Dynamical Systems: New Applications
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en-us2014 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 23 Apr 2014 22:16:07 GMTWed, 23 Apr 2014 22:16:07 GMTNew applications in the Dynamical Systems categoryhttp://www.mapleprimes.com/images/mapleapps.gifDynamical Systems: New Applications
http://www.maplesoft.com/applications/category.aspx?cid=189
Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies
http://www.maplesoft.com/applications/view.aspx?SID=153515&ref=Feed
<p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p><img src="/view.aspx?si=153515/Elastic_Collision_image1.jpg" alt="Descartes & Mme La Marquise du Chatelet And The Elastic Collision of Two Bodies" align="left"/><p><strong><em> ABSTRACT<br /> <br /> The Marquise</em></strong> <strong><em>du Chatelet in her book " Les Institutions Physiques" published in 1740, stated on page 36, that Descartes, when formulating his laws of motion in an elastic collision of two bodies B & C (B being more massive than C) <span >having the same speed v</span>, said that t<span >he smaller one C will reverse its course </span>while <span >the more massive body B will continue its course in the same direction as before</span> and <span >both will have again the same speed v.<br /> <br /> </span>Mme du Chatelet, basing her judgment on theoretical considerations using <span >the principle of continuity</span> , declared that Descartes was <span >wrong</span> in his statement. For Mme du Chatelet the larger mass B should reverse its course and move in the opposite direction. She mentioned nothing about both bodies B & C as <span >having the same velocity after collision as Descartes did</span>.<br /> <br /> At the time of Descartes, some 300 years ago, the concept of kinetic energy & momentum as we know today was not yet well defined, let alone considered in any physical problem.<br /> <br /> Actually both Descartes & Mme du Chatelet may have been right in some special cases but not in general as the discussion that follows will show.</em></strong></p>153515Fri, 07 Mar 2014 05:00:00 ZDr. Ahmed BaroudyDr. Ahmed BaroudyDynamics of the Battlefield: The Lanchester Model
http://www.maplesoft.com/applications/view.aspx?SID=146801&ref=Feed
<p>Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Frederick W. Lanchester, became fascinated by the dynamics of the battlefield. Various mathematical models were proposed in an effort to explain--and to predict--how military forces interacted on the battlefield. During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima. Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered many casualties and injuries.</p><img src="/view.aspx?si=146801/army2.JPG" alt="Dynamics of the Battlefield: The Lanchester Model" align="left"/><p>Around the time of World War I, July 28, 1914 to November 11, 1918, many mathematicians and engineers, including Frederick W. Lanchester, became fascinated by the dynamics of the battlefield. Various mathematical models were proposed in an effort to explain--and to predict--how military forces interacted on the battlefield. During World War I these mathematical investigations were mainly academic, although during World War II the United States government actually applied these models to make important decisions about the Battle of Iwo Jima in which the American forces seized control of the Japanese island of Iwo Jima. Outnumbered and outgunned by the Americans, the Japanese were defeated even before the battle began although the American forces suffered many casualties and injuries.</p>146801Mon, 06 May 2013 04:00:00 ZDouglas LewitDouglas LewitBifTools - 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 DimitrovaAn Epidemic Model (for Influenza or Zombies)
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<p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p><img src="/view.aspx?si=127836/Cholera.jpg" alt="An Epidemic Model (for Influenza or Zombies)" align="left"/><p>Systems of differential equations can be used to model an epidemic of influenza or of zombies. This is an interactive Maple document suitable for use in courses on mathematical biology or differential equations or calculus courses that include differential equations. No knowledge of Maple is required.</p>127836Thu, 17 Nov 2011 05:00:00 ZDr. Robert IsraelDr. Robert IsraelThe Orbit of Kepler 16b
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<p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a double-star system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p><img src="/view.aspx?si=126766/kepler16b.png" alt="The Orbit of Kepler 16b" align="left"/><p>NASA's Kepler space telescope recently made the news by finding a planet that orbits a double-star system, a situation that brought to mind the fictional planet Tatooine of the movie Star Wars. On such a planet, if it had a solid surface, you could see a double sunset. <br /><br />This worksheet explores the orbital mechanics of such a system.</p>126766Tue, 18 Oct 2011 04:00:00 ZDr. Robert IsraelDr. Robert IsraelVarInt
