We investigate the implications that a sum rule for the average velocity of a geological fault has on the distribution of earthquake sizes. Under general conditions the exponent B of the Gutenberg-Richter law for the distribution of small seismic moments is shown to obey 8 & 1. This result does not rely on any particular earthquake model and should be equally applicable to the case of friction between two sliding blocks as well as to computer simulations of earthquake dynamics. For the distribution of large earthquakes we specifically consider three diA'erent possible models: (a) A single power law exists over the entire range of magnitudes; then B= l. (b) The large events do not fall in the scaling region; in this case we are able to derive a relationship between the exponents relating their frequency and seismic moment to the total size of the system. (c) The large earthquakes also show scaling behavior but with a diAerent exponent B'; then the sum rule implies that B'& l. PACS number(s): 91.30.8i, 46.30.Jv, 05.70.Jk Along a single geological fault there will be earthquakes of many different sizes. The seismic moment m is commonly used as a measure of the size of an earthquake [1]. It is defined here as the amplitude of the motion integrated over the active region m =J xds',where x is the relative displacement of the two sides of the fault and s is the area of the active region of the fault during an earthquake. The rate of occurrence (number of events per unit time) p of an earthquake of seismic moment m is observed to obey the empirical Gutenberg-Richter law [2] p(m) =Amwhere A and 8 are constants [3]. For small to-intermediate magnitudes all available data [4] indicate that the distribution of B values for many different regions around the world is highly concentrated around I, except for volcanic regions, where 8 values between 1 and 2 are common [5]. Also microearthquakes are found to obey Eq.(2) with 8= I [6]. The relation (2) has been used to characterize both the seismicity of a single fault as well as the seismicity in a broad region that includes many faults.A variety of earthquake models have been introduced. The so-called spring-block models [7] and their cellularautomata counterparts [8,9] have recently attracted some attention in the physics community.To a large extent their success has been judged by whether or not they reproduce the Gutenberg-Richter law with the value 8 = l. An essential ingredient of all these models is that the two sides of the "fault" are forced to move at a constant average velocity with respect to one another to mimic the driving mechanism of plate tectonics. The existence of these different models indicates that it is possible to start with quite different assumptions and still produce a power law for the distribution of earthquake sizes. What remains to be understood is why the real earthquake data taken from many different faults as well as the data from computer models which have very different starting assumptions give approximately the same exponent. %'e will argue b...

We investigate the synchronization between two systems consisting ofcoupled circle maps that have a common drive, which may be chaotic or regular. We observe several new aspects ofchaotic and regular synchronization. In the chaotic regime the transition from synchronization to nonsynchronization corresponds to the transition from one to two Liapunov exponents. We find regions in the parameter space with periodic motion where synchronization is always achieved, never achieved, or, depending on the initial conditions, sometimes achieved. The nonsynchronization or synchronization are stable in the presence of a weak chaotic (or noisy) signal.

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