Definitions from Stability of systems

A discrete-time system describes values $x_n$ at times $n=0,1,2,\ldots$ through a recurrence

A first-order system uses $x_{n+1}=f(n,x_n)$ and needs one initial value; a second-order system uses $x_{n+1}=f(n,x_n,x_{n-1})$ and needs two.

A homogeneous second-order discrete linear system has the form

Here $a,b,c\in\mathbb C$, $a\ne0$ and the coefficients are constant, so a characteristic-root calculation can replace the recurrence by an algebraic equation.

For the linear systems considered here, a system is stable when every solution satisfies $|x_n|\to0$ as $n\to\infty$ and unstable when some solution is unbounded.

A continuous-time system describes a function $x(t)$ through a differential equation with $t\in\mathbb R$. For $a,b,c\in\mathbb C$ with $a\ne0$, consider the second-order homogeneous equation

The exponential trial function $x(t)=Ke^{rt}$ leads to the algebraic equation

For the linear systems considered here, a continuous system is stable when every solution decays to zero as $t\to\infty$ and unstable when some solution is unbounded.