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<p><strong>VarInt</strong> - <em>Computer Algebra Aided Design of Variational Integrators</em>.</p>
<p>Create, design and analyse (new) variational integrators for autonomous dynamical systems (with non-conservative forces) up to arbitary order with VarInt, a package for Maple.</p><img src="/view.aspx?si=88830/VarIntSmall.jpg" alt="VarInt" align="left"/><p><strong>VarInt</strong> - <em>Computer Algebra Aided Design of Variational Integrators</em>.</p>
<p>Create, design and analyse (new) variational integrators for autonomous dynamical systems (with non-conservative forces) up to arbitary order with VarInt, a package for Maple.</p>88830Tue, 25 Jan 2011 05:00:00 ZChristian HellstrÃ¶mChristian HellstrÃ¶mAttracteurs de Gumowski-Mira
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<p class="MsoNormal">En 1980, deux physiciens I. Gumowski et C. Mira, du centre de recherche CERN de Genève en Suisse utilisèrent le système d'équation</p>
<p class="MsoNormal"><span> </span>x<sub>n+1</sub> = By<sub>n</sub> + f(x<sub>n</sub>)<sub><o:p></o:p></sub></p>
<p class="MsoNormal"><span> </span>y<sub>n+1</sub> = - x<sub>n</sub> + f(x<sub>n+1</sub>)</p>
<p class="MsoNormal"><span> </span>où f(x) = Ax + 2(1 - A) x<sup>2</sup> / (1 + x<sup>2</sup>) et A, B sont des constantes.</p>
<p><span>pour simuler la trajectoire de particules se déplaçant à haute vitesse dans un accélérateur de la forme d'une mince boîte cylindrique de plusieurs mètres de long. Ils découvrirent à leur grande surprise </span><span>que les trajectoires issues de ce système et portées sur un plan cartésien produisent des images surprenantes.</span></p><img src="/view.aspx?si=87666/G_M.png" alt="Attracteurs de Gumowski-Mira" align="left"/><p class="MsoNormal">En 1980, deux physiciens I. Gumowski et C. Mira, du centre de recherche CERN de Genève en Suisse utilisèrent le système d'équation</p>
<p class="MsoNormal"><span> </span>x<sub>n+1</sub> = By<sub>n</sub> + f(x<sub>n</sub>)<sub><o:p></o:p></sub></p>
<p class="MsoNormal"><span> </span>y<sub>n+1</sub> = - x<sub>n</sub> + f(x<sub>n+1</sub>)</p>
<p class="MsoNormal"><span> </span>où f(x) = Ax + 2(1 - A) x<sup>2</sup> / (1 + x<sup>2</sup>) et A, B sont des constantes.</p>
<p><span>pour simuler la trajectoire de particules se déplaçant à haute vitesse dans un accélérateur de la forme d'une mince boîte cylindrique de plusieurs mètres de long. Ils découvrirent à leur grande surprise </span><span>que les trajectoires issues de ce système et portées sur un plan cartésien produisent des images surprenantes.</span></p>87666Sun, 16 May 2010 04:00:00 ZAndre LevesqueAndre LevesqueDynamical 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 LynchOptimal Control Design of a Voice Coil Actuator Head in a Hard Drive
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<p>This document examines the steps involved in quickly and accurately controlling the position of the voice coil actuator head. The design of the closed-loop system, including the approach used to investigate the dynamics of the plant model and design the mathematical model of the controller, will be explained in detail.</p><img src="/view.aspx?si=6877/thumb.jpg" alt="Optimal Control Design of a Voice Coil Actuator Head in a Hard Drive" align="left"/><p>This document examines the steps involved in quickly and accurately controlling the position of the voice coil actuator head. The design of the closed-loop system, including the approach used to investigate the dynamics of the plant model and design the mathematical model of the controller, will be explained in detail.</p>6877Sun, 09 Nov 2008 05:00:00 ZMaplesoftMaplesoftControl Systems Design Tools: Creating and Working with System Objects
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Maple provides a series of controls systems design tools that give you the ability to work analytically with linear time-invariant dynamic systems. The DynamicSystems package is a collection of procedures for creating, manipulating, simulating, and plotting linear time-invariant systems models. In this Tips and Techniques, you will learn how linear systems are modeled using the DynamicSystems package, how to create System Objects, and how to transform your models between a variety of different representations.<img src="/view.aspx?si=6587/TransferFunction.JPG" alt="Control Systems Design Tools: Creating and Working with System Objects" align="left"/>Maple provides a series of controls systems design tools that give you the ability to work analytically with linear time-invariant dynamic systems. The DynamicSystems package is a collection of procedures for creating, manipulating, simulating, and plotting linear time-invariant systems models. In this Tips and Techniques, you will learn how linear systems are modeled using the DynamicSystems package, how to create System Objects, and how to transform your models between a variety of different representations.6587Thu, 28 Aug 2008 00:00:00 ZMaplesoftMaplesoftFrequency Domain System Identification
