Wikipedia / Linear dynamical system
Wikipedia / Linear dynamical system
Linear dynamical systems are dynamical systems whose evolution 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
-dimensional vector. This variation can take two forms: either as a flow varies continuously with time
varies in discrete steps
are two valid solutions, then so is any linear combination are any two scalars 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
is aligned with a right eigenvector of the matrix, the dynamics are simple
is the corresponding eigenvalue; the solution of this equation is
as may be confirmed by substitution.
is diagonalizable-dimensional space can be represented by a linear combination of the right and left eigenvectors.
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 - I) are the eigenvalues of A, to each other may be used to determine the stability of the dynamical system
is the trace is the determinant of A. Thus the two roots are in the form:
,
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).