The Butterfly Effect

Surely many people have heard of the Butterfly Effect. Behind this name we find the formulation of the theory of chaos. After the discovery that the universe has a story (Big Bang) and we certainly two key theories of physics (quantum and relativity) the formulation of the unpredictability of events that changed the conception of our world now. We are facing a revolution totally ignored by majority of the population, which is easier to think about God's plans, conspiracies and other fantasies when he sees the events of our day; but the reality is that man proposes and random features.

Pierre Simon de Laplace certainly was one of the key figures in the scientific worldview to the early twentieth century. He was defending that the laws of physics were above the divine will, he maintained that the Causal Determinism was the thread which moved events.

The principle of Causal Determinism postulates that any outcome is due to a cause and this is the previous state under the action of physical laws. Thus for one particular initial state is only compatible one single final state. This had been confirmed by principle very extended in the scientific world of the time, which explained that the world was conceived from simple basic laws that could be expressed as simple algebraic formulas, presumably analytical. A better-known example is the second law of Newtonian dynamics (F = ma) that you simply conceptual and algebraic, all the physical preparation of the XVI, XVII and XVIII are sums, differences, multiplication and division, exponentiation with a fixed (squares, cubes ...) of some variable, nothing compared to the current physical expressions.

Lagrange, contemporary of Laplace, established from its system of equations one analytical and universal way to solve mechanical systems.  The system of equations gave a unique solution provided that could provide enough initial boundary conditions. Would only be possible the multievaluation if unknown or only partially known some initial conditions. Chance was simply lack of knowledge of all initial conditions, summarizing when we are throwing the craps is as Julius Caesar crossing the Rubicon: the result is already given regardless of the events that happen later, because these will be determined by the laws of physics and the first action. Yes Atheist Lagrange and other eighteenth and nineteenth century physics (believers or not) the fate existed; it was a logical consequence of what happened in the second 0 of history.

But after the discovery of the planet Neptune in the mid- nineteenth King Oscar II of Sweden was interested in the stability of the solar system, giving a prize to those who demonstrate the stability of this or not. We do not know if the king Oscar was doing for philanthropy or he had real fears that the solar system will be collapsing, but in the same way that Neptune was discovered by perturbations in the orbit of Uranus, it was plausible that a combined disturbance could lead to a loss of stability of the set of objects which are orbiting the Sun. The French mathematician Henri Poincare found a stable solution won the award but a friend asked him to see that he had made a mistake, when he knows he returned the prize until to find a solution; the solution had be finding: he demonstrates there was no solution. From 'no solution' to the stability of the solar system produce a first Chaos Theory is based on the fact that the propagation of errors made temporary diverges in long-term extrapolation was small divergence in origin. Whether the solution was not so strange, first we don’t come out of the first formulation of luck where imprecision in the initial values is simply unknowing of the exact value and second from Newton tried to solve systems of three bodies in gravitation no general solution and hence on all of the solar system was nothing more than a Chimera; however the result was disturbing not so much for danger of collapse (unlikely), but the fact of initial values toward the end result had to maintain proportionality, which isn’t happening.

The true theory of chaos will come with Edward Lorenz. From his studies on climatic simulations showed that, as simple as it was the model, it evolved temporarily in a way drastically different for small variations in the initial values. In 1979 he exposes their studies at the annual meeting of the American Association for the Advancement of Science, in a paper originally called "Small mistakes to start can be fatal" but to turn it into paper change the title "Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" to paraphrase a Chinese proverb "beating wings of a butterfly can be seen in the whole world." The title of this article is the popular notion of Butterfly effect although in fact the name was due to the previous drawing similar to a butterfly that was obtained graph generated by the evolution of two variables; as shown in the illustration in the begin of write. 

What we perceive as equilibrium is only a steady state that depends of more than two variables with nonlinear relationships crossed. Thus the time evolution of reduced set of variables will move more or less delimited (it would be one of the wings of the butterfly) in a region, that we call attractor, until some of these values out of the zone of influence of the attractor; then evolution is totally chaotic until it falls into a new attractor or the same where it was before, going to have a new steady state. Chaos isn’t the steps between attractors properly but rather from the fact that negligible order variations in the initial conditions produce big changes in evolution. An example are the illustrations that follow; three numeric simulations that in the illustration start in the same physical location as the variation is so small that the graph can’t represent, their trajectories in the begin of simulation are identical, when some time has passed they go out of the first loop (it is much smaller than it looks in the picture but that would be the beginning of the path in the middle ear of the second great illustration) and enter a second loop where we begin to separate the paths each in a different color; at the moment that shows the second picture we can see that a particle returns to the first loop (red) second still orbiting the second loop (blue) and the third has taken a path of escape (green). In the last picture we see each particle follows its own path by a number of independent random orbits and then jumping to another loop.

What happens in reality is that for certain ranges of values, systems have more than one possible behavior. The system may be in a stable or unstable regime but only one when it is outside this range, in contrast it is within this interval can jump randomly to any of the possible behaviors. To make a comparison as if it is a flat set of holes, through which runs a ball delivery which can be introduced randomly in any of these. To get the ball in the hole may have enough energy to overcome holes; it will be trapped in the hole otherwise. Suppose however that the walls of the hole are tunnels where the ball can move freely between holes without having to overcome the potential; then that the ball remains in the hole no longer depend only on energy but if the ball passes over one of these holes. If this tunnel there are many with a very small impact parameter, it causes falling in a tunnel or another depend of almost punctual values on the initial conditions when ball downed into hole, as well as the time that it remains in the hole which is depending of tunnel which finally introduced. If we experiment was sending balls infinitesimal variations of the initial values, surely we can establish a statistical behavior which we relate to the number of holes and impact parameter, but it will be impossible to establish a predictable behavior. Extrapolating it to any system into one-evaluable area, it passes between stable and unstable states completely predictable, but when in areas of multi evaluation then it can be in any of the stable states, introducing randomly in any of unstable states which will lead it to another stable state compatible.

The great news is that the future is not written, and that any disturbance for stupid insignificant it may seem to change the course of events is important. Everything lefts their mark in this world, but on average most effects are canceled by others against the many millions of actions that occur at any time. As we all know everyone, lucky is to be in the right time and place right, and so insignificant an event at a specific time and specific place could condition the fate of the entire universe going to be spread over time. Yes a butterfly really can trigger Nuclear Apocalypse if it takes the wrong direction, but it's impossible to know what might be.