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System identification deals with the problem of identifying a model describing some physical system by measuring the response of the system. This example illustrates a problem where the structure of the model is known, and parameters of the model are identified. This is done by first designing the input signal, which is applied to the system. The measured output is converted to the frequency domain and the parameters are estimated in this domain.<img src="/view.aspx?si=1460/thumb.gif" alt="Frequency Domain System Identification" align="left"/>System identification deals with the problem of identifying a model describing some physical system by measuring the response of the system. This example illustrates a problem where the structure of the model is known, and parameters of the model are identified. This is done by first designing the input signal, which is applied to the system. The measured output is converted to the frequency domain and the parameters are estimated in this domain.1460Tue, 06 May 2008 00:00:00 ZMaplesoftMaplesoftDynamics in Spherical Coordinates
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A brief introduction to dynamics in spherical coordinates.<img src="/view.aspx?si=4892/Dynamics in Spherical Coords Sketch 1.jpg" alt="Dynamics in Spherical Coordinates" align="left"/>A brief introduction to dynamics in spherical coordinates.4892Thu, 05 Apr 2007 00:00:00 ZJ. M. RedwoodJ. M. RedwoodStandard Map on a Torus
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Maple worksheet for producing a standard map on a torus is developed. A brief introduction of Hamiltonian mechanics using a simple pendulum system as an example is provided, followed by a discussion of a kicked rotor system which consists of a simple pendulum with the potential energy turned on in delta-function pulses. The integration of the kicked rotor problem yields the standard map, which has characteristics of a large class of systems. Plotting the standard map on a torus facilitates a three-dimensional visualization the Kolmogorov-Arnold-Moser (KAM) theory of chaos in a Hamiltonian system. An animation demonstrating chaos created by homoclinic tangle near a hyperbolic point is contained in this worksheet.<img src="/view.aspx?si=1703/SMTorus.jpg" alt="Standard Map on a Torus" align="left"/>Maple worksheet for producing a standard map on a torus is developed. A brief introduction of Hamiltonian mechanics using a simple pendulum system as an example is provided, followed by a discussion of a kicked rotor system which consists of a simple pendulum with the potential energy turned on in delta-function pulses. The integration of the kicked rotor problem yields the standard map, which has characteristics of a large class of systems. Plotting the standard map on a torus facilitates a three-dimensional visualization the Kolmogorov-Arnold-Moser (KAM) theory of chaos in a Hamiltonian system. An animation demonstrating chaos created by homoclinic tangle near a hyperbolic point is contained in this worksheet.1703Tue, 10 Jan 2006 00:00:00 ZDr. Frank WangDr. Frank WangA Double Pendulum Model using DynaFlexPro and Maple
http://www.maplesoft.com/applications/view.aspx?SID=1682&ref=Feed
This model is an extension of the Simple Pendulum exercise in the DynaFlexPro manual (see Related Link above). It was a simple task to add another pendulum at the end of the first pendulum, and then generate a simulation using DynaFlexPro and Maple. You can see the DynaFlexPro model by following the related link above(note requires Java Runtime Engine, V1.5.0_04).
Try creating animations for different pendulum lengths and damping coefficients in the MapleNet application above.<img src="/view.aspx?si=1682/dblpendulum.JPG" alt="A Double Pendulum Model using DynaFlexPro and Maple" align="left"/>This model is an extension of the Simple Pendulum exercise in the DynaFlexPro manual (see Related Link above). It was a simple task to add another pendulum at the end of the first pendulum, and then generate a simulation using DynaFlexPro and Maple. You can see the DynaFlexPro model by following the related link above(note requires Java Runtime Engine, V1.5.0_04).
