____
 Never in the annals of science and engineering has there been a phenomenon so ubiquitous, a paradigm so universal, or a discipline so multidisciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg, for beneath it lies a much finer structure of immense complexity, a geometric labyrinth of endless convolutions, and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once wantonly linearized by the engineers and applied scientists of yore, thereby forefeiting their only chance to grapple with reality.        Leon O. Chua



BOUNCING  BALL
    

A simple variation of Fermi's model will be used to show the effect of collision process in the rich dynamical response of impact systems. A dynamic schemata of the model is shown on top left of this page, which shows a ball falling freely under the force of gravity and "bouncing" of a "periodically driven" table. Well, it is not a 'real' simulation but it does create a visual illusion of the simple model described above!! It is not a realistic simulation in the sense that the motion of the ball is not determined by Newton's law and the collision process is not modelled by any rule. Having said that now if we assume that we do simulate a realistic model, we want to find out how does the motion of the ball depend upon the system parameters. Consider the motion of a ball bouncing on a periodically vibrating table. To keep the matters simple, we will assume that the ball's motion is confined to the vertical direction and that, between impacts, the ball's height is determined by Newton's law for a freely falling particle under the influence of gravity. Thus, between the impacts, the position and the velocity of the ball at any given time is given by

x(t) = x0 + v*t
v(t) = v0 - g*t
where x0 and v0 are the initial position and the velocity and 'g' denotes the acceleration due to gravity. The collision process is modelled using coefficient of restitution rule. It is assumed that the table's mass is much greater than the ball's mass and the impact between the ball and the table is instantaneous. The implication of these assumptions is that the table's motion is unaffected by the collisions. Assuming that motion of the table is igiven by a sinusoidal form The equations of motion are reset, at the time of impact by the following expression
v+ = K*(u - v-) + u
where u is the velocity of the table at the instant the collision occurred and the parameter K called the coefficient of restitution, is a measure of amount of energy lost in the process of collision (actually 1-K2 ). The value of K ranges between 0 and 1. Values of K less than 1 imply inelastic collision (dissipative system) and K = 1 implies elastic collision (conservative system). Note, in either case there is a nonlinear force that is applied to the ball when it hits the table. At impact, the ball's velocity suddenly reverses from the downward to the upward direction which appears as a sharp discontinuity in the phase space describing the motion of the ball (position velocity plot).

It turns out simulating this simple model one observes, for a wide range of the drive frequency (a system parameter), keeping the other parameter fixed, this simple dynamical system shows chaos!! This is surprising, given the fact that if we assume the table to be held fixed the ball bouncing of the table would either eventually come to rest on the table (if energy is lost as a result of collision -- dissipative system) or the ball oscillates with the same amplitude (if there is no loss of energy as a result of the collision -- conservative system).

Of the several equivalent ways that exists to represent and study the dynamics of bouncing ball system, we will show the motion of the ball by measuring the time between impacts and see how the eventual behavior of the ball changes as we change one of the system parameters. All plots shown here will be for the case of inelastic collisions, note for a driven bouncing ball system with elastic collision, the motion is unbounded and thus dismissed for being uninteresting!



ATTRACTOR RECONSTRUCTION

So finally, here it is!! The chaotic attractor of the bouncing ball system for the A = 0.012, K = 0.5 and drive frequency 2*PI*6 rads/sec, reconstructed using the time between impacts. The attractor is embedded in 3-D space. The following pictures are for same parameter values as above except that the views are different.

Figure 1: In all figures time between impact is used to reconstruct the attractor with the attractor embedded in 3-D space. The parameter values chosen are A=0.012, K = 0.5 and drive frequency = 2*PI*6.0 rad/sec. The plot is ti-2, ti-1, and ti for X-, Y- and Z- axis, respectively. Thus, the plot essentially is a return map in 3-D and the intersection of the body diagonal passing through the origin with any point on the chaotic attractor gives the location of the embedded unstable fixed-point attractor of the map. The view used for plotting the above attractor are top left (0 0), top right (45 45), bottom left (90 180) and bottom right (45 255). The X-axis, before the rotation, is the horizontal axis pointing to the right, the Y-axis the vertical axis pointing vertically up and the Z-axis is the axis pointing out of the plane (towards you!). Then the first angle represents the rotation about the X-axis and the second angle represents the rotation about the new Z-axis.

Note the embedding of the attractor in 3-D space was important from the point of view of controlling the unstable fixed point embedded in the chaotic attractor by precluding the false nearest neighbors when the control constants are calculated using the linear dynamics around the fixed point.



BIFURCATION PLOT

Following is the bifurcation plot for the bouncing ball system as the drive frequency is changed from 2*PI*5.5 rad/sec. to 2*PI*6.1 rad/sec. The plots show the time between impacts and the relative velocity of the ball with respect to the velocity of the table at the time of the impact. It is clear from the plot that for the parameter values chosen (K=0.5 and A=0.012) the bouncing ball goes through period doubling and eventually becomes chaotic. The chaotic invariant set shown in figure 1 goes through crisis at drive frqeuency sightly above 2*PI*6.0 and it destroys the chaotic attractor. Beyond this crisis point the system either shows long transient chaos or the orbit follows the shadow of the chaotic attractor before it eventually settles down to a sticking solution.

Figure 2: Bifurcation plot for bouncing ball system as a function of the drive frequency. Top: impact interval against the drive frequency. Below: Relative velocity against the drive frequency.



STICKING SOLUTIONS - GRAZING IMPACTS

As shown for the case of impact oscillator system, the grazing collision of the orbit leads to extreme stretching and contraction in region of the phase space near the colliding boundaries and leads to destruction of the orbit that grazes the wall and is sometime followed by transient chaos or intermittency. The lower plot in figure 2 clearly shows the effect of grazing collisions (when the relative velocity is zero just before the impact).

Figure 3: Reconstructing the chaotic attractor (shown in green for drive frequency 2*PI*6.0) using the impact time interval, just before the attractor goes through crisis. The points in red show the shadowing of the chaotic attractor by the orbits that goes through grazing collisions. The drive frequency for points plotted in green is 2*PI*6.03987. The range of impact time interval is from 0.0 to 2.2. All other system parameters are set as before.



BOUNCING BALL CODE

All figures on this page were produced from this simple C-code that generates the data file and could either be piped through my data to ps converter (dps) program which outputs a postscript file or use any of the plotting program to plot the data. Once in postscript format, use the following command line to convert the file into gif format.

gs -sDEVICE=ppmraw -sOutputFile=- -sNOPAUSE -q infile.ps -c showpage -c quit | pnmcrop | pnmmargin -white 10 | ppmtogif > outfile.gif

The file infile.ps is the ps file supplied by you. You need to have Pbmplus utilities installed for this commandline to work. Well all said and done there is an XWindow and DOS version of the program that plots the data as it is generated, allows to view the attractor in 3-D at any view angle. The program also implements the recursive proportional control strategy in conjunction with adaptive learning algorithm to stabilize the unstable periodic orbit of the impact system. The program can be used as template to simulate any system modelled by ordinary differential equations or maps. The programs are set up to control a sinusoidally driven nonlinear pendulum.



REFERENCE

[1]
C. N. Bapat, S. Sankar, and N. Popplewell, Repeated impacts on a sinusoidally vibrating table reappraised, J. Sound Vib. 108 (1), 99-115 (1986).

[2]
Nicholas B. Tufillaro, Jeremiah Reilly, and Tyler Abbott, An experimental approach to nonlinear dynamics and chaos, Addison-Wesley, 1992.

[3]
J. Thomas, M. A. Rhode, and R. W. Rollins, Chaos in an impact oscillator: experiment, simulation, and control, presented at Dynamics Days Arizona, 16-th Annual International Conference, Scottsdale, AZ, January 8-11, 1997.