Quantum chaos

• Dr. Martin Gutzwiller, Yale University, New Haven, CT

Quantum Chaos describes and tries to understand the nature of the wave-like motions for the electrons in atoms and molecules (quantum mechanics), as well as electromagnetic waves and acoustics, etc.. To a limited extent, these waves are like the chaotic trajectories of particles in classical mechanics, including the light rays in optical instruments and the sound waves in complicated containers.

Introduction

The need for a scientific connection

Quantum Chaos (QC) tries to understand the connection between two phenomena in physics, call them Q and C. The word quantum (Q) comes from the physics of small systems like atoms and molecules, where the energy very often appears only in a well defined amount, called quantum. Very surprisingly, the movement of a small particle like the electron in a molecule looks more like a wave on the surface of a pond than the scratch of a dot on some plate. Wave phenomena of this kind describe the propagation of light, and quite generally most electronagnetism, as well as sounds in any kind of medium. These waves obey linear partial differential equations, whose solutions have smooth shapes, and are quite pleasant to behold.

C stands for the chaos, i.e. unexpected and nearly unpredictable behavior of very simple mechanical devices like the double pendulum, or the motion of a billiard ball on an imaginary table with a more complicated than rectangular shape. The double pendulum does unexpected turns and loops, and the exact direction of the ball after few bounces is difficult to predict. The motion is controlled by ordinary differential equations, whose solutions are extremely sensitive to the initial conditions. The resulting shape of the trajectories is confusing, although it can be computed rather simply to arbitrarily many decimals.

These two phenomena contradict our expectations because we try to find a simple explanation for the behavior of many interesting and useful objects. An electron as a wave in a molecule makes a pleasant picture, but its computation is tricky, particularly if one has to understand several elctrons acting simultaneously. The same is true for elctromagnetic and sound waves. Trajectories for electrons and rays for light and sound seem more in touch with our experience, therefore direct and satisfying. But for the complete explanation, the trajectories and rays are not always helpful. Nevertheless our intuition has to be prepared with the help of simple models that fall back on what our senses and intelligence can grasp. Quantum and Chaos look unrelated, and almost contradictory in spite of our scientific efforts.

Classical Mechanics (CM)

This difference in appearance has required very different scientific explanations. Before the 20-th century, the physical laws of Isaac Newton were able to explain the motion of the planets and moons in the solar systems, but also solve many problems of everyday life. This approach came to be called Classical Mechanics (CM). It is based on the solution of ordinary differential equations. They are able to explain what we now call chaos, although finding the best methods for each case is not easy. At the end of the 19-th century, Henri Poincare invented new treatments for chaotic systems, and his work was continued by many outstanding mathematicians and astronomers. E.g. he used “Surface of Section” where the same trajectory cuts through a fixed surface, over and over again, leaving a dot wherever it crossed. In the simplest cases this leads to smooth curves, while chaos produces a wild scatter of isolated points.

Quantum Mechanics (QM)

The idea of the quantum of energy, however, and the electrons moving like waves, was found to be incompatible with classical mechanics. During the first 25 years of the 20-th century, the best physicists tried to find some compromise with classical mechanics, but only with limited success. The big breakthrough came in 1925, and within four years there was a new kind of mechanics, that is capable of solving all atomic, molecular, and optical problems. Schroedinger’s equation has to be solved to get the wave function of the system, and that is the most convenient expression of quantum mechanics (QM). It is a partial differential equation very much like the wave equation for the explanation of sound, radio and light waves, etc. But in many-body systems quantum mechanics goes way beyond our familiar kind of wave phenomena.

Building a bridge between CM and QM

Quantum Chaos (QC) tries to build a bridge between QM and CM. This bridge provides a transition from QM to CM, as well as from CM to QM. The existence of such a bridge puts limits on CM and on QM. An ever smaller tennis-ball bounces differently from moving surfaces, and it looks more like an electron. Similarly an ever larger molecule eventually may become a big crystal that does no longer move like a wave.

Now imagine a tennis ball bouncing off the hard walls in a closed court. The trajectory of an ideal ball keeps on going around the court. Two motions with initially close directions may eventually have no similarity, depending on the exact shape of the court. This process keeps on going in CM as long as we want. The only limit to the precision in CM is the size of the computer. In QM, however, a built-in lower limit for the description of the motion prevents the chaos from getting too wild. Chaos in QM is mild compared to chaos in CM, but its appearance is not as clear cut as in CM.

