procedures for determining the random and chaotic earthquake - - PowerPoint PPT Presentation
Analysis of different numerical procedures for determining the random and chaotic earthquake properties R. Magaa*, A. Hermosillo*, M. Prez** *Instituto de Ingeniera Universidad Nacional Autnoma de Mxico Ciudad Universitaria, 04510
*Instituto de Ingeniería Universidad Nacional Autónoma de México Ciudad Universitaria, 04510 México, Distrito Federal e-mail: rmat@pumas.iingen.unam.mx ** Centro Tecnológico Aragón FES Aragón, UNAM Email: marcelo@tigre.aragon.unam.mx
September 2010
geotechnical design.
appropriate mathematical tools (not limited to criteria used in stationary linear dynamic).
was realized for three earthquakes in Mexico, occurred in: Aguamilpa dam in Nayarit, Acapulco and Mexico City. The procedure followed is based on concepts of chaotic Hamiltonian mechanics, which is a generalization of the classic mechanics, and it lies on iterative equation systems, called maps.
dynamic systems, and criteria for identifying the chaotic time series content as well as the application of these criteria to the earthquakes mentioned.
can be understood as a process in which the action of the some system components over other ones. Feedback is the starting point for understanding the 'chaoticity' and complexity of many natural and social phenomena.
Hamiltonian dynamics. A wide class of physical phenomena can be described by Hamiltonian equations. This class includes particles, fields, classical and quantum objects, and it makes up a significant part of our knowledge of the basics of dynamics in nature. Hamiltonian dynamics is very different from, for example, dissipative dynamics, and its analysis uses specific tools that cannot be applied in other cases. Discovery of chaotic dynamics is a result of discovering new features in Hamiltonian dynamics and new types of solutions of the dynamical equations.
A Hamiltonian system with N degrees of freedom is characterized by a generalized coordinate vector , generalized momentum vector , and a Hamiltonian H = H(p, q) such that the equations of motion are:
i i i
q H dt dp p
i i i
p H dt dq q
The space (p, q) is 2N-dimensional phase space and a pair (pi, qi). The Hamiltonian can depend explicitly on time, i.e. H = H(p, q t). Then the system can be considered in an extended space of 2(N + 1) variables.
It what follows some mathematical models of chaotic physical systems are presented, which can also be modeled by iterative equations systems. Physical models of chaos. A discrete form of the time evolution equations will be called maps; generally speaking, they can be written in a form of iterations:
n n n n n
1 1
where the time-shift operator Tn is (2N x 2N) matrix that depends on n. 1 There are many typical physical models. The Poincare map is most often used in physical applications. Other examples are the Sinai Billiard model, etc
The following is an example of a chaotic system which is a particular case of Hamiltonian potential function, which includes shocks (given as a series of pulses). Consider a Hamiltonian:
t (x) f K + Ho(P) = H n
in which perturbation is a periodic sequence of δ function type pulses (kicks) following with period T= 2π/v, K is an amplitude of the pulses, is a frequency and (f(x)≤1) is some function. The equations of motion, corresponding to (2), are
t (x) ' f K
p n
T p x x
n n n 1 1
There is a special case for and f(x) = -cos(x) and w(p) = p Take into account the before we can derive the iteration equation:
n n n
x KTf p p '
1
Figure 1. Solution of pendulum equation in phase space
Any kind of equation is an approximate way to describe an ensemble of trajectories or particles, while neglecting some details of dynamics. All this means that, depending upon the information about the system we would like to preserve, the type and specific structure of the kinetic equation depends on
graining of trajectories. These properties of dynamics require a new approach to kinetics (based on fractional differential equations) when the scaling features of the dynamics dominate others and, moreover, do not have a universal pattern as in the case of Gaussian processes, but instead, are specified by the phase space topology and the corresponding characteristics of singular zones.
Structuring in the phase space. As has been noted, the solutions in the phase space for chaotic systems give rise to specific structures, which are induced by attractors, and are classified to classical and chaotic dynamics as follows. Stable attractors (or classic dynamic). In the phase diagrams, these converge on stable points, whereas in periodic signals, the trajectories have well-defined paths. Strange attractors (chaotic dynamics). These movements correspond to unpredictable, irregular and seemingly random curves in the phase diagram, but are located according to some probabilistic distribution within a certain
attractor is called chaotic.
