In controllability of continuous-time systems the map given by plays the role that plays in discrete-time.

These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor.

The definition for discrete-time systems is almost identical to that for continuous-time systems.

In particular, a linear continuous-time system governed by Eqn.

The last one is known as the bilinear transform, or Tustin transform, and preserves the (in)stability of the continuous-time system.

The zeros of the continuous-time system are in the right-hand side of the complex plane.

This exponential smoothing property matches the exponential decay seen in the continuous-time system.

It is necessary, however, to compensate for the frequency warping by pre-warping the given frequency specifications of the continuous-time system.

The derivation for a continuous-time system is similar, with summations replaced with integrals.

Almost everything in continuous-time systems has a counterpart in discrete-time systems.