The goal of our project is to use rigorous computations to prove the existence of chaotic symbolic dynamics inside the attractor of the LPA model. To this date no other proofs of chaos for this model exist. The LPA model is a three dimensional population model of flour beetles. Much work has been done studying the dynamics of this system, and experiments even seemed to imply chaos(See http://caldera.calstatela.edu/nonlin/ for data and papers published about the model).

So, how are we going about proving the existence of chaos in this system? First, we discretize our phase space into 3-D cubes at a certain resolution(also referred to as depth) to obtain a finitely computable represention of our system. We use a program called GAIO to obtain a collection of these boxes(http://math-www.uni-paderborn.de/%7Eagdellnitz/gaio/). We can see how our model maps these boxes forward, and from this map a transition matrix can be obtained, which states the boxes that any given box maps to. From this transition matrix one can find candidate boxes for fixed points, n-cycles, and connections between them. 

To prove the existence of invariant objects(i.e. fixed points, n-cycles, chaotic horseshoes,..) we use a mathematical tool called the Conley Index. For a discussion of the ideas behind Conley Index see http://www.math.gatech.edu/~mischaik/pub_page/paperlist_index.html. When we want to investigate an invariant object, we first try to isolate it from the other objects in the system. Once isolated, what is called an index pair can be constructed, and then using a program called CHomP (http://www.math.gatech.edu/~chomp/) we can compute the index, from which a rigorous proof of the invariant objects existence can be constructed. For an example of this procedure and a proof of the existence of a chaotic horseshoe using symbolic dynamics in the Henon map see http://www.math.cornell.edu/%7Esday/thesis.pdf.

Most applications start off with a root box at a depth of 1, then subdivide this box by calculating the maximal invariant set in the box. We originally sought to do just this. However, we found that the computations to do this would be so intensive that it would not be worth proceeding in this direction, because we need to go to a very fine resolution to get results. So we decided to manually insert boxes around the attractor, and then try to say something about the system. At first we could find no low periodic orbits in the attractor. We later proved that there are no periodic orbits of period smaller than 11. This further complicates matters when trying to find a chaotic horseshoe. 

As of now, we are trying to first grow a complete isolating neighborhood around the attractor. We are confident that once we have done this, discovery of the chaotic horseshoe, and thus the proof of chaos in the LPA model, will follow shortly.