COMPUTATION OF LYAPUNOV EXPONENT FOR CHARACTERIZING THE DYNAMICS OF EARTHQUAKE

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1 COMPUTATION OF LYAPUNOV EXPONENT FOR CHARACTERIZING THE DYNAMICS OF EARTHQUAKE Folasae. L. Aeremi an Olatune. I. Popoola 1 Department of Physics, University of Ibaan, Ibaan, Nigeria. ABSTRACT Earthquakes forecasting an preiction is a global challenge, several precursors an methos of earthquake preiction have been propose, but lack consistency an are not reliable for preiction hence occurrence of earthquakes is sometimes assume to be ranom. This stuy was esigne to investigate an characterise the occurrence of earthquakes using chaos theory. The Lyapunov Exponent an its spectrum were obtaine from earthquake ata using moifie two imensional system metho from Sprott s proceures. The results show that the values of the Lyapunov exponent were positive but the magnitue varies for all regions consiere an the Lyapunov exponent spectrum exhibit an asymptotic behaviour in all the regions.this stuy showe that although seismicity exhibit apparent ranomness but earthquake occurrence is not stochastic but a non-linear eterministic ynamical process. KEYWORDS Earthquake preiction, non-linear ynamic, chaos theory, computation,lyapunov exponent 1. INTRODUCTION Earthquake can be efine as a suen release of accumulate strain energy in a confine region of the Earth interior or along a fault or fracture in the earth's crust, The strain is the result of stress which occurs along plate bounaries ue to the relative motion of the lithospheric plates.earthquake preiction is still a global challenge because, the hypocenters of earthquakes are inaccessible an the state of the lithosphere at seismogenic epth can not be observe irectly [1]. The earth crust where most earthquakes occur is highly heterogeneous in istribution of strength an store elastic strain energy. No satisfactory theory of earthquake source process exists at present []. Application of ynamics theory is limite because response of rocks to stress is highly complex an non linear. The interaction of faults is highly complex an non linear. The relationship is complex an may iffer in seismogenic zones [3]. Laboratory an experimental moels are conucte on a limite scale an o not replicate the complex an heterogeneous conitions of the problem or phenomenon in situ. Determining the history of earthquakes is ifficult in some areas because the historic recor of such area is short compare with the average time between major earthquakes [4]. Because little is known about the physics of faulting many attempts to preict earthquakes have searche for precursors [5].Several precursors have been propose such as: Foreshocks,Seismic cycle, Seismic Doughnut, Seismic quiescence [5]. Raon concentration an temperature ecrease in grounwater; Grounwater rise in well [6,7]. However DOI : /ijrap

