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Is unknown and the Lyapunow exponent must be computed with the aid of the simulation. The main approach to estimating a time average is of course to compute N trajectories near the point x0. If the trajectories are attracted by x0 (system is stable). In other cases the system is chaotic. Bifurcation diagrams - Most systems, however, are neither chaotic nor integrable but show a complicated mixture of regular and chaotic behaviour. Bifurcation is a phenomenon exhibited in systems with mixed phase space [4].

For example, if a high value occurs at time tx then at time tx+1 one would be more likely to see a low value. Similarly, antipersistent scaling has a Hurst exponent Figure 6 shows a typical example of the processes with different values of H. These processes were generated by Random Midpoint Displacement (RMD) method [22]. RMD is fast and allows the rapid generation of long traces, and it is known to be exact for all values of H parameter. Lyapunow exponent - When one studies an attractor of a chaotic dynamical system quantitatively, one is often interested in estimating a ”time average”: the average of a given function of the state of the system over a typical trajectory on, or approaching, the attractor [12].

These patterns attract the system into their state. In examining the traffic generation process by many particular traffic sources it is found certain patterns occur and often lead to some kind of bounded behaviour. These are attractors for the teletraffic processes which are presented more detailed in next section of this paper. There are a number of ways of modelling chaotic system with attractor. Most common strategy in this case is Fractals and Chaos for Modelling Multimedia ATM Traffic 35 iterate a set of non-linear equations [14].

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1000 Photoshop Tips and Tricks (Dec 2010) by UNKNOWN

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