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.

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