2 these precursors o not reprouce themselves [8-10]. Lack of consistency in seismic precursors an inability to preict earthquake has mae many seismologists to conclue that earthquake occurrence is stochastic an its preiction is a Gambler s fallacy[11] Occurrence of earthquakes is assume to be ranom when these methos prouce unreliable pattern[1-15]. The fact that these patterns consiere to be ranom coul be chaotic (preictable but ifficult) has not been investigate. For earthquake to be preicte or preictable there is a nee to evelop a physics base theory of the seismogenesis, nucleation an precursory process of the earthquake. The ynamic nature of earthquake must be clearly unerstoo in a scientific manner anmetho, not just by speculation hence there is the nee to know an not assume whether the occurrence of earthquake is perioic, chaotic or ranom in space an time.this stuy was esigne to investigate an characterize the ynamics of earthquakes occurrence using chaos theory.. BACKGROUND A stochastic or ranom process is a probabilistic process as oppose to a eterministic process because there is some ineterminacy such that even if the initial conition or starting point is known, there are several many irections in which the process may evolve. A similar initial conition may give entirely ifferent result or output.in science it is a non-linear ynamical system that lies between regular eterministic system an stochastic system. A chaotic system is eterministic but ifficult to preict because it is sensitive to initial conition (the butterfly effect)..1 CHAOTIC SYSTEM Chaos is a type of motion that lies between regular eterministic trajectory that are preictable an the unpreictable stochastic behaviour that is characterize by complete ranomness, Nonlinear ynamical systems that appear to have ranom, unpreictable behaviour, in other wors the chaotic system follows eterministic rule but its evolution appear ranom[16]. Chaotic system is a ynamic system that is eterministic, non-linear, sensitive to initial conition hence ifficult to preict or a stochastic behaviour occurring in a eterministic system. Chaos is the stuy of eterministic systems that are so sensitive to measurement that their output appears ranom [16, 17].. LYAPUNOV EXPONENT The Lyapunov exponent is a quantitative measurement of the rate of exponential ivergent or convergent of a ynamic system an it is a basic inicator of eterministic chaos [18]. The sign of the Lyapunov exponents provie a qualitative picture of a system s ynamics. Its sign inicates whether the system is chaotic or not. The magnitue of the Lyapunov exponent reflect the time scale in which the system ynamic become unpreictable while its sign inicates sensitivity to initial conition[19,0].the Lyapunov exponents quantify the exponential ivergence of initially close state-space trajectories an estimate the amount of chaos in a system[1]. The Lyapunov exponent is positive when neighbouring trajectories iverge from each other at large n, which correspons to chaos. However, if the trajectories converge to a fixe point or limit cycle, they will get closer together, which correspons to negative Lyapunov exponents. Hence we can etermine whether or not the system is chaotic by the sign of the Lyapunov exponent. It is a way of istinguishing between a stochastic process an a eterministic system[].obtaining the Lyapunov exponents from a system with known 10

3 ifferential equations is no real problem but ifficult or nearly impossible for experimental ata without known ifferential equations[0].experimental ata typically consist of measurements of a single observable. There is a nee to employ reconstruction an locating the nearest neighbour on the trajectory. The nearest neighbour is at a minimal point or istance from the reference point. 3. MATERIAL AND METHOD 3.1 DATA ACQUISITION The ata use in this stuy were obtaine from Earthquake catalog of Avance National Seismic System (ANSS), Northern California Earthquake Data Centre, an USA, Perio of catalogue The catalog contain the origin time, ate, epicentre, epth in kilometre, latitue, longitue, magnitue an magnitue type 3. METHODS Locations an magnitues of large earthquakes globally were ientifie from 1899 to 009. Large earthquake in the circum-pacific zone were mappe an ivie into three regions: Region 1 is within Latitue 48 to 60 an Longitue -179 to -160, Region is within Latitue 3 to 44 an Longitue 134 to 148 while Region 3 is within Latitue -4 to -9 an Longitue -80 to -66.The Lyapunov exponent an spectrum were computebase on the following proceure. If two orbits are separate by a small istance o at time t = 0, then at a later time t, their separation is given by λt ( t) = 0e (1) If the system evolves through an iteration process then n λn = 0e () Where n is the number of iteration hence λis imensionless. n 1 j λ n = ln, n 0 (3) n j= 1 j o ( x x ) + ( y y ) ) = (4) a b ( x x ) + ( y y ) ) 0 = a bo a bo (5) a b For a two imensional system with variables x an y, the separation is efine by equation 4 where a an b enote the two orbits [3].If a an b enote two orbits, j is efine as the separation between the j th pair of nearest neighbours while o is the separation between the first pair of nearest neighbours, this is also known as the initial separation. In this stuy, since 11

4 earthquake is a single observable, x is efine as the latitue while y is efine as the longitue of epicentre of the earthquake. x a an y a are efine as the latitue an longitue of a reference large earthquake respectively while x b an y b are the latitue an longitue of the subsequent earthquakes respectively. The initial pair of nearest neighbours are the reference large earthquake an the first subsequent earthquake (in time an space) within the region hence o is efine as the initial separation between the first subsequent earthquake an the reference large earthquake while j is the separation between the j th subsequent earthquake an the reference large earthquake, this is necessary because earthquake is a single observable 4. RESULTS AND DISCUSSION Figure 1; Lyapunov exponent spectrum for Region 1 Figure. Lyapunov exponent spectrum for Region 1