Two examples from physics

The eigenstates of a quantum system

In order to appreciate the problem of making the connection between QM and CM, there will be first a simple presentation of some examples, where the connection was established successfully. In describing these examples it is important to be aware of some basic differences between CM and QM with respect to atoms and molecules. In CM there are almost no conditions where the nuclei and electrons with their electrostatic interactions can find some kind of equilibrium, because they are bound to collapse.

In QM there is usually a whole set of eigenstates with precise energies, starting with the “ground state” that is absolutely stable. The “excited states” can decay only if the system is allowed to interact with the electromagnetic field, and emit or absorb photons to change its energy. If these eigenstates are limited in space, they can be enumerated with integers, starting with 0 for the ground state, and positive integers \(n\) in the order of increasing energy \(E_n\ .\) The set of these energies is the spectrum.

Energy levels of the donor impurity in a silicon or germanium crystal

A donor impurity replaces an atom of Si or Ge in the crystal lattice; it has an effective nuclear charge of 1 higher than Si or Ge, and brings along an extra electron which tries to stay nearby if there is no outside electric field. Therefore, there is a local hydrogen atom in the crystal, with one trouble: the inertial mass of the electron in the x-direction is effectively much larger than in the y- and z-direction, by a factor 5 in Si and by a factor 20 in Ge.

An ordinary hydrogen atom near ionization in a strong magnetic field

At first the only electron stays near the nucleus in the ground state. But then it is exposed to some ultraviolet light, of sufficient and well controlled frequency to almost throw out the electron, i.e. ionize the atom. The electron ends up far away, but is still weakly held by the nucleus in one of the great number of eigenstates at a great distance. That leads to a measured spectrum that looks as if the lines of absorption were arranged totally at random. The eigenstates near ionization are random creatures!

The Path Integral (PI)

The Path Integral of Dirac and Feynman

In order to make the transition from CM to QM, a very general procedure is required. A natural concept of “physical length” \(L\) for a trajectory in CM was found about a century after Newton’s time. Then two of the most imaginative theoretical physicists, P.A.M. Dirac and Richard Feynman, before and after WWII, suggested a new approach to QM, and a bridge to CM. A short explanation of their idea has to do the job at this point.

You can ask the question in CM: How does the electron get from the place \(x\) to the place \(y\) in the fixed time \(t\) while it is subject to some known forces. Answer: Consider any smooth path \(z(s)\) with \(0 < s < t\) from \(x\) to \(y\), compute for this path the “physical length” \(L\ .\) In order to calculate the physical length of a particular path, the total time available is divided into small intervals. For each interval the difference of the kinetic energy minus the potential energy is multiplied with the duration of the time interval, and all these contributions are added up for the whole path. The path with the smallest length, say \(L_0\ ,\) is the simplest classical trajectory that connects the two fixed endpoints in the given time. Let me confess that this idea of the physical length \(L\ ,\) based on the difference between kinetic and potential energy, does not catch my intuition.

For QM: Any path from \(x\) to \(y\) in the given time \(t\) carries a wave, where the phase is the physical length \(L\ ,\) divided by Planck’s constant \(h\ .\) Then let all these waves interfere with each other, and add up. This “path integral” (PI) is difficult to calculate. If the lengths \(L\) are large compared to \(h\ ,\) however, most contributions cancel one another. Any classical trajectory is then favored, because paths with small deviations from \(L_0\) are numerous in its neighborhood.

Simplification of the path integral for complicated problems

In order to get the spectrum without the wave functions, the time \(t\) is replaced by the energy \(E\) with the help of a Fourier transform. The dependence on the space coordinates \(x\) and \(y\) is eliminated be setting \(x = y\ ,\) and then integrating over all available space \(x=y\ .\) The result in QM is the trace, simply the sum over the resonance denominators \(1/(E-E_n)\) over the spectrum.

In semi-classical evaluation of the PI, all the trajectories from \(x\) to \(y\) in time \(t\ ,\) i.e. all stationary points in the variation of the physical length \(L\ ,\) are used. If \(x = y\ ,\) the classical trajectories close themselves, but initial and final momentum do not agree. After the summation over available space \(x=y\ ,\) the trace accepts only those closed orbits where initial and final momenta agree. The result is a periodic orbit (PO).

