Classifying Exponentials

QUIZ QUESTION::

Consider motion given by the following equation:

y(t) = e^{\lambda t}

Match the following values of \lambda to the written description of motion.

ANSWER CHOICES:

Value of Lambda	Description of motion
Growing oscillation
Pure exponential decay
Pure exponential growth
Pure oscillation
Decaying oscillation

SOLUTION:

Value of Lambda	Description of motion
Growing oscillation
Pure exponential decay
Pure exponential growth
Pure oscillation
Decaying oscillation

Stability vs Instability

We can also classify a function as stable, unstable, or neutrally stable based on what happens to the function as time progresses.

Stable Functions

Intuitively, a system is stable when small perturbations eventually go away. This means small deviations from an equilibrium state eventually go to to zero. More formally, we can define stability for a function as:

\lim_{t \rightarrow \infty} f(t) =0

Unstable Functions

A system is unstable when small perturbations are amplified with time.

\lim_{t \rightarrow \infty} f(t) =\infty

Neutrally Stable Functions

A system is "neutrally stable" when small perturbations don't go away, but stay bounded. Pure oscillations are a common example of neutral stability.

Relationship to Complex Exponentials

The real part of a complex exponential dictates the stability of the system. When the real part is positive, the system is unstable. When the real part is negative, the system is stable.