Analysis of different numerical procedures for determining the random and chaotic earthquake properties R. Magaña*, A. Hermosillo*, M. Pérez** *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
Introduction • The purposes of this paper are: • - To consider nonlinear dynamical aspects, in the criteria for structural and geotechnical design. • - show that some earthquakes have chaotic content in addition to the random one. Taking into account that are non-stationary processes, so they must use appropriate mathematical tools (not limited to criteria used in stationary linear dynamic). • Like examples of application of these concepts, the chaotic content analysis 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. • This article briefly discusses some of the mathematical models of nonlinear dynamic systems, and criteria for identifying the chaotic time series content as well as the application of these criteria to the earthquakes mentioned.
Theoretical Fundaments A characteristic of systems with a periodic time evolutions is feedback, which 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.
Solutions of differential equations in phase space. 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.
Hamiltonian equation 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: dp H dq H i i p q 1 i i , , dt q dt p i N i i 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.
Modeling chaotic systems 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: ˆ p , q T p , q n 1 n 1 n n n 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
Universal and standard map 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 H = Ho(P) + K f (x) n T - in which perturbation is a periodic sequence of δ function type pulses (kicks) following with period T= 2 π /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 p = - K f ' (x) n x = H' (P) p T 0 - Take into account the before we can derive the iteration equation: ' x x p T p p KTf x n 1 n n n 1 n n 1 There is a special case for and f(x) = -cos(x) and w(p) = p
For small K << 1 we can replace the difference equations with the differential ones: This is the pendulum equation, and its solutions are presented in figure 1. Figure 1. Solution of pendulum equation in phase space
Fractals and chaos 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 our choice of the reduced space of variables and on the level of coarse- 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.
Fractals and chaos 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 structure. A dynamical systems that converge in the long run to a strange 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 of orbits A is a subset of , then the orbits will have the following behavior: - Dissipative system: If A shrinks over time. - System expansion: If A expands over time. - System conservative: If A is maintained over time.
Typical chaotic oscillators 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, x 10 x 20 y Rössler, Verhulst, and Duffing (Laurent et al., 2010). Only the first is presented below. The y 28 x y xz Lorenz system was designed for convection 8 analysis and is not generally used to study z z xy population data. 3 Lorenz attractor standard values for the constants were set as follows (through an iterative equation system):
Detection Algoritthms 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: • it must be sensitive to initial conditions, • it must be topologically mixing, and • its periodic orbits must be dense.
Detection Algoritthms 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 or fluids is an example of a chaotic system. 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.
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