In this article, we analyze the chaotic behavior of finite difference operators associated with certain differential equations. Our examples range from numerical schemes for a birth-and-death model with proliferation to a class of second-order partial differential equations that includes the hyperbolic heat transfer equation, the telegraph equation, and the wave equation. We provide sufficient conditions on the spatial and time steps of the scheme that guarantee chaos for the corresponding operators, and we compare them with the conditions needed to ensure chaotic analytical solutions.