Dynamics theory has its origin in newtonian mechanics that studies systems of particles moving under the effect of external forces. Despite the upheavals that physics caused by the discoveries of relativity and quantum theory the ideas of Newtonian Dynamics remain still alife and they have given to mathematics and physics, and more recently, to chemistry, bilogy, computer science, ..., lots of analytic tools to study their behavior. Scientists have always tried to understand processes in order to predict effects from causes. Mathematician and astronom Pièrre Simon de Laplace (1749-1827) announced that if we could imagine an enough large conscience able to know all the exactly localizations of all the objects of the univers and their speeds, as also all the forces, then no secrets would be in this conscience. It could compute anything about the past and the future from cause-effect laws (this idea is known as Laplace Demon). Here rises the main idea of determinism, in which the universe was comparable with a very precise clock.
So past-present-future are tied by causal relations. So obtaining all relevant data we can do an exact prediction. All these ideas related to determinism ended with the claim of the uncertainty principle of Werner Heisenberg (1927). According to Heisenberg, we can not know the present in all its details and all perception is a selection from an abundance of possibilities and a limitation of future possibilities [...]. Because all experiments are left to quantum mechanics laws and so to uncertainty principle too, invalidity of causality law is definitely set through quantum mechanics. But, was that the end of determinism ? the answer is no. Heisenberg's uncertainty principle have not finished with determinism, it has only change it. The most accurate experiment can not be totally isolated of influences caused by the surrounding world, and the state of a system can not be known in a precise way at any time. Absolute mathematical precission proposed by Laplace is not physically possible. But Laplace and Heisenberg claimed theories bringing us to the two extremes where all can be known and nothing can be known about systems.

What we could say is that approximately the same causes follow approximately the same effects. If this wasn't true we would be unable to cite natural laws or build working machines. Predictability depends on the system under study. And not always systems formed by lots of variables or freedom degrees are the less predictables. As stated dy Aristotle: the whole is more than the sum of its parts (Metaphysica 10f, 1045a). Aristotle visioned the paradigm of complexity, where nonlinear interactions must be considered in order to analyze complex systems. But not all the multivariable systems show unpredictability. For instance, tides are processes caused by lots of factors (shape of the coastline, sea temperature, salinity, sea pressure, waves on sea surface, the position of the Sun and the Moon, to cite some of them) but we can predict them easily. What is really important is the way in which variables interact, or in other words, the degree of nonlinearity. For example, Edward N. Lorenz showed, in 1962, that a simple set of three continuous equations involved in cellular convection related to the atmospheric system could unfold extremely complex dynamics. He actually realized that numerical solutions of the differential equations of his model were nonperiodic and unstable i.e. sensitive to initial conditions (chaos). He really demonstrated that long-term weather forecasts can not be exact although if we were able to collect climatic data every second in weather stations in each square metre on the Earth. From that paper, that was published in a meteorological scientific magazine, was initially ignored by physicists and mathematicians, but, in fact, a new science had just already born. Nonlinear dynamics, chaos theory and more generally talking, complexity science, have seduced lots of people in the world, and of course scientists too. A science related to complex dynamics, to fractals, to chaos, to strange creatures opened up in phase space, to imagination, to the limits of the mind and to the most wild and inexplored subject in mathematics, physics, ecology, language or molecular biology. Mathematically, dynamics are studied from Dynamical systems theory. Such theory deals with continuous as well as discrete systems (i.e. equations, maps or computation algorithms). A very important part of dynamical systems is topology. In this context, the solutions of the previously mentioned systems i.e. continuous and discrete ones, are interpreted as a state variables of an N dimensional abstract space with N as the degrees of freedom or number of variables involved in the dynamics under study. Such abstract space is the so-called phase space or state space.

Mathematics is the art of giving the same name to different things
Jules Henri Poincaré