There is another round of discussion of the *Chaos* aspects of software development projects and the models of this chaos.

- The
*chaotic*ones that are totally unpredictable like a*double pendulum* **A double pendulum (two pendulums attached to each other) is also a simple system**. It is easy to make and easy to understand. And yet, it undergoes unpredictable chaotic motion due to a high sensitivity to the initial setup of the pendulum.

The common paradigm of the *Double Pendulum *is misunderstood and misapplied to the problems of modeling systems.

A double pendulum, is one pendulum suspended from another, shown below. The path of the bottom element (m_{2}) is a
potentially chaotic system. When certain parameter ranges undergo a slight
change in one of the initial starting conditions there can be dramatic effect on the
subsequent motion of the pendulum. As a result the motion of a double pendulum
extremely difficult to predict without a *model *of the system. The external observation is the pendulum exhibits * seemingly* random or chaotic behavior.

The first step in solving for the motion of m_{2} is to understand the difference between chaotic motion and unpredictable motion. The double pendulum *can* behave in a chaotic manner. But the motion of m_{2} as well as m_{2} is predictable. These terms get mixed up and we need to keep them straight if we're going to apply them to paradigms outside of the double pendulum.

Chaos can described as:

- Being sensitive to initial conditions.
- Possessing topological mixing, meaning the system evolves over time.
- Possessing a density of periodic orbits, meaning every point in the space is approached arbitrarily closely by periodic orbits.

**Why Is This Important for Project Management**

When we use analogies in project management to explain a situation, we need to actually have the right analogy. The Double Pendulum is a common explanation of chaotic behaviour. But the Double Pendulum, while behaving chaotically, is a deterministic system driven by the solution to the Hamiltonian equations of motion. This is called *Hamiltonian Chaos*.

The Hamiltonian description of motion uses momentum as the variable.

Depending on the initial conditions the system behaves regularly or show chaotic behaviour. The solution to this equation can be found in Java at Physics Department of the University of Buffalo.

This solution plots the path of the double pendulum in its normal mode or in its chaotic mode. Solutions in Mathematica can be found in Matematica for Theoretical Physics: Classical Mechanics and Nonlinear Dynamics.

So we can see that the *solution* to the double pendulum is not *unknown*. The plot of the path can be *known* by running the simulation for the needed period of time.

What this means is using the Double Pendulum as an analogy for *chaotic* behaviour of projects requires careful wording and application. Human systems don't not have Hamiltonian's, double pendulums do. With Hamiltonian, we can write a simualtion to show the motion of the pendulum, run it and see whatit does - Exactly what it does. We can run the simulation forward in time and *Predict* what it will do sometime in the future.

No mysteries about the path of the pendulum parts, they are completly specified by the solutions to the Hamiltonian. They may look chaotic, unpredictable, and wandering all over the place. But the equations of motion are explictly defined in the Hamiltonian solution and its simulaiton in our favorite language, Java or Mathematica.