More advanced details: Let us consider a pinball (a prototype of "dynamical systems") made in the following way. In a sloped plane, we put a series of arrow of nails, one below the other, taking care that given two nails in one arrow the nail in the arrow below is placed forming an equilateral triangle (see figure). We drop balls from the top of the pinball, the balls bounce it the nails and we collect them in boxes, one box for each column, placed in the bottom. Can we predict where the balls will go? Can we win? Can we localize a ball in the top of the pinball in a way to send it to a selected box? The experience shows us that it is extremely difficult. No one can flip the ball with complete precision. Two shots which hit a nail only a hair’s breadth apart will bounce away at slightly different angles. As they drop, that difference in angle will move them slowly apart, so that when they hit the second nail, the separation will be more than slight. On the second bounce, each will go its separate way. Small changes in your activity result in big changes in the result. Mathematicians call this a chaotic system. The tendency of paths to separate over time is called sensitivity to initial conditions and is the defining characteristic of chaotic systems. In these cases, no useful deterministic prediction is allowed. Why this happens? What is the mechanism that leads to this behavior? Can we get some useful information? Can we obtain at least an statistical information? Can we make probabilities? Can we recover this information from the experience? What happen if we change a little bit the pinball? Can we model those systems? Are the "qualitative mechanism" that leads to the "chaos" that we could observe in the pinball, a general mechanism that appears in any systems that we see as chaotic?

These questions have directed the theory of dynamical systems.

Let us choose a class of "more mathematical" dynamical systems: for instances, differential equations, iteration of transformation or maps. Can we know the trajectories? What can we say of their "dynamics"? Can we predict? Even if we can not know the behavior of all dynamical systems, we would like to understand a large class of then, i.e.: we would like to describe the dynamics of "big sets" (generic or residual, dense, etc) in the space of all dynamical systems.

In the direction to understand this, we could pick up a dynamical property that we could consider essential (for instance, the transitivity which means that there is dense trajectory, or that the dynamic is "extremely mixed") and roughly speaking we could split the set of dynamics into the one´s that robustly exhibit this behavior and the one´s that do not fall in this category.

So, we would try to understand on one hand, which are the structure that yields systems to be (for instance) robustly transitive or chaotic, and from this to obtain a deeper understood of the dynamic. In others words, from the simple information that the systems looks chaotic (which in particular make hopeless any deterministic description) and remains chaotic when we change it a little bit, we would like to know how the trajectories "are mixed" and also we would like to know if it is at least possible to get an statistical description. On the other hand, we should try to know, the dynamical mechanism that leads to the lack of transitivity or chaoticity and also to extract the dynamical consequences of these mechanism.

In the case of differentiable dynamics, this approach leads to study the interplay between the the dynamic in the ambient space and dynamic of the tangent application ("the derivative"). In few words (and reducing the problem to maps), how the dynamics of the tangent map controls or determines the underlying dynamics.

To be more precise, on one side we see that robust dynamical properties imply the existence of some structure in the the tangent dynamic. Though, to see if these structure are the mechanisms that characterize the dynamical property involved and to get more information, we have to study which is the feedback from the structure introduced.

On the other side, we want to know if the lack of this structure leads to the lack of the dynamical property that we are considering. These kind of problem push the studies in the direction to understand bifurcation and their dynamical consequences.