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<title>Dynamical system simulation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Dynamical system simulation</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For classical dynamics simulations, see <a href="Dynamical_simulation" title="Dynamical simulation">Dynamical simulation</a>.</div>

<p><b>Dynamical system simulation</b> or <b>dynamic system simulation</b> is the use of a computer program to model the time-varying behavior of a <a href="Dynamical_system" title="Dynamical system">dynamical system</a>. The systems are typically described by <a href="Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a> or <a href="Partial_differential_equations" class="mw-redirect" title="Partial differential equations">partial differential equations</a>. A simulation run solves the state-equation system to find the behavior of the state variables over a specified period of time. The equation is solved through numerical integration methods to produce the transient behavior of the state variables. Simulation of dynamic systems predicts the values of model-system state variables, as they are determined by the past state values. This relationship is found by creating a model of the system.<sup id="cite_ref-FOOTNOTEKorn20072_1-0" class="reference"><a href="#cite_note-FOOTNOTEKorn20072-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

<p>Simulation models are commonly obtained from discrete-time approximations of continuous-time mathematical models.<sup id="cite_ref-FOOTNOTEKleeAllen20163_2-0" class="reference"><a href="#cite_note-FOOTNOTEKleeAllen20163-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

As <a href="Mathematical_model" title="Mathematical model">mathematical models</a> incorporate real-world constraints, like gear <a href="Backlash_(engineering)" title="Backlash (engineering)">backlash</a> and rebound from a hard stop, equations become nonlinear. This requires numerical methods to solve the equations.<sup id="cite_ref-FOOTNOTEKleeAllen201693_3-0" class="reference"><a href="#cite_note-FOOTNOTEKleeAllen201693-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>A <a href="Numerical_simulation" class="mw-redirect" title="Numerical simulation">numerical simulation</a> is done by stepping through a time interval and calculating the integral of the derivatives through <a href="Numerical_integration" title="Numerical integration">numerical integration</a>.

Some methods use a fixed step through the interval, and others use an adaptive step that can shrink or grow automatically to maintain an acceptable error tolerance. Some methods can use different time steps in different parts of the simulation model.

</p><p>There are two types of system models to be simulated: difference-equation models, and differential-equation models. Classical physics is usually based on differential equation models. This is why most old simulation programs are simply differential equation solvers and delegate solving difference-equations to “procedural program segments.”Some dynamic systems are modeled with differential equations that can only be presented in an implicit form. These differential-algebraic-equation systems require special mathematical methods for simulation.<sup id="cite_ref-FOOTNOTEKleeAllen20163_2-1" class="reference"><a href="#cite_note-FOOTNOTEKleeAllen20163-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Some complex systems’ behavior can be quite sensitive to initial conditions, which could lead to large errors from the correct values. To avoid these possible errors, a rigorous approach can be applied, where an algorithm is found which can compute the value up to any desired precision. For example, the constant e is a computable number because there is an algorithm that is able to produce the constant up to any given precision.<sup id="cite_ref-FOOTNOTEGalatoloHoyrupRojas2011_4-0" class="reference"><a href="#cite_note-FOOTNOTEGalatoloHoyrupRojas2011-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<p>The first applications of computer simulations for dynamic systems was in the aerospace industry.<sup id="cite_ref-FOOTNOTEKleeAllen2016xiii_5-0" class="reference"><a href="#cite_note-FOOTNOTEKleeAllen2016xiii-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Commercial uses of dynamic simulation are many and range from nuclear power, steam turbines, 6 degrees of freedom vehicle modeling, electric motors, econometric models, biological systems, robot arms, mass-spring-damper systems, hydraulic systems, and drug dose migration through the human body to name a few. These models can often be run in <a href="Real-time_simulation" title="Real-time simulation">real time</a> to give a virtual response close to the actual system. This is useful in <a href="Process_control" class="mw-redirect" title="Process control">process control</a> and <a href="Mechatronic" class="mw-redirect" title="Mechatronic">mechatronic</a> systems for tuning the <a href="Automatic_control" class="mw-redirect" title="Automatic control">automatic control</a> systems before they are connected to the real system, or for human training before they control the real system.

</p><p>Simulation is also used in computer games and animation and can be accelerated by using a <a href="Physics_engine" title="Physics engine">physics engine</a>, the technology used in many powerful <a href="Computer_graphics" title="Computer graphics">computer graphics</a> <a href="Software" title="Software">software</a> <a href="Computer_program" title="Computer program">programs</a>, like <a href="3ds_Max" class="mw-redirect" title="3ds Max">3ds Max</a>, <a href="Maya_(software)" class="mw-redirect" title="Maya (software)">Maya</a>, <a href="LightWave_3D" title="LightWave 3D">Lightwave</a>, and many others to simulate physical characteristics. In computer animation, things like <a href="Hair" title="Hair">hair</a>, <a href="Cloth" class="mw-redirect" title="Cloth">cloth</a>, <a href="Liquid" title="Liquid">liquid</a>, <a href="Fire" title="Fire">fire</a>, and <a href="https://en.wiktionary.org/wiki/particles" class="extiw external" title="wiktionary:particles">particles</a> can be easily modeled, while the human <a href="Animator" title="Animator">animator</a> animates simpler objects. Computer-based dynamic animation was first used at a very simple level in the 1989 <a href="Pixar" title="Pixar">Pixar</a> <a href="Short_film" title="Short film">short film</a> <i>Knick Knack</i> to move the fake snow in the snowglobe and pebbles in a fish tank.