Dynamical systems can be classified according to the behaviour of their orbits (Espinosa, 2005). These orbits correspond to the movement in which the system evolves over time. Thus, if the system moves in a set such that the set
There are well-known chaotic oscillators, which are characterized by iterative systems of equations, due to space limitations in this study, only one is discussed in what follows. The changes over time of four well-known low- dimensional chaotic systems are studied: Lorenz, Rössler, Verhulst, and Duffing (Laurent et al., 2010). Only the first is presented below. The Lorenz system was designed for convection analysis and is not generally used to study population data. Lorenz attractor standard values for the constants were set as follows (through an iterative equation system):
The possibility of reaching chaotic trajectories in nonlinear dynamical systems leads naturally to the empirical question of how to distinguish such trajectories of other really random time series (Gimeno et al., 2004). The topic about the chaotic detection has attracted the attention of scientists from different disciplines that have used different statistical procedures to measure chaos. In common usage, "chaos" means "a state of disorder", but the adjective "chaotic" is defined more precisely in chaos theory (Wikipedia, 2010). Although there is no an universally accepted mathematical definition of chaos, a commonly-used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:
Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour. Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes
Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may not be chaotic.
There are different procedures to detect chaos, in what follows some of them are commented. Peters (Peters, 1994) tries to find evidence of a series with chaotic behaviour by graphic analysis and notes that the series of financial asset prices have graphically the same structure, whatever the timescale studied (Espinosa, 2005). The fact that these series have the same appearance on different time scales is an indication that this is a fractal. For the reconstruction of the recurrence maps is necessary to find hidden patterns and structural changes in the data or similarities in patterns across the time series under study. Thus, a signal off determinism will be when more structured is the recurrence map. A random signal is when the recurrence map is more uniform distributed on the phase space and does not have an identifiable pattern.
Traditional methods of time series analysis come from the well-established field of digital signal processing. Most traditional methods are well- researched and their proper application is understood. One of the most familiar and widely used tools is the Fourier transform. However, these methods are designed to deal with a restricted subclass of possible data. The data is often assumed to be stationary, that is, the dynamics generating the data are independent of time. With experimental nonlinear data, traditional signal processing methods may fail because the system dynamics are, at best, complicated, and at worst, extremely noisy. In general, more advanced and varied methods are often required. Another tool for analyzing time series is the wavelet transform (WT). The WT has been introduced and developed to study a large class of phenomena such as image processing, data compression, chaos, fractals, etc. The basic functions of the WT have the key property of localization in time (or space) and in frequency, contrary to what happens with trigonometric functions. In fact, the WT works as a mathematical microscope on a specific part of a signal to extract local structures and singularities. This makes the wavelets ideal for handling non-stationary and transient signals, as well as fractal-type structures
Chaos indicators. Among these are: correlation dimension, Lyapunov exponents, Kolmogorov entropy, etc. Correlation Dimension. A clear indicator that a system is chaotic is to have a small correlation dimension. Lyapunov exponents. The most important indicator of chaos in a nonlinear system is the Liapunov
system converges or diverges. They are calculated,
will have (n-1) exponents. The most important is the greatest of them. If the greatest of all is negative, the system will converge over time. However, if it is positive, the error will grow exponentially over time, and the system will exhibit the sensitive dependence
The Lyapunov exponent characterises the extent of the sensitivity to initial conditions (Wikipedia, 2010). Quantitatively, two trajectories in phase space with initial separation diverge . where λ is the Lyapunov exponent. The rate
to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic. Kolmogorov entropy. The entropy of a dynamical system can be thought of as the “disorder” to which the system tends with time. In this case the attractors, if any, does not tend to disappear but to perpetuate itself, so the system is
periodic if its entropy is close to 0%; Chaotic if it is between 0 and 100% and Random if it is near of 100%.
