the steady states bifurcations with a single zero eigenvalue of the Jacobian;</li>
</ul>
<ul>
<li><strong>BifTools [calcHopfBifPoints]</strong> calculates the Andronov-Hopf bifurcation points of an ODE system, using the method of resultants;</li>
</ul>
<ul>
<li><strong>BifTools [calcHopfBif]</strong> calculates the normal form of the Andronov-Hopf bifurcation of the equilibrium points;</li>
</ul>
<ul>
<li><strong>BifTools [calcBTBif]</strong> calculates the normal form of the Bogdanov-Takens (double zero) bifurcation, using the projection or the direct method for center manifold reduction.</li>
</ul>128951Fri, 23 Dec 2011 05:00:00 ZNeli DimitrovaNeli DimitrovaSuperfractal forms
http://www.maplesoft.com/applications/view.aspx?SID=35009&ref=Feed
<p>The aim of this worksheet is to present Maple users some experiments related to superfractals which are a new kind of fractal objects recently discovered by M. Barnsley. Classical fractals are self-similar objects that are uniquelly determined by their IFSs whereas in case of superfractals some probabilistic mechanism is implemented.</p><img src="/applications/images/app_image_blank_lg.jpg" alt="Superfractal forms" align="left"/><p>The aim of this worksheet is to present Maple users some experiments related to superfractals which are a new kind of fractal objects recently discovered by M. Barnsley. Classical fractals are self-similar objects that are uniquelly determined by their IFSs whereas in case of superfractals some probabilistic mechanism is implemented.</p>35009Wed, 23 Dec 2009 05:00:00 ZDr. Agnieszka LisowskaDr. Agnieszka LisowskaSnowflake Fractal
http://www.maplesoft.com/applications/view.aspx?SID=34945&ref=Feed
<p>A fractal object is any geometric object that posses the property of self-similarity. Self-similarity is a term attributed to Benoît Mandelbrot, to describe any object that appears roughly the same at any level of magnification. Fractal objects are readily prevalent in nature and can be easily seen by examining the intricate shape of sea shells, snowflakes and lightning bolts.</p>
<p>This application illustrates how Maple can be used to generate the “Snowflake” fractal.</p><img src="/view.aspx?si=34945/thumb.jpg" alt="Snowflake Fractal" align="left"/><p>A fractal object is any geometric object that posses the property of self-similarity. Self-similarity is a term attributed to Benoît Mandelbrot, to describe any object that appears roughly the same at any level of magnification. Fractal objects are readily prevalent in nature and can be easily seen by examining the intricate shape of sea shells, snowflakes and lightning bolts.</p>
<p>This application illustrates how Maple can be used to generate the “Snowflake” fractal.</p>34945Thu, 10 Dec 2009 05:00:00 ZMaplesoftMaplesoftDynamical Systems with Applications using Maple
http://www.maplesoft.com/applications/view.aspx?SID=1701&ref=Feed
<p>Companion software for "Dynamical Systems with Applications using Maple 2nd Edition", Birkhäuser (2009). ISBN 978-0-8176-4389-8.</p>
<P><A HREF="http://www.maplesoft.com/books/books_detail.aspx?sid=102355">More information about this book is available here</A>.</P><img src="/view.aspx?si=1701/thumb.jpg" alt="Dynamical Systems with Applications using Maple" align="left"/><p>Companion software for "Dynamical Systems with Applications using Maple 2nd Edition", Birkhäuser (2009). ISBN 978-0-8176-4389-8.</p>
<P><A HREF="http://www.maplesoft.com/books/books_detail.aspx?sid=102355">More information about this book is available here</A>.</P>1701Thu, 08 Oct 2009 04:00:00 ZDr. Stephen LynchDr. Stephen LynchThe Read-Bajraktarevic Functional Equation and Selfsimilarity
http://www.maplesoft.com/applications/view.aspx?SID=7064&ref=Feed
The aim of this worksheet is to show to Maple users, following [8], [10], that fractal curves may be obtained as solutions to some functional equations with the so-called Read-Bajraktarevic operator. Additionally, it will be demonstrated the power of Maple to handle with the problem that needs to its solving a symbolic approach.<img src="/view.aspx?si=7064/thumb.gif" alt="The Read-Bajraktarevic Functional Equation and Selfsimilarity" align="left"/>The aim of this worksheet is to show to Maple users, following [8], [10], that fractal curves may be obtained as solutions to some functional equations with the so-called Read-Bajraktarevic operator. Additionally, it will be demonstrated the power of Maple to handle with the problem that needs to its solving a symbolic approach.7064Tue, 23 Dec 2008 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiA Note On Subdivision Curves
http://www.maplesoft.com/applications/view.aspx?SID=7066&ref=Feed
In this worksheet we present subdivision method applied to three non-collinear control points. It will be demonstrated that depending on the form of subdivision matrices quadratic B-spline, quadratic Bézier and fractal- like curves or other fractal objects can be generated.<img src="/view.aspx?si=7066/1.jpg" alt="A Note On Subdivision Curves" align="left"/>In this worksheet we present subdivision method applied to three non-collinear control points. It will be demonstrated that depending on the form of subdivision matrices quadratic B-spline, quadratic Bézier and fractal- like curves or other fractal objects can be generated.7066Tue, 23 Dec 2008 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiSierpinski Gasket with Control Points
http://www.maplesoft.com/applications/view.aspx?SID=5564&ref=Feed
In this worksheet we show how one can easily change the shape of the well-known Sierpinski gasket. The method is based on relationship between subdivision schemes and IFS (Iterated Function System) that contains information needed for fractal rendering. We present examples of deformed Sierpinski gaskets obtained using different subdivision matrices and differently placed control points. The worksheet is the eighth one in the series of the authors' works published earlier on Maplesoft web page.<img src="/view.aspx?si=5564/Sierpinki_gasket_24.jpg" alt="Sierpinski Gasket with Control Points" align="left"/>In this worksheet we show how one can easily change the shape of the well-known Sierpinski gasket. The method is based on relationship between subdivision schemes and IFS (Iterated Function System) that contains information needed for fractal rendering. We present examples of deformed Sierpinski gaskets obtained using different subdivision matrices and differently placed control points. The worksheet is the eighth one in the series of the authors' works published earlier on Maplesoft web page.5564Wed, 19 Dec 2007 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiEnsemble de Mandelbrot
http://www.maplesoft.com/applications/view.aspx?SID=5054&ref=Feed
Une procédure pour visualiser l'ensemble de Mandelbrot.<img src="/view.aspx?si=5054/mandelbrot_27.jpg" alt="Ensemble de Mandelbrot" align="left"/>Une procédure pour visualiser l'ensemble de Mandelbrot.5054Mon, 25 Jun 2007 00:00:00 ZProf. Samir KADDOURIProf. Samir KADDOURIFractal Dimension and Space-Filling Curves (with iterated function systems)
http://www.maplesoft.com/applications/view.aspx?SID=4869&ref=Feed
By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.<img src="/view.aspx?si=4869/thumb.png" alt="Fractal Dimension and Space-Filling Curves (with iterated function systems)" align="left"/>By using complex numbers to represent points in the plane, and the concept of iterated function system, we efficiently describe fractal sets of any dimension from 0 to 2 and continuous curves that pass through them. Maple's animation feature allows us to make "movies" that show the transition through different dimensions.4869Fri, 16 Feb 2007 00:00:00 ZProf. Mark MeyersonProf. Mark MeyersonFractal Volumes
http://www.maplesoft.com/applications/view.aspx?SID=4847&ref=Feed
In this worksheet we show how one can render fractally solid bodies. The method is based on relationship between subdivision schemes and IFS (Iterated Function System) that contain information needed for rendering volumes during iteration process. We present examples of trilinear, bilinear-quadratic and triquadratic fractal volumes based on 8, 12 or 27 control points.<img src="/view.aspx?si=4847/thumb2.jpg" alt="Fractal Volumes" align="left"/>In this worksheet we show how one can render fractally solid bodies. The method is based on relationship between subdivision schemes and IFS (Iterated Function System) that contain information needed for rendering volumes during iteration process. We present examples of trilinear, bilinear-quadratic and triquadratic fractal volumes based on 8, 12 or 27 control points.4847Wed, 29 Nov 2006 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiProbability Games - Parrondo's Paradox
http://www.maplesoft.com/applications/view.aspx?SID=1761&ref=Feed
This Worksheet contains a detailed explanation and analysis of Parrondo's paradox, which states that two losing games can couple into a winning one. It also includes an interactive simulation of the two original games used by J.M.R. Parrondo himself. It is written in Spanish.
Esta hoja contiene un estudio y un analisis detallados sobre la paradoja de Parrondo, que establece que dos juegos desfavorables pueden combinarse y producir resultados favorables para el jugador. Asi mismo, incluye una simulación interactiva de los dos juegos originales usados por J.M.R. Parrondo.<img src="/view.aspx?si=1761/Probability Games - Parrondo's Paradox (Spanish)_8.gif" alt="Probability Games - Parrondo's Paradox" align="left"/>This Worksheet contains a detailed explanation and analysis of Parrondo's paradox, which states that two losing games can couple into a winning one. It also includes an interactive simulation of the two original games used by J.M.R. Parrondo himself. It is written in Spanish.
Esta hoja contiene un estudio y un analisis detallados sobre la paradoja de Parrondo, que establece que dos juegos desfavorables pueden combinarse y producir resultados favorables para el jugador. Asi mismo, incluye una simulación interactiva de los dos juegos originales usados por J.M.R. Parrondo.1761Wed, 05 Jul 2006 00:00:00 ZMario MartínezMario MartínezFractal Teapot from Utah
http://www.maplesoft.com/applications/view.aspx?SID=1740&ref=Feed
In this worksheet we demonstrate fractally generated the famous teapot from Utah. Additionally fractal teacup and teaspoon are also presented. Theory that stands behind fractal modeling of 3D shapes is described in our previous worksheet "Fractal rendering of 3D patches" and the references given there. Data describing control points of bicubic Bezier patches that are used for modeling of the teapot, teacup and teaspoon can be found here: ftp://ftp.funet.fi/pub/sci/graphics/packages/objects/teaset.tar.Z.<img src="/view.aspx?si=1740/teapot.jpg" alt="Fractal Teapot from Utah" align="left"/>In this worksheet we demonstrate fractally generated the famous teapot from Utah. Additionally fractal teacup and teaspoon are also presented. Theory that stands behind fractal modeling of 3D shapes is described in our previous worksheet "Fractal rendering of 3D patches" and the references given there. Data describing control points of bicubic Bezier patches that are used for modeling of the teapot, teacup and teaspoon can be found here: ftp://ftp.funet.fi/pub/sci/graphics/packages/objects/teaset.tar.Z.1740Tue, 30 May 2006 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiFractal Rendering of 3D Patches
http://www.maplesoft.com/applications/view.aspx?SID=1739&ref=Feed
This worksheet presents examples of fractal rendering of 3D patches and more complex shapes (pyramid, wine glass, vase and a rabbit). To obtain IFS for fractal generation of 3D patches self-similarity and subdivision strategies have been used. Having IFSs for fractal rendering of patches one may use them for fractal generation of any 3D shape that may be presented as a finite collection of single patches. Precisely speaking, we use the so-called PIFS (Partitioned IFS), not IFS, because every patch is modeled separately by its own IFS. This worksheet is the fifth one in the series of earlier authors' worksheets (see:
http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1651 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1657 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1667 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1862 )
devoted to fractal modeling of shapes of different kinds.<img src="/view.aspx?si=1739/Patches.jpg" alt="Fractal Rendering of 3D Patches" align="left"/>This worksheet presents examples of fractal rendering of 3D patches and more complex shapes (pyramid, wine glass, vase and a rabbit). To obtain IFS for fractal generation of 3D patches self-similarity and subdivision strategies have been used. Having IFSs for fractal rendering of patches one may use them for fractal generation of any 3D shape that may be presented as a finite collection of single patches. Precisely speaking, we use the so-called PIFS (Partitioned IFS), not IFS, because every patch is modeled separately by its own IFS. This worksheet is the fifth one in the series of earlier authors' worksheets (see:
http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1651 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1657 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1667 , http:/www.maplesoft.com/applications/app_center_view.aspx?AID=1862 )
devoted to fractal modeling of shapes of different kinds.1739Tue, 30 May 2006 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiOn Fractal Modeling of 3D Curves and Wireframes
http://www.maplesoft.com/applications/view.aspx?SID=1646&ref=Feed
In this worksheet methodology and examples of fractal rendering of 3D curves are presented. Deterministic and probabilistic approaches for obtaining attractor for a given collection of Iterated Function System (IFS) are used. This Maple application can be treated as the fourth one prepared by the same authors. Some improvements of Maple code for fractal rendering of linear segments are described.<img src="/view.aspx?si=1646/FractalCurves.gif" alt="On Fractal Modeling of 3D Curves and Wireframes" align="left"/>In this worksheet methodology and examples of fractal rendering of 3D curves are presented. Deterministic and probabilistic approaches for obtaining attractor for a given collection of Iterated Function System (IFS) are used. This Maple application can be treated as the fourth one prepared by the same authors. Some improvements of Maple code for fractal rendering of linear segments are described.1646Tue, 02 Aug 2005 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiOn Geometric Chaikin's Approach to Fractal Modeling of Contours