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<ul><li><a href="Comparison_of_system_dynamics_software" title="Comparison of system dynamics software">Comparison of system dynamics software</a> — includes packages not listed below</li>

<li><a href="Simulink" title="Simulink">Simulink</a> — A MATLAB-based graphical programming environment for modeling, simulating and analyzing dynamical systems</li>

<li><a href="MSC.Adams" class="mw-redirect" title="MSC.Adams">MSC Adams</a> — A multibody dynamics simulation software</li>

<li><a href="SimulationX" title="SimulationX">SimulationX</a>— Software for simulating multi-domain dynamic systems</li>

<li><a href="AMESim" class="mw-redirect" title="AMESim">AMESim</a> — Software for simulating multi-domain dynamic systems</li>

<li><a href="AGX_Multiphysics" class="mw-redirect" title="AGX Multiphysics">AGX Multiphysics</a> — A <a href="Physics_engine" title="Physics engine">physics engine</a> for simulating multi-domain dynamic systems</li>

<li><a href="Dymola" title="Dymola">Dymola</a> — Software for simulating multi-domain dynamic systems using the Modelica language</li>

<li><a href="EcosimPro" title="EcosimPro">EcosimPro</a> — A simulation tool for modeling continuous-discrete systems</li>

<li><a href="Hopsan" title="Hopsan">Hopsan</a> — Software for simulating multi-domain dynamic systems</li>

<li><a href="MapleSim" title="MapleSim">MapleSim</a> — Software for simulating multi-domain dynamic systems</li>

<li><a href="Modelica" title="Modelica">Modelica</a> — A non-proprietary, object-oriented, equation-based language for dynamic simulation</li>

<li><a href="Physics_engine" title="Physics engine">Physics engine</a></li>

<li><a href="VisSim" title="VisSim">VisSim</a> — A visual language for nonlinear dynamic simulation</li>

<li><a href="PottersWheel" title="PottersWheel">PottersWheel</a> — A Matlab toolbox to calibrate parameters of dynamic systems</li>

<li><a href="Simcad_Pro" title="Simcad Pro">Simcad Pro</a> — A dynamic and interactive discrete event simulation software</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-FOOTNOTEKleeAllen20163-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKleeAllen20163_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKleeAllen20163_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKleeAllen2016">Klee & Allen (2016)</a>, p. 3.</span>

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<li id="cite_note-FOOTNOTEKleeAllen201693-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleeAllen201693_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleeAllen2016">Klee & Allen (2016)</a>, p. 93.</span>

</li>

<li id="cite_note-FOOTNOTEGalatoloHoyrupRojas2011-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGalatoloHoyrupRojas2011_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGalatoloHoyrupRojas2011">Galatolo, Hoyrup & Rojas (2011)</a>.</span>

</li>

<li id="cite_note-FOOTNOTEKleeAllen2016xiii-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEKleeAllen2016xiii_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFKleeAllen2016">Klee & Allen (2016)</a>, p. xiii.</span>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFGalatoloHoyrupRojas2011" class="citation cs2">Galatolo, Stefano; Hoyrup, Mathieu; Rojas, Cristóbal (2011), <i>Dynamical systems, simulation, abstract computation.</i>, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1101.0833">1101.0833</a></span></cite></li>

<li><cite id="CITEREFKorn2007" class="citation cs2">Korn, Granino A. (2007), <i>Advanced dynamic-system simulation: model-replication techniques and Monte Carlo simulation</i>, John Wiley & Sons</cite></li>

<li><cite id="CITEREFKleeAllen2016" class="citation cs2">Klee, Harold; Allen, Randal (2016), <i>Simulation of dynamic systems with MATLAB and Simulink</i>, Crc Press</cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720114730/http://sbel.wisc.edu/Courses/ME751/2010/index.htm">Textbook and lectures on dynamic simulation</a></li>

<li><a rel="nofollow" class="external text" href="https://www.embedded.com/dynamic-system-simulation">Dynamic System Simulation</a></li></ul>

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<li><a href="Reservoir_simulation" title="Reservoir simulation">Reservoir simulation</a></li></ul>

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<ul><li><a href="Social_simulation" title="Social simulation">Social simulation</a></li>

<li><a href="Computational_sociology" title="Computational sociology">Sociology</a></li>

<li><a href="Computer_simulation_and_organizational_studies" title="Computer simulation and organizational studies">Organizational studies</a></li></ul>

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