The Hurst coefficient indicates the persistence or non-persistence in a time series (Espinosa, 2005). Of being persistent, this would be a sign that this series is not white noise and, therefore, there would be some kind of dependency between the data. The calculation of Hurst coefficient reveals that is given in the following power law shown in equation (6):
H
where a is a constant; N is the number of observations; H is the Hurst exponent, is the statistic depends on the size series and is defined as the coefficient of variation of the series divided by its standard deviation. The Hurst coefficient is used to detect long-term memory in time series.
To calculate the correlation dimension Grassberger and Procaccia (Grassberger et al, 1983) developed an efficient algorithm that suggest that , where D is the capacity dimension. The idea is to replace the algorithm to calculate , called box-counting, by the estimation of distances between points (which representing positions of the system along an orbit) in the attractor set.
Between that is an algorithm developed by Wolf (Wolf et al, 1985), which implements the theory in a very simple and direct fashion (Kodba et al., 2004). The whole program package that can be downloaded from our Web page (User) consists of five programs (embedd.exe, mutual.exe, fnn.exe, determinism.exe and lyapmax.exe) and an input file ini.dat, which contains the studied time series. All programs have a graphical interface and display results in the forms of graphs and drawings. For these reasons the Nonlinear Dynamics Toolbox was created (Reiss, 2001). The Nonlinear Dynamics Toolbox (NDT) is a set of routines for the creation, manipulation, analysis, and display of large multi-dimensional time series data sets, using both established and original techniques derived primarily from the field of nonlinear dynamics. In addition, some traditional signal processing methods are also available in NDT.
A common methodology used to determinate if a system have chaotic behavior is the next: firstly, it is used the embedding delay of coordinates in
space); for this purpose, both, the embedding delay (t) and embedding dimension (m) have to be calculated. Two methods are used: the mutual information method to estimate the appropriate embedding delay and the false nearest neighbor method (FNN) to estimate the embedding dimension. Next, a determinism test is performed to determine if the series were obtained of chaotic or random systems. Finally, the computation of the maximal Lyapunov exponent is performed to determinate if chaos in the phenomenon is present
Figure 2: Signal Acapulco Figure 3: Signal Aguamilpa Figure 4: Signal CDA Figure 5: Mutual inf. t=6
Chaotic analysis in three accelerograms signals corresponding to three sites in Mexico: a) Acapulco, Aguamilpa and Central de Abastos (CDA) ( in Mexico City) were performed (see figures 2-4). In what follows, a chaotic analysis of the Acapulco signal is presented.
The software used to calculate the parameters m and t, the phase space reconstruction, the determinism test and the estimation of the maximal Lyapunov exponent can be consult and download from the site on the web (user). Phase space reconstruction Below, the estimation of the parameters m and t which are necessary for the phase space reconstruction using the mutual information and the FNN methods is presented (see figures 5 and 6). In the figure 7 a projection in a plane of the attractor system is presented. Figure 8 shows the graph corresponding to the determinism (determ) test and finally, in figure 9, the estimation of the maximal Lyapunov exponent is presented.
Figure 6: FNN. m=4 Figure 7: Phase space Figure 8. Determ =0.638 Figure 9. Maximal Lyapunov exponent=0.9
Table 1. Resume of results
Signal m Determ Max Exp Lyap Acapulco 6 4 0.638 0.90 Aguamilpa 1 5 0.397 0.68 CDA 10 10 0.723 0.60 From the analysis presented it can be concluded that the three signals have a chaotic behavior.
In table 1 a resume of the analysis made to the three signals is presented.
biological, economic, etc., represented by systems of differential equations of integer and fractional order, which can be replaced by iterative equations systems, and have movement histories which when are to be represented in a phase diagram have complex topological structures (including fractal type).
time series and deduce whether these come from deterministic chaotic systems or are either purely random kind.
usually the design procedures are based on linear dynamic mathematical models or purely random, but this strategy is not entirely appropriate because most natural processes are not stationary, like earthquakes, and therefore it is necessary to develop a more consistent design methodology.
signals were generated from a deterministic chaotic phenomenon because the vector field turned out to be deterministic and the maximal Lyapunov exponent was positive, suggesting that the reconstructed system have an important deterministic chaotic component.
the signals detected in each place, have effects of the geologic system where were registered.