Classical and Quantum Chaos

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Classical and Quantum Chaos


A Fredholm Determinant for Semi-classical Quantization 0.01truecm

We investigate a new type of approximation to quantum determinants, the ``Quantum Fredholm Determinant", and test numerically the conjecture that for Axiom A hyperbolic flows such determinants have a larger domain of analyticity and better convergence than the quantum Selberg products derived from the Gutzwiller trace formula. The conjecture is supported by numerical investigations of the 3-disk repeller, a normal-form model of a flow, and a model 2- map [][]. -0.1truecm P. Cvitanovic, P.E. Rosenqvist, G. Vattay, and H.H. Rugh 0.3truecm

Entire Fredholm Determinants for Evaluation of Semi-classical and Thermodynamical Spectra 0.01truecm

Proofs that Fredholm determinants of transfer operators for hyperbolic flows are entire can be extended to a large new class of multiplicative evolution operators. We construct such operators both for the Gutzwiller semi-classical quantum mechanics and for classical thermodynamic formalism, and introduce a new functional determinant which is expected to be entire for Axiom A flows, and whose zeros coincide with the zeros of the Gutzwiler-Voros zeta function. []. -0.1truecm P. Cvitanovic and G. Vattay 0.3truecm

Creeping Orbits in Semi-classical Quantization 0.01truecm

We try to incorporate the diffraction phenomena in the semi-classical description of classically chaotic systems. This leads us to introduction of periodic creeping orbits which are created by diffraction and thus have no classical particle trajectory analogue. We are studying their effects in simple 2-disk and 3-disk scattering systems, and are comparing our results to exact quantum mechanical results. -0.1truecm P. Cvitanovic, P.E. Rosenqvist, G. Vattay, and (A. Wirzba) 0.3truecm

Trace Formula for Dynamic Systems in the Presence of Noise 0.01truecm

Evolution operator methods for evaluation of averages in chaotical dynamical systems require arbitrarily fine symbolic dynamics partition of the phase space; at the same time, the attainable accuracy in physical dynamical systems is always limited by external noise. We use external noise as a parameter which defines the smallest phase-space partitions that need to be taken into account in a given computation, and generate finite Markov diagram approximation to the dynamics. -0.1truecm P. Cvitanovic and (R. Mainieri) 0.3truecm

Advection of Vector Fields by Chaotic Flows 0.01truecm

We have introduced a new transfer operator for chaotic flows whose leading eigenvalue yields the dynamo rate of the fast kinematic dynamo, and applied cycle expansion of the Fredholm determinant of the new operator to evaluation of its spectrum. The theory has been tested on a normal form model of the vector advecting dynamical flow. If the model is a simple map with constant time between two iterations, the dynamo rate is the same as the escape rate of scalar quantities. However, a spread in Poincaré section return times lifts the degeneracy of the vector and scalar advection rates, and leads to dynamo rates that dominate over the scalar advection rates. For sufficiently large time spreads we have even found repellers for which the magnetic field grows exponentially, in contrast to the scalar densities which decay exponentially []. -0.1truecm (N.J. Balmforth), P. Cvitanovic, (G.R. Ierley), (E.A. Spiegel), and G. Vattay 0.3truecm

Periodic Orbit Expansions for Power Spectra of Chaotic Systems 0.01truecm

We develop the periodic orbit theory of power spectra for chaotic dynamical systems. The theory is tested numerically on several 1-dimensional mappings. -0.1truecm P. Cvitanovic, (M.J. Feigenