Advances in Conception about Earthquakes - Earthquake Engineering for Structural Design - page 125

a starting of initial conditions with 2.000000 values, the final result can be entirely
different from the same system with a start of 2.000001 values. Starting from this
revelation, it is impossible to predict the weather accurately. In 1963, Lorenz
published a paper describing what he had discovered in a meteorological journal,
because he was a meteorologist, not a mathematician or physicist. As a result,
Lorenz's discoveries were not acknowledged until years later, when others
rediscovered them. Many scientists believe that the 20

th

Century will be known for
only three revolutionary theories: relativity, quantum mechanics and Chaos.
A comparison of a linear and nonlinear system is shown in (Figure 4.15). In a
linear system the left or right fluctuations have no great influence. For this system,
Newtonian laws show how it is possible, given an initial point in phase space, to
plot out the trajectory for all future times. In other words, given the full
specification of the system, it is possible to determine its future behavior. Any
external force or perturbation will produce a predictable change, while tiny
external fluctuations have a negligible effect.
In exchange, for a complex nonlinear system, the fluctuations produce
increasing effects as the parameter increases in value, but the fluctuations have
periodical characteristics. At the same value of parameter, the curve parameter-
effect breaks in two in a point named bifurcation. The fluctuations increase, but
remain in the periodical field. Raising the growth rate a little more, a new
bifurcation occurs in four different values. As the parameter rose further, the curve
bifurcated again. The bifurcation comes faster and faster, until suddenly, chaos
appears.
In this field it is impossible to predict the behavior of the Complex Nonlinear
System , because it strongly depends on the initial value from which the chaos
occurs.
The great influence of a small perturbation in the initial conditions can grow to
have unpredictably large consequences. The meteorologist Lorenz called this
phenomenon the butterfly effect. The sensitivity to these initial conditions is
described in the following metaphor:

“… A flap of a butterfly's wings inBrazil starts a tornado in Texas…”

showing, in a philosophical signification, the impossibility to determine the
behavior of nonlinear complex system in the high field of parameters. Even the
deterministic systems, which are not stochastic or random systems, could present,
under some conditions, the chaotic motions.
So, a fundamental question arises, however in this pessimistic sentence: Is it
possible to predict the main factors influencing the chaotic motions, even if no
exact description is possible? Even if there are still much more unsolved than
solved problems, the answer is positive. There are some properties of the chaotic
phenomena, discovered by the new Theory of Chaotic Dynamics , which allow the
prediction of the chaotic motions. One of them is the deterministic feature of the
parameters of the initial conditions, studied in the frame of the Theory of
Deterministic Chaos (Moon, 1987, Schuster, 1988).