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.

In a linear dynamical system, the variation of a state vector (an

) equals a constant matrix (denoted

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.,

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.

If the initial vector

is the corresponding eigenvalue; the solution of this equation is as may be confirmed by substitution.

-dimensional space can be represented by a linear combination of the right and left eigenvectors (denoted

is a linear combination of the individual solutions for the right eigenvectors Similar considerations apply to the discrete mappings.

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

− τ λ +

Thus the two roots are in the form: and

then the eigenvalues are of opposite sign, and the fixed point is a saddle.

then the eigenvalues are of the same sign.

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).