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
N
x
A
x
x
d | |
dt |
x(t)=Ax(t)
or as a mapping, in which
x
xm+1=Axm
These equations are linear in the following sense: if
x(t)
y(t)
z(t) \stackrel{def
\alpha
\beta
A
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
x0 \stackrel{def
rk
A
d | |
dt |
x(t)=Ark=λkrk
where
λk
x(t)=rk
λkt | |
e |
If
A
N
lk
A
x0=
N | |
\sum | |
k=1 |
\left(lk ⋅ x0\right) rk
Therefore, the general solution for
x(t)
x(t)=
n | |
\sum | |
k=1 |
\left(lk ⋅ x0\right) rk
λkt | |
e |
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,
λn
d | |
dt |
x(t)=Ax(t).
λ2-\tauλ+\Delta=0
\tau
\Delta
λ | ||||
|
λ | ||||
|
\Delta=λ1λ2
\tau=λ1+λ2
\Delta<0
\Delta>0
\tau>0
\tau<0