The Trace formula

Connecting the quantum spectrum with a semi-classical spectrum

The result of the whole program in the preceding section is expressed in a relatively simple formula, now generally called the trace formula (TF). On the left is the trace \(g(E)\) as obtained from QM. It is the sum of the resonance denominators for the spectrum of the quantum system, \[g(E)=\Sigma_n 1/(E-E_n)\ .\] On the right is the semi-classical approximation \(g_C(E)\) of \(g(E)\ ,\) i.e. the sum over all periodic orbits (PO) in the corresponding classical system, \[g_C(E) = \Sigma_\nu A_\nu exp(iL_\nu/h + i\lambda_\nu\pi/2)\ .\] The amplitude \(A_\nu\) for each PO reflects its stability; the phase depends on the length \(L_\nu\) of the PO, and a multiple \(\lambda_\nu\) (Morse index) of \(\pi/2\) for each classical bounce off a dynamical wall. These are all classical quantities.

The TF can be given an intuitive interpretation: The open parameter \(E\) represents a small perturbation with a constant frequency \(\mu = E/h\) that works on the system from the outside, where \(h\) is always Planck's constant. The reaction of the system is a forced motion of the same frequency, with the amplitude \(g(E)\ .\) The closer \(E\) is to one of the eigenvalues \(E_n\ ,\) the larger is the response of the system; we get a resonance! The external perturbation of frequency \(\mu\) can be described also by its period \(\tau\ ,\) the reciprocal of \(\mu\ .\) The classical particle gets chased around in its space, and it is critical where it lands after one period \(\tau\ .\) The effect on the classical particle will be larger if it comes back to its starting point after one, or perhaps two or three such periods. Therefore, the classical description of a quantum resonance depends on the PO's. The physical length of a PO, \(L_\nu\) in the TF, yields the period in time by taking the derivative w.r.to the energy \(E\) of the PO.

A chaotic motion where the trace formula is correct

This correspondence between the set of energies \(E_n\) in QM and the set of periodic orbits in CM is a deep mathematical result, even if the proposed derivation of the TF is sloppy by mathematical standards. The result was first derived as an equality by the mathematician Atle Selberg in 1952 for the motion on a 2-dimensional surface of constant negative curvature.

Surfaces of constant negative curvature are products of the non-Euclidean geometry, starting in the first half of the 19-th century. It was then discovered at the end of the 19-th century that their geodesics, equivalent to the trajectories of a small ball rolling freely on the surface, were very chaotic. But it was also understood that these surfaces came in very many, very symmetric varieties, i.e. like polygones, they were tiling all the available space. The Euclidean plane has relatively few regular triangles, squares, hexagones, without any chaotic behavior of the straight lines. The sphere, of constant positive curvature, is trivial.

Selberg tried to find a relation between Riemann’s zeta-function, which holds all the secrets of the prime numbers, and geometry. The zeroes of the zeta-function would play the role of the eigenvalues, and the logarithm of the primes are the corresponding PO’s, unstable as on Selberg’s surfaces. But no real quantum problem for the zeta-function is known.

With 2 as well as with 3 dimensions, with constant negative curvature, there is an incredible variety of geometric models. They have different topologies, and then within each topology there are continuous parameters available to generate surfaces that are metrically different. With constant positive curvature, however, there is only one surface up to a scale factor, the sphere of 2 or 3 dimensions. Evidently chaotic motions are much more numerous, than the regular motion, even in pure geometry.

Figure 1: The 2 simplest periodic orbits for the electron in the neighborhood of the donor impurity in Si at the center of the circle; its radius corresponds to the energy of the electron set to 1.

Figure 2: The 2 next simplest PO's, each characterized by a binary code that indicates the order of intersection with the horizontal axis.

Figure 3: A periodic orbit of code length 10, without much symmetry, and therefore hard to find.

The results for the 2 examples of Atomic Chaos

The spectrum of a donor impurity

The extra electron does not stay very close to the place of the donor impurity, because the neighboring atoms of Si and of Ge get pushed out of their ordinary positions by the presence of the impurity. The effective attraction of the electron gets weakened by factors 11 for Si and 15 for Ge. The ordinary Coulomb force gets divided by 11 or 15, and the radius of the impurity increases by that factor. Figure 1 and Figure 2 show the 4 shortest PO´s. and their codes, i. e. intersections with the x-axis. All of them have some symmetry, and finding them is easy. Figure 3 shows a PO of code length of 10, and no symmetry. Finding it requires patience because this PO is very unstable.