5 Figure 3. Lyapunov exponent spectrum for Region 3 Table 1: the Lyapunov exponents for the three regions Region LEX The Lyapunov exponent is positive when neighbouring trajectories iverge from each other at large n, which correspons to chaos. However, if the trajectories converge to a fixe point or limit cycle, they will get closer together, which correspons to negative Lyapunov exponents. Hence we can etermine whether or not the system is chaotic by the sign of the Lyapunov exponent. It is a way of istinguishing between a stochastic process an a eterministic system [].The Lyapunov exponents quantify the exponential ivergence of initially close state-space trajectories an estimate the amount of chaos in a system [1]. It is a basic inicator of eterministic chaos [18].Figure 1 showe the spectrum of the Lyapunov exponent for region 1, Observation showe that the Lyapunov exponent is very sensitive to iscontinuity in the catalogue an exhibits an asymptotic behaviour before the occurrence of a large earthquake. Lyapunov exponent spectrum for region an region 3 are shown in Figures an 3 respectively, the Lyapunov exponent spectrum exhibit an asymptotic behaviour in all the regions. Table 1 showe the result of the 13

6 Lyapunov exponent compute for the all the regions. The value of the Lyapunov exponent is positive for each of the region but its magnitue varies from one region to another. The lowest value of was obtaine in Region 3 while the highest value of.688 was obtaine in Region. 5. CONCLUSION The Lyapunov exponent using two-imensional system metho was positive for all the regions, inicating the chaotic nature of the earthquakes occurrence. This was highest for Region (.688) an lowest for Region 3 (0.688). The Lyapunov exponent spectrum behave asymptotically prior to the occurrence of the large earthquake, these can serve as a precursor for earthquake forecasting in hazar management. ACKNOWLEDGEMENTS Avance National Seismic System (ANSS), Northern California Earthquake Data Centre, USA, for earthquake catalogue. REFERENCES [1] Kossobokov, V. (009) Practise of preicting large earthquake on global an regional scales. Avance School of Non Linear Dynamics an Earthquake preiction. Trieste, Italy. [] Geller, R. J. (1997) Earthquake preiction: A critical review. Geophysical Journal International. 131, , oi: /j x.1997.tb06588.x. [3] Biagi, P. F. (1999) Nature Debate, Nature Macmillan Publishers Lt 1999 Registere No Englan. [4] Michael, A. 1999a. How well can we preict earthquakes? Unite States Geological Survey, Menlo Park, California, USA. Nature Debate, Nature Macmillan Publishers Lt 1999 Registere No Englan. [5] Stein, S. an Wysession, M. (003) An Introuction to Seismology, Earthquakes an Earth Structure. Blackwell Publishing, Lt. [6] IASPEI, (1994) International Association of Seismology an Physics of the Earth s Interior, Preliminary list of Significant Precursors. [7] Roeloffs, E. an Langbein, J. (1994)The earthquake preiction experiment at Parkfiel, California, Reviews of Geophysics. 3, [8] Roeloffs, E. an Quilty, E. (1997) Water level an strain changes preceing an following the August 4, 1985 Kettleman-Hills California earthquake. Pure an Applie Geophysics. 149, [9] Scholz, C.H. (1990)The Mechanics of Earthquakes an Faulting.Cambrige University Press. [10] Wyss, M. (1991) Secon roun of evaluation of propose earthquake precursors, Pure an Applie Geophysics 149, 3-16 [11] Campbell, W.H A misuse of public funs: UN support for geomagnetic forecasting of earthquakes an meteorological isasters.eos Trans. Am. Geophys. Union 79, [1] Geller, R. J. Jackson, D. D. Kagan, Y. Y. an Mulargia, F. (1997) Earthquakes cannot be preicte. Science. 75, [13] Geller R.J. (1999) Earthquake Preiction: What shoul we be ebating? Nature Debate, Nature Macmillan Publishers Lt 1999 Registere No Englan. [14] Michael, A. 1999b. Realistic preictions: are they worthwhile? Unite States Geological Survey, Menlo Park, California, USA Nature Debate, Nature Macmillan Publishers Lt 1999 Registere No Englan. 14

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