Try creating animations for different pendulum lengths and damping coefficients in the MapleNet application above.1682Tue, 08 Nov 2005 00:00:00 ZMaplesoftMaplesoftCreating an Animation of a Spatial Crank-Slider Mechanism using DynaFlexPro and Maple
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The following Maple Worksheet details the steps involved in creating an animation of a Spatial Crank-Slider Mechanism using Maple 10 and DynaFlexPro, a third party product offering from Maplesoft.<img src="/view.aspx?si=1681/Slider 3D - spatial.JPG" alt="Creating an Animation of a Spatial Crank-Slider Mechanism using DynaFlexPro and Maple" align="left"/>The following Maple Worksheet details the steps involved in creating an animation of a Spatial Crank-Slider Mechanism using Maple 10 and DynaFlexPro, a third party product offering from Maplesoft.1681Tue, 08 Nov 2005 00:00:00 ZMaplesoftMaplesoftMobile Robot Modeling and Simulation
http://www.maplesoft.com/applications/view.aspx?SID=1467&ref=Feed
<p>Derive the model of a two wheel differential drive mobile robot and simulate its trajectory response to various inputs. A 3-D trajectory animation of the mobile robot has been created (shown above) based on the open loop system response of the derived mobile robot model. Various open loop and closed loop responses of this system have been generated in a separate application example application entitled, <a href="/applications/view.aspx?SID=1466" >Mobile Robot</a>.</p><img src="/view.aspx?si=1467/MobileRobotModel_3.gif" alt="Mobile Robot Modeling and Simulation" align="left"/><p>Derive the model of a two wheel differential drive mobile robot and simulate its trajectory response to various inputs. A 3-D trajectory animation of the mobile robot has been created (shown above) based on the open loop system response of the derived mobile robot model. Various open loop and closed loop responses of this system have been generated in a separate application example application entitled, <a href="/applications/view.aspx?SID=1466" >Mobile Robot</a>.</p>1467Mon, 16 May 2005 04:00:00 ZMaplesoftMaplesoftInverted Pendulum on an Oscillating Table
http://www.maplesoft.com/applications/view.aspx?SID=4490&ref=Feed
We model the motion of an inverted pendulum supported on a table that oscillates vertically with motion y=A cos(wt). We show that for high enough frequency, the pendulum does not fall.<img src="/view.aspx?si=4490/1294.jpg" alt="Inverted Pendulum on an Oscillating Table" align="left"/>We model the motion of an inverted pendulum supported on a table that oscillates vertically with motion y=A cos(wt). We show that for high enough frequency, the pendulum does not fall.4490Thu, 25 Mar 2004 13:46:08 ZFrank WangFrank WangPendulums Coupled by a Spring
http://www.maplesoft.com/applications/view.aspx?SID=4405&ref=Feed
We model the motion of two identical pendulums swinging in parallel planes, attached by a spring. We describe the motion by the pendulums' angles of deflection over time.
We assume that the pendulums swing in the x-z plane, their hinges are d units apart on the y-axis, they have unit length and unit mass, and their mass is concentrated at the ends. Pendulum 1 swings about the origin, and pendulum 2 swings about the point (0, d , 0). We assume the spring has natural length d .<img src="/view.aspx?si=4405/pendulums.gif" alt="Pendulums Coupled by a Spring" align="left"/>We model the motion of two identical pendulums swinging in parallel planes, attached by a spring. We describe the motion by the pendulums' angles of deflection over time.
We assume that the pendulums swing in the x-z plane, their hinges are d units apart on the y-axis, they have unit length and unit mass, and their mass is concentrated at the ends. Pendulum 1 swings about the origin, and pendulum 2 swings about the point (0, d , 0). We assume the spring has natural length d .4405Thu, 07 Aug 2003 16:48:49 ZJason SchattmanJason SchattmanAnalysis and Simulation of Simple Dynamic Systems
http://www.maplesoft.com/applications/view.aspx?SID=1388&ref=Feed
Dynamic Systems (as they are used in this worksheet) are mechanical systems comprising of masses and constraints. Constraints are a set of rules that have to be followed as the masses are allowed to move freely. For example, a very simple dynamic system might be that of a pendulum. The constraint of a pendulum is that one mass has to stay at a fixed distance from a fixed mass (rigid-rod constraint). Examples of more complex system are a double pendulum and a triple pendulum.
4 case studies are presented in this work sheet:
Damped Oscillation,
Resonance,
Bead on wire, and
Vibrating Tower.
A practice problem can be found at the end of the work sheet.<img src="/view.aspx?si=1388/1103.gif" alt="Analysis and Simulation of Simple Dynamic Systems" align="left"/>Dynamic Systems (as they are used in this worksheet) are mechanical systems comprising of masses and constraints. Constraints are a set of rules that have to be followed as the masses are allowed to move freely. For example, a very simple dynamic system might be that of a pendulum. The constraint of a pendulum is that one mass has to stay at a fixed distance from a fixed mass (rigid-rod constraint). Examples of more complex system are a double pendulum and a triple pendulum.
4 case studies are presented in this work sheet:
Damped Oscillation,
Resonance,
Bead on wire, and
Vibrating Tower.
A practice problem can be found at the end of the work sheet.1388Mon, 07 Oct 2002 13:58:04 ZForhad AhmedForhad AhmedModeling of Atmospheric Motion and Astrodynamics
http://www.maplesoft.com/applications/view.aspx?SID=4309&ref=Feed
Equations of motion are systems of differential equations. Most practical systems are only solvable numerically. We show applications to movement in the atmosphere and the dynamics of space ships.<img src="/view.aspx?si=4309/1095.jpg" alt="Modeling of Atmospheric Motion and Astrodynamics" align="left"/>Equations of motion are systems of differential equations. Most practical systems are only solvable numerically. We show applications to movement in the atmosphere and the dynamics of space ships.4309Fri, 20 Sep 2002 14:42:03 ZStanislav BartonStanislav Barton