http://www.maplesoft.com/applications/view.aspx?SID=1451&ref=Feed
This worksheet can be treated as the third part of our previous Maple applications
(<a href="http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651">On Fractal Modeling Of Contours</a> and <a href="http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657">Probabilistic Approach To Fractal Modeling Of Shapes</a>)
in which we used analytical representation of Bezier curves in 2D contour modeling. Here we demonstrate how one can model fractally any contour basing on purely geometric Chaikin's approach.<img src="/view.aspx?si=1451/Chaikin_20.gif" alt="On Geometric Chaikin's Approach to Fractal Modeling of Contours" align="left"/>This worksheet can be treated as the third part of our previous Maple applications
(<a href="http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651">On Fractal Modeling Of Contours</a> and <a href="http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657">Probabilistic Approach To Fractal Modeling Of Shapes</a>)
in which we used analytical representation of Bezier curves in 2D contour modeling. Here we demonstrate how one can model fractally any contour basing on purely geometric Chaikin's approach.1451Tue, 22 Mar 2005 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiProbabilistic Approach To Fractal Modeling Of Shapes
http://www.maplesoft.com/applications/view.aspx?SID=1441&ref=Feed
This worksheet is the continuation of On Fractal Modeling Of Contours, by the same authors. We recall that based on the astonishing result by Goldman it is possible to construct Iterated Function System (IFS) for Bezier curves and render them as attractors of their IFS. That IFS is uniquely determined by Bezier control points. Unfortunately, Goldman's method fails for linear segments that are in fact Bezier curves of degree 1. In On Fractal Modeling Of Contours we described in detail how to construct IFS both for quadratic Bezier curves and for linear segments. Quadratic Bezier arcs and linear segments create a set of parts of which one can made every shape with a sufficient accuracy. Then in On Fractal Modeling Of Contours we demonstrated some examples of contours (heart, leaves) generated fractally basing on a suitable set of IFS using deterministic method.<img src="/view.aspx?si=1441/ProbabilisticModellingShapes_35.gif" alt="Probabilistic Approach To Fractal Modeling Of Shapes" align="left"/>This worksheet is the continuation of On Fractal Modeling Of Contours, by the same authors. We recall that based on the astonishing result by Goldman it is possible to construct Iterated Function System (IFS) for Bezier curves and render them as attractors of their IFS. That IFS is uniquely determined by Bezier control points. Unfortunately, Goldman's method fails for linear segments that are in fact Bezier curves of degree 1. In On Fractal Modeling Of Contours we described in detail how to construct IFS both for quadratic Bezier curves and for linear segments. Quadratic Bezier arcs and linear segments create a set of parts of which one can made every shape with a sufficient accuracy. Then in On Fractal Modeling Of Contours we demonstrated some examples of contours (heart, leaves) generated fractally basing on a suitable set of IFS using deterministic method.1441Mon, 21 Feb 2005 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiOn Fractal Modeling Of Contours
http://www.maplesoft.com/applications/view.aspx?SID=1435&ref=Feed
This worksheet demonstrates how to create contours using fractals. In the development of the curves many concepts are presented such as attractors, bezier curves and fractal modeling.<img src="/view.aspx?si=1435/fractals.gif" alt="On Fractal Modeling Of Contours" align="left"/>This worksheet demonstrates how to create contours using fractals. In the development of the curves many concepts are presented such as attractors, bezier curves and fractal modeling.1435Mon, 07 Feb 2005 00:00:00 ZProf. Wieslaw KotarskiProf. Wieslaw KotarskiColor Plate: Julia Sets
http://www.maplesoft.com/applications/view.aspx?SID=1391&ref=Feed
This application proves the artistic side of mathematics by showing several graphs of Julia sets.<img src="/view.aspx?si=1391/JuliaSets_33.gif" alt="Color Plate: Julia Sets" align="left"/>This application proves the artistic side of mathematics by showing several graphs of Julia sets.1391Tue, 25 Mar 2003 12:33:20 ZMaplesoftMaplesoftRossler attractor
http://www.maplesoft.com/applications/view.aspx?SID=4239&ref=Feed
This worksheet illustrates the Rossler Attractor with animations<img src="/view.aspx?si=4239/attractor.gif" alt="Rossler attractor" align="left"/>This worksheet illustrates the Rossler Attractor with animations4239Tue, 12 Mar 2002 09:11:03 ZYufang HaoYufang Hao