One of the keys to profitable trading is to take only those trades with the most potential and stay out of situations where there is marginal potential. My trading group has completed fifteen years of intensive research into applying the new Science of Chaos, along with quantum physics, holography, cybernetics, nonlinear dynamics, information theory, and fractal geometry, to the world of stock and commodity trading. When this research started, we were using super computers. Through a process of literally millions of iterations, we devised a process that will work on ordinary personal computers. The Alligator is basically a compass to keep you trading in the appropriate direction no matter which way the immediate price is moving. The Alligator is a personification of this process and influences every signal in my trading group’s arsenal. We will demonstrate both the setup and how to use this valuable tool while trading. In this chapter, we will describe the Alligator, what it does, how to construct it, and how to trade it.