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A concrete case of Chaotic motion



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A concrete case of Chaotic motion. We consider a concrete case for the parameters 5 and . The aperiodic appearance of (see Fig.4) suggests that the system under consideration is chaotic.



To verify that motion realized at is chaotic, we need to show the sensitivity to initial conditions on its attractor. We choose two points separated by close to the attractor and examine initiated evalutions from them. Figure 5 illustrates the variation of the separation d with time t. The exponential growth of separation d for is clearly noticeable. The separation saturates the size of the attractor for t > 120. Therefore, there is a positive Liapunov exponent associated with the chaotic orbit at and its approximate value is 0.0495. Much more insight can be gained from a Poincaré section (Figure6) consisting of stroboscopic points at instants 0.782), 0,1,2 ... of the orbit of the system (9) in the space . Figure 6 shows the next 10000 points after the transition decays about the first 1000 periods. The corresponding attractor of the chaotic solution is presented in Figure 3(b).






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