Time-invariant dynamical systems

The second problem from (8.20) is to determine an expression for ${\dot x}(t)$. This is where the laws of physics, such as the acceleration of rigid bodies due to applied forces and gravity. The most common case is time-invariant dynamical systems, in which ${\dot x}$ depends only on the current state and not the particular time. This means each component $x_i$ is expressed as

$${\dot x}_i = f_i(x_1,x_2,\ldots,x_n) ,$$ (8.25)

for some given vector-valued function $f = (f_1,\ldots,f_n)$. This can be written in compressed form by using $x$ and ${\dot x}$ to represent $n$-dimensional vectors:

$${\dot x}= f(x) .$$ (8.26)

The expression above is often called the state transition equation because it indicates the state's rate of change.

Here is a simple, one-dimensional example of a state transition equation:

$${\dot x}= 2 x - 1 .$$ (8.27)

This is called a linear differential equation. The velocity ${\dot x}$ roughly doubles with the value of $x$. Fortunately, linear problems can be fully solved ``on paper''. The solution to (8.27) is of the general form

$$x(t) = \frac{1}{2} + c e^{2 t} ,$$ (8.28)

in which $c$ is a constant that depends on the given value for $x(0)$.