Figure 4: The semi-classical spectrum of the donor impurity in Si, plotted as intersections with the horizontal E-axis; upper diagram on the basis of only the 8 shortest PO's, and lower diagram with the 71 shortest PO's.

Figure 5: Numerical computations for the spectrum of the donor-impurity: Names for the levels in first column, QM in 1969 in second, then trace formula in 1980, and QM in high precision in 1988.

The trace \(g(E)\) can be written as a converging product of factors \((E-E_n)\ ,\) and \(g_C(E)\) becomes something similar with respect to the PO’s. Figure 4 shows \(g_C(E)\ ,\) the upper diagram for only the 8 shortest PO’s, and the lower for the 71 PO’s. The approximate energies \(E_n\) are the intersections of the curve with the \(E\)-axis. The correct values are indicated by short lines crossing the \(E\)-axis. The low energies come out very well. Figure 5 shows the energy levels E_n , each with the usual description in the hydrogen atom, level and angular momentum in the first column, the second column computed with QM in 1969, third column computed with trace formula in 1980, fourth column computed with QM and high precision.

Figure 6: Resurgence Spectroscopy of an "Ordinary hydrogen atom near ionization in a strong magnetic field", cf. main text. The coordinates are in the upper diagram: relative absorption versus increasing energy, and in the lower: correlation versus time (period).

Ordinary hydrogen atom near ionization in a strong magnetic field

The upper diagram in Figure 6 shows the measured absorption in a high precision experiment. The width of the individual lines depends on the stability of the laser light. There are no sensible names for the lines in this spectrum, like we had in the donor impurity. The question arises whether this spectrum is truly random. The answer depends on all kinds of tests one could try; and then one would have to interpret the result. It then came as a great surprise: the trace formula suggests that the Fourier transform of this spectrum, from energy \(E\) to time \(t\ ,\) yields strong lines whenever there is a PO with that time for its period. The lower diagram in Figure 6 shows the PO’s for this random spectrum. This method of explaining a random looking spectrum was only discovered by the work on the trace formula; it is now called Resurgence Spectrocopy. Although this analysis does not always work, it is marvelous result of QC.

Figure 7: A liquid is contained in an open container with the shape of a stadium, which is lightly shaken at various fixed frequencies; photographic pictures of the vibrating surface show the same well defined patterns as can be found in the stadia of figures 8 to 11.

Beyond atomic physics

All kinds of ordinary waves inside hard walls

The history of optics is well known for the battles between rays and waves as the fundamental way of propagating light. The mathematics of these waves and their relation to the corresponding rays is almost identical to the relation between CM and QM. Chaos in the optical rays is just as complicated as in the motion of electrons. A popular model in 2 dimensions is a flat area surrounded by a hard wall. The rays inside such a cavity are straight lines with ideal reflection at the wall. Chaos comes from the shape of the wall, a simply closed curve. The equations for an electron in such a model are the same as for light, sound, fluid motion as in Figure 7. It shows the surface waves of a liquid due to the shaking of its container.

Russian mathematicians distinguished themselves after WWII by studying in great detail certain classes of geometric models to determine the nature of the trajectories. The measure of chaos is called its entropy, and the main results show that it is not zero. Among them is the “stadium”, 2 parallel lines of equal length that are connected with half circles at each end. It is instructive to look at some work with this system.

Figure 8: Resonances in a microwave cavity between 17 and 18 GHz at temperatures of 300 K (upper) and 2 K (lower diagram) to reduce the damping in the metal of the cavity, increasing the Q-factor and the resolution of the spectrum.

Figure 9: Fourier analysis (resurgence spectroscopy) of the resonances for the lower part of figure 8. Each one of the various peaks can be associated with the period of the various periodic orbits in the stadium.

Microwaves in the stadium and light in a oval-shaped cavity

Microwaves with a wave length of several centimeters are interesting to watch in a stadium-like cavity of about 1 m, but no more than 2 cm thick. At room temperature the resistance of the metal of the cavity only allows subdued resonances, while at 2 degrees Kelvin they are very clear as in Figure 8. With Resurgence Spectroscopy, i.e. Fourier transform from \(E\) to \(t\ ,\) the PO’s of the stadium are shown in Figure 9, just as in Figure 6.

