Detection of the chaotic behaviour of a bouncing ball by the 0–1 test
Introduction
A bouncing ball system consisting of the ball following a free fall in the Earth gravitational field and impacting the kinematically forced plate (Fig. 1) was first studied by Holmes [1]. He proposed the scenario of a horseshoe transition to the chaotic vibration. Continuing these investigations Everson [2] provided a bifurcation diagram and calculated the maximal Lyapunov exponent showing the regions of chaotic vibrations. Later it became one of the simplest model of original non-smooth mechanical systems which were predicted theoretically and confirmed experimentally to follow chaotic trajectories [3], [4].
At the same time, because of impacts, the case of a bouncing ball is the typical example of non-smooth mechanical systems. In general for discontinuous systems [5], [6], [7] the standard method of the maximal Lyapunov exponent estimation scheme by Benettin et al. [8], [9] and Wolf et al. [10] using the equations of motion cannot be simply applied. Another approaches are based on attractor reconstruction from the time series [10], [11], [12], [13]. Alternatively, for arbitrary dynamical systems including discontinuous one, the 0–1 test can be applied [14], [15], [16], [17], [7]. This method is based on the spectral properties of chaotic system quantifying the system response by a single number: 0 for the regular and 1 for the chaotic solution. The procedure of its estimation relies on universal mean square displacement scaling in time for chaotic systems [18], [19].
Fortunately the present case of a bouncing ball can be partially integrated between impacts. Consequently, instead of differential equations we study a smooth impact map. This is an ideal situation to compare directly both methods, based on the maximal Lyapunov exponent and the 0–1 test.
The velocity of the plate can be written asNeglecting an air drag one can examine the free fall motion of the ball, with a gravity acceleration g, between impacts with a restitution coefficient . This procedure leads to the difference equations for impacts:where is the time interval between impacts. Writing the velocity in units and introducing a phase one gets the following dimensionless mapHere is the dimensionless driving frequency. In the limit of the above equations can be simplified [20] toIn Fig. 2 we show the regular (Fig. 2a) and chaotic (Fig. 2b) responses of the examined system (Eq. (4)). The corresponding velocities have been plotted against consecutive impact instants . In simulations we have used the same initial conditions were . While the system parameters were , for chaotic solution and and , for regular one. Note that regular time series is the period two solution. For better clarity, in Fig. 3a and b we show also the corresponding phase diagrams for the same conditions and system parameters. Here one can clearly see the fractal structure of an attractor composed of multiple lines (Fig. 3a) and the two points attractor (Fig. 3b), respectively.
Section snippets
Estimation of the Lyapunov exponent
The maximal Lyapunov exponent (MLE) can be defined using the corresponding Jacobian matrices defined for the impact map (Eq. (4)):where i denotes the impact number.
Following Eckmann and Ruelle [21], and also Ladeira and Leonel [22] one has to define a new matrix as the product of the Jacobian matrices :MLE for a two-dimensional map is simplywhere corresponds to eigenvalues of
Application of the test 0–1
Starting from one of the initial map coordinates (here ) we follow [14], [15], [17], [7] by defining new coordinates and aswhere c is a certain constant (here ).
If for a given periodic series ,represents random number and we are going to get Brownian motions [17], [23], [24], [25] asand the total mean square displacement is scaled as n.
On the other hand for a periodic or quasi-periodic signal
Summary and conclusions
In summary we would like to mention that we performed the test 0–1 to the system showing different solutions to quantify them by a single parameter K. Namely for regular and chaotic solutions we obtained or 1, respectively. These results were successfully confirmed by direct estimation of the maximal Lyapunov exponent (Fig. 6).
By applying the new coordinates ( and ) we obtained an unbounded drift for a chaotic solution and a motion around a certain circle in case of regular motion.
Acknowledgements
This research has been partially supported by the 6th Framework Programme, Marie Curie Actions, Transfer of Knowledge, Grant No. MTKD-CT-2004-014058 and by the Polish Ministry of Science and Higher Education, Grant No. 9008/B/T02/2007/33. G.L. and A.S. would like to thank Prof. M. Wiercigroch for hospitality during their visits to Aberdeen and Prof. G. Rega for helpful discussions.
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