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.
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
are two valid solutions, then so is any linear combination
of the two solutions, e.g.,
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.
Linear approximation of a nonlinear system: classification of 2D fixed point according to the trace and the determinant of the Jacobian matrix (the linearization of the system near an equilibrium point).
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).