My book (Advanced Engineering Mathematics 2nd Edition - MD Greenberg) states in Theorem 3.4.3 that in order for a system of Differential Equations to be stable "it is necessary and sufficient that the characteristic equation have no roots to the right of the imaginary axis in the complex plane and that any roots on the imaginary axis be nonrepeated."
I understand the part about the roots of the characteristic equation being on the left of the $Imaginary$ axis only but wouldn't it also hold if the roots were within the unit circle on the Argand plane?
The negative half-plane is the condition for the exponent, being inside the unit circle is a condition for the basis.
That is, in linear differential equations with constant coefficients one usually uses a trial solution $y(t)=ce^{\lambda t}$. This is then bounded for positive times if the real part of the exponent factor $\lambda$ is non-positive.
In other contexts like difference equations (linear recursion equations) with constant coefficients or numerical ODE integration methods, especially the linear multi-step methods, the trial solution takes the form $y_n=cq^n$. This is bounded for positive integer $n$ if the basis $q$ is inside or on the unit circle.
Both can be connected if the exponential function is sampled on an equidistant sequence $t_n=nh$. Then $e^{\lambda t_n}=(e^{\lambda h})^n$, and, as $h>0$, $|e^{\lambda h}|\le 1$ is equivalent to $Re(\lambda)\le 0$.