Introduction to dynamical systems using billiards

Billiards as dynamical systems

Dynamical systems are defined as the set of prescribed rules to evolve certain state in time. If the rules involve some random or probability feature (e.g., lottery) we say that the dynamical system is stochastic. Otherwise the system is called deterministic. Billiards are very illustrative example of dynamical system. In this case pointwise particles move in straight lines and experience specular collisions in the boundaries (like light in a mirror). The main general properties of billiards motion are illustrated below in a mushroom billiard (semi-circular stem placed on top of a triangular foot, see references at the end).

Magnetic Billiards

Another kind of dynamical systems are the so called magnetic billiards. In this case the straight line movement between collision is replaced by a the movement on semi-circles. The physical motivation is the movement of electric charges under the influence of a perpendicular magnetic field.

Chaos in billiards

Non-linear dynamical systems typically present chaos. Its most striking characteristics is its sensitivity to initial conditions. In the simulation below this effect is illustrated: the same position was chosen for the red and blue balls (no interaction between them) and a difference of 0.5% in the direction of movement (first case), and 0.5% in the position (second case). One sees that after some bounces the two trajectories are far away from each other. Chaos introduces unpredictability in fully deterministic dynamical systems.

Chaos and Order coexist

Trajectories in billiards may be chaotic (red) or regular (blue), depending on the initial condition. Regular trajectories perform a periodic or quasi-periodic movement and are restricted to the stem of the mushroom. If one waits long enough, every chaotic trajectories visits every point of the mushroom billiard table. This property is a generic property of Hamiltonian systems (to whom billiards belong). We say that such systems have a mixed phase space.

Intermittent Chaos

One interesting problem of nonlinear dynamics which is not completely solved is how the movement of chaotic trajectories in a mixed phase space look like. When chaotic trajectories approach the regular region (located at the mushrooms stem) they spent a very long time close to them before visiting again the rest of the chaotic region (foot of the billiard). During this time the movement is very regular and we call thus the full movement as intermittent (alternates between chaos and regular). This can be seen also in another paradigmatic example of Hamiltonian systems, the [standard map], whose phase space is shown at the right.


How to generate such animations? Making gif animations using Xmgrace.

About mushroom billiards

  • L. Bunimovich, Mushrooms and other billiards with divided phase space, Chaos 11 (2001), 802. Kinematics, equilibrium, and shape in Hamiltonian systems: The "lab"effect, Chaos 13 (2003), 903.

About chaos in Hamiltonian systems

  • E. Ott, Chaos in dynamical systems, Cambridge University Press, Cambridge, 2002.
  • A. M. Ozorio de Almeida, Hamiltonian systems: Chaos and quantization, Cambridge University Press, Cambridge, 1992.
  • Mackay and Meiss (Eds.), Hamiltonian Dyanmical systems, Adam Hilger, Bristol, 1987.

About intermittent chaos and stickiness

  • J.D. Meiss, Symplectic maps, variational principles and transport, Rev. Mod. Phys. 64 (1992), 795.
  • J. D. Meiss and E. Ott, Markov-tree model of intrinsic transport in Hamiltonian systems, Phys. Rev. Lett. 55 (1985), 2741.Markov-tree model of transport in area-preserving maps, Physica D 20 (1986), 387.
  • G. M. Zaslavsky, Chaos, fractinal kinetics, and anomalous transport, Phys. Rep. 371 (2002), 461.
  • E. G. Altmann, Intermittent chaos in Hamiltonian dynamical system [Ph.D. Thesis] (2007).

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