“Linear dynamical system”的意思、由来-开放百科全书

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Linear dynamical system

释义

Introduction
Solution of linear dynamical systems
Classification in two dimensions
Linear systems and higher-order differential equations
Linearizing nonlinear systems
See also
External links

Linear dynamical systems are dynamical systems whose evaluation functions are linear. While dynamical systems, in general, do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the equilibrium points of the system and approximating it as a linear system around each such point.

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

or as a mapping, in which

varies in discrete steps

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.

Solution of linear dynamical systems

If the initial vector

is aligned with a right eigenvector of

the matrix , the dynamics are simple

where is the corresponding eigenvalue;

the solution of this equation is

as may be confirmed by substitution.

If is diagonalizable, then any vector in an -dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted ) of the matrix .

Therefore, the general solution for is

a linear combination of the individual solutions for the right

eigenvectors

Similar considerations apply to the discrete mappings.

Classification in two dimensions

The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A. The sign and relation of these roots, , to each other may be used to determine the stability of the dynamical system

For a 2-dimensional system, the characteristic polynomial is of the form where is the trace and is the determinant of A. Thus the two roots are in the form:

Note also that and . Thus if then the eigenvalues are of opposite sign, and the fixed point is a saddle. If then the eigenvalues are of the same sign. Therefore, if both are positive and the point is unstable, and if then both are negative and the point is stable. The discriminant will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).

Introduction

Solution of linear dynamical systems

Classification in two dimensions

See also