Dynamical system simulation

Overview

Simulation models are commonly obtained from discrete-time approximations of continuous-time mathematical models. As mathematical models incorporate real-world constraints, like gear backlash and rebound from a hard stop, equations become nonlinear. This requires numerical methods to solve the equations.

A numerical simulation is done by stepping through a time interval and calculating the integral of the derivatives through numerical integration. 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.

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.

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.