Figure 10: Resonances in the conductivity for electrons inside a two-dimensional stadium (upper) and circle (lower diagram) of mesoscopic size, as a function of an applied very large magnetic field and at very low temperature.

Figure 11: Picture explaining the laser function inside a glas cavity; the oval shape (not an ellipse) has been carefully chosen for the light to escape at either end tangentially, as shown in the lower picture.

===The stadium in the real world===Figure 10 shows the electric resistance versus an applied magnetic field in a conducting layer between two semiconductors in two configurations. The mean free path of the electrons is larger than the stadium or the circle; the temperature is extremely low, and the resonances are very sharp. The statistical distribution for fhe chaotic stadium has only one broad peak, whereas the nonchaotic circle has many resonances beyond 2 symmetric minima. Figure 11 shows laser light caught inside a stadium of glass with an oval cross section. The light is forced out at the ends tangentially by the curvature, and only there.

Spectral Statistics and more Applications

Applications in nuclear physics

In contrast to the use of QM in atomic and molecular physics, the atomic nucleus is not well understood, because the forces between the nucleons, i.e. proton and neutron, are much more complicated than the simple Coulomb forces between nuclei and electrons. Nuclear physicists have to work with empirical models. Nevertheless the spectrum of nuclear energy levels is very rich, and therefore, complicated.

Neutron resonance spectroscopy provides a unique situation where, in a narrow energy window, successive eigen-energies (in the compound nucleus region) around, say, the one-hundred-thousandth level in a heavy nucleus, can be detected very accurately one by one (cf. fig. 12). It is then natural to adopt a statistical approach. Such statistics were discussed ever after WWII under the assumption that the fluctuation properties of the energy levels come from finite, but large matrices of various kinds. The random choice of the matrix elements was investigated and compared with the experiments. This "random matrix theory" became the foundation for understanding large parts of the nuclear spectra. In the beginning of the 1980’s the origin of these empirical random matrices was finally explained by the important conjecture that the origin of the distributions is the result of Quantum Chaos.

If such a connection was in fact to be expected, one could check it in other systems with a rich spectrum. A partial proof of this general conjecture in some special cases has since been found on the basis of the trace formula. Some special features of the PO’s in CM are limiting the statistics of the system in QM. In the 1970's, some mathematicians observed that the statistics of the "mysterious" zeroes for Riemann´s zeta-function have strong similarities with the eigenvalues of random-matrices. Some physicists like to talk about “Riemannium” as a new element with characteristic features in the “spectrum” of its zeroes.

Figure 12: Total cross section for the reactions n + 232 Th as a function of the neutron energy (from the compilation 'Neutron Cross Section', 1964). Notice the neutron energies, which are given to single eV, as well as the sharpness of the lines.

Some generalizations of the trace formula

• Sofar we have studied only how the spectrum of some wave phenomenon arises approximately with the help of the PO's. The path-integral also tells us how a particle starts in the point \(x\) and ends up in the point \(y\ .\) One can even give to \(x\) and \(y\) certain distributions to reflect the conditions of the experiment.

• A particular PO can depend on the energy or on the time available. For instance, it can appear or disappear as one increase the time of the energy. In that case the TF needs some additional details to be worked out in the neighborhood of the transition in time or energy.

• In the case of light rays, but just as well in the presence of steep rises in the potential energy, the ray or the trajectory may simultaneously split into reflection and into refraction on a wall. Such a possibility increases the number of PO's greatly.

• Many simple problems in molecular physics require the electron to tunnel, i.e. overcome a mountain of potential energy that is higher than the available total energy. In the simplest chemical bond, two protons being held together by either one or two electrons, the electron cannot move "classically" from the neighborhood of one proton to the neighborhood of the other proton. Therefore we have to allow classical trajectories with stretches of negative kinetic energy, where the time is a purely imaginary quantity, i.e. its square is negative.

• The angular momentum with a spin of h/2 is a very important attribute for the electron. The description by the Pauli matrices characterizes its local direction by 3 real components, i.e. a vector S of fixed length at each point in space. There are essentially 2 waves spreading at the same time over the same volume; together they determine exactly the 3 components of S. The motion of the electron through any electric or magnetic field will then lead to a motion of S along its motion in space.

• Scattering of electrons and photons from atoms and molecules can be treated. For this purpose the process is most usefully considered as in a Feynman diagram, where a light ray hits the electron trajectory. Energy and momentum have to be conserved at the point of collision.

• For quite a while it was not clear whether it is possible to get reliable results from TF. But with some better understanding, the precision of the bound states depends on a chosen upper limit \(E_n\) of the energy. Quite unexpectedly, if the upper limit is chosen relatively low, the TF will yield a few of the lowest states quite well, contrary to the general assumption that semiclassical results are good only for large energies.

• The TF arises from the second order correction to the propagator, or path integral PI, because we took into account the second order variation to the appropriate classical trajectories in the PI. The lowest order is the famous formula of Hermann Weyl, which yields the density of the eigenstates for any linear differential operator. For quantum mechanics the TF implies a correct term of order 2 in Planck's quantum h. By including third- and higher order variations in the PI, one can get a formal expansion to higher order for the spectrum and other properties.

• The time dependence in QM has not as yet been studied in great detail for many systems. It turns out to be a very difficult mathematical problem, with many unexpected features even in very simple systems such as the reflexion of the wave from a steep wall of finite height. Strangely, the PI is defined for a fixed time interval t; the energy E arises only with the help of a Fourier integral. The time dependence in QM should be easy to obtain directly from the PI, or its semiclassical approximation. But it is quite tricky, even numerically in an oval-shaped stadium. There is still much work to do that might have many practical applications, and compare directly with experiments.

Various technical areas of application

• Bound states and scattering in chemistry.
• Intra- and inter-molecular dynamics due to molecular vibrations.
• 2-dimensional electron traps on a metal surface.
• Shell structure of crystals depending on the lattice vibrations.
• Magnetic susceptibility in anti-dot arrays.
• Spin-orbit coupling for electrons in GaAs/GaAlAs interface.
• Cohesion and stability of metal nanowires.
• Concert halls, drums, church bells, tsunamis, etc.

Some Reading

Postmodern Quantum Mechanics, by Eric J. Heller and Steven Tomsovic, Physics Today (American Institute of Physics) July 1993, 38-46.

Einstein's Unknown Insight and the Problem of Quantizing Chaos, by A. Douglas Stone, Physics Today (American Institute of Physics) August 2005 37-43.

Celestial Mechanics on a Microscopic Scale, by T.Uzer, D.Farrelly, J.A.Milligan, P.E.Raines, and J.P.Skelton, Science 253 (1991) 42-48.

The Culture of Quantum Chaos, by M. Norton Wise and David C. Brock, Stud. Hist. Phil. Mod. Phys., Vol.29, No.3 (1998) 369-389.

References

Resource Letter ICQM-1: The Interplay between Classical and Quantum Mechanics, by Martin C. Gutzwiller, American Journal of Physics 66 (1998) 304-324. Same Title with Same Editor, Collection of reprints, published 2001 by AAPT (American Association of Physics Teachers), College Park, MD 20740-3845.

Quantum Chaology (The Bakerian Lecture 1987), by M. V. Berry, in Dynammical Chaos, Proceedings of the Royal Society, edited by Michael V. Berry, I.C. Percival, and N.O. Weiss, A 413, 1-198.

Semiclassical Physics, by Matthias Brack and Rajat K. Bhaduri, Addison-Wesley Inc., Reading MA, 1997, 444 p.

Quantum Signatures of Chaos, by Fritz Haake, Springer-Verlag, Berlin-Heidelberg, 2nd ed 2001, 479 p.

Quantum Chaos - An Introduction, by Hans-Juergen Stoeckmann, Cambridge University Press, 1999.

Quantum Chaos Y2K, Proceedings of Nobel Symposium 116, edited by Karl-Fredrik Berggren and Sven Aberg, in Physica Scripta, Kungl. Vetenskapsakademien and World Scientific, Singapore, 2001.

Internal references

• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

See Also

Dynamic Billiards, Chaos, Dynamical Systems, Periodic Orbit, Quantum Mechanics

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

Reviewed by: Anonymous

Reviewed by: Prof. Gregor Tanner, School of Mathematical Sciences, University of Nottingham

Accepted on: 2007-09-24 11:05:44 GMT