Bouncing balls and invariant measures
Consider a ball bouncing on a sinusoidally forced moving table. Let \( s_t \) represent the phase of the table at impact at time \( t \), and \( v_t \) represent the exit velocity of the ball immediately after impact with the table.
The evolution of the ball/table system from one impact to the next can be completely described by these two measurements. If we denote by \( s_{t+1} \) and \( v_{t+1} \)the table phase and ball exit velocity respectively, then at time \( t+1 \) these quantities may be calculated from the previous impact as follows: \begin{align*} s_{t+1} &= s_t + v_t,\\ v_{t+1} &= a \times v_t - g \times \cos(s_t + v_t). \end{align*}
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The constant \( a \) is a factor between 0 and 1 describing how 鈥渂ouncy鈥 the ball is, and the constant \( g \) is a positive number describing the force applied by the table. The bouncing ball evolution defined by the above formula is an example of a dynamical system. From this formula, and some initial measurements \( s_0 \) and \( v_0 \), we can in principle compute the table phases and exit velocities of all future impacts: \( s_1, s_2, s_3, \ldots \) and \( v_1, v_2, v_3, \ldots \)
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By plotting the point \( (s_t,v_t) \) in an xy-plane for each \( t = 1,2,3,\ldots \), we can obtain a two-dimensional 鈥減icture鈥 of the dynamics. For some choices of \( a \) and \( g \), the bouncing ball is a chaotic dynamical system, and this 鈥減icture鈥 of dots fills out a strange attractor.
The distribution of points in this two-dimensional strange attractor is not uniform. Some parts of the attractor are more densely covered with dots than others, as some particular table phase/exit velocity combinations are more common than others. There is a well defined distribution of points on the attractor. This distribution measures how frequently table phase/exit velocity combinations occur. This distribution is also invariant under the evolution of the dynamical system. Being invariant means that this distribution looks the same, no matter when you observe it. The formal name for this special distribution is an invariant measure, and it is the basis for probabilistic and statistical descriptions dynamical systems.
This figure is a three-dimensional representation of the invariant measure of the bouncing ball鈥檚 strange attractor. The height and colour of the ridges indicate the frequency of table phase/exit velocity combinations. Higher and redder regions corresponds to more frequent combinations.
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To make things even more interesting, let鈥檚 say that our ball now has a hard side and a soft side. This means that our constant a will depend upon which side of the ball impacts the table. We can make a rule that says that every time it lands on the soft side, it will surely land on the hard side next time, while after landing on the hard side, there is a 50/50 chance of which side it next lands on.
This figure is a representation of the invariant measure for this two-sided bouncing ball system. Because our constant a is varying from impact to impact, the invariant measure is different to before. This means that the frequency of table phase/exit velocity combinations has changed in a precise way.
Lorenz's Butterfly Wings and Almost-Invariant Sets
The Lorenz equations are a three-dimensional set of ordinary differential equations. \begin{align*} \frac{dx}{dy} &= - s \times (y-x)\\ \frac{dy}{dt} &= r \times x - y - x \times z,\\ \frac{dz}{dt} &= -b \times z + x \times y.\end{align*}
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They represent a very simplified model of convection in the upper atmosphere, and they too describe a dynamical system. The constants \( s \), \( r \) and \( b \) are parameters that affect the behaviour of the system. Evolution of the three quantities \( x \), \( y \) and \( z \)can be tracked for al \( t \) into the future by solving Lorenz鈥檚 equations. Like the Bouncing Ball, the dynamical system defined by the Lorenz equations is also chaotic. If we are presented with the starting conditions \( x(0), y(0) \) and \( z(0)\),we can in principle calculate \( x(t), y(t) \) and \( z(t) \)for all future \( t\). If we draw a curve \( (x(t), y(t), z(t))\) in three-dimensional space, we will trace out a strange attractor known as the Lorenz attractor.
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An invariant set is a piece of an attractor that does not move in time. Invariant sets are very important, as each one is a building block of the system. In chaotic dynamical systems, there are no invariant sets; there is no way to break the system up into simpler pieces. The best that one can do is to look for almost-invariant sets. These sets break the system up into pieces which are individually highly chaotic, but which have only weak chaotic connections between themselves.
These figures show a primary numerical calculation of almost invariant sets. The three three-dimensional shapes are the Lorenz attractor viewed from three different angles. The colouring indicates almost-invariant sets by colouring each set a similar colour.
These figures show a secondary numerical calculation of almost invariant sets. The three three-dimensional shapes are again the Lorenz attractor viewed from three different angles, and the colouring indicates almost-invariant sets by colouring each set a similar colour.
By combining the primary and secondary numerical calculations, we arrive at three almost-invariant sets, indicated by the colours red, green, and blue. Each of these three almost-invariant sets as describes a collection of upper atmosphere weather conditions. Each of these almost-invariant sets has the special property that if a particular combination of weather conditions exist at the present time in the red (respectively, green and blue) set, it is very likely that future weather conditions will also belong to the red (respectively, green and blue) set.
Almost invariant sets are relevant to applications in dynamical astronomy. They can be used to characterise resonance regions in multi-body systems, and to compute transport rates of biological material between planetary objects.
Almost invariant sets are also relevant to molecular dynamics and drug design. Identifying almost-invariant sets numerically eliminates costly and time consuming laboratory experiments.
References
- J Guckenheimer and P Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 1983.
- E Lorenz. Deterministic Nonperiodic Flow. Journal of Atmospheric Sciences, Volume 20, 1963.
- G Froyland and M Dellnitz. Detecting and Locating Near-Optimal Almost-Invariant Sets and Cycles. SIAM Journal on Scientific Computing, 24(6), pp. 1839-1863, 2003.
- M Dellnitz, O Junge, WS Koon, F Lekien, M Lo, JE Marsden, K Padberg, R Preis, SD Ross, and B Thiere: Transport in Dynamical Astronomy and Multibody Problems. To appear in the International Journal of Bifurcation and Chaos, 2004.
- P Deuflhard, W Huisinga, A Fischer, and Ch Sch眉tte. Identification of almost invariant aggregates in nearly uncoupled Markov chains, Linear Algebra and its Applications. 315, pages 39-59, 2000.
Further Information
- GAIO (software used to perform computations)
- M Dellnitz, G Froyland, and O Junge. The Algorithms behind GAIO 鈥 Set Oriented Numerical Methods for Dynamical Systems. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp 145-174. Springer, 2001.
- (software used to create the images).
- , School of Mathematics and Statistics, UNSW.