Linear Dynamical System - Introduction

Introduction

In a linear dynamical system, the variation of a state vector (an -dimensional vector denoted ) equals a constant matrix (denoted ) multiplied by . This variation can take two forms: either as a flow, in which varies continuously with time

\frac{d}{dt} \mathbf{x}(t) = \mathbf{A} \cdot \mathbf{x}(t)

or as a mapping, in which varies in discrete steps

\mathbf{x}_{m+1} = \mathbf{A} \cdot \mathbf{x}_{m}

These equations are linear in the following sense: if and are two valid solutions, then so is any linear combination of the two solutions, e.g., where and are any two scalars. The matrix need not be symmetric.

Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.

Read more about this topic:  Linear Dynamical System

Famous quotes containing the word introduction:

    “We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.”
    —Angela Carter (1940–1992)

    “Such is oftenest the young man’s introduction to the forest, and the most original part of himself. He goes thither at first as a hunter and fisher, until at last, if he has the seeds of a better life in him, he distinguishes his proper objects, as a poet or naturalist it may be, and leaves the gun and fish-pole behind. The mass of men are still and always young in this respect.”
    —Henry David Thoreau (1817–1862)

    “The role of the stepmother is the most difficult of all, because you can’t ever just be. You’re constantly being tested—by the children, the neighbors, your husband, the relatives, old friends who knew the children’s parents in their first marriage, and by yourself.”
    —Anonymous Stepparent. Making It as a Stepparent, by Claire Berman, introduction (1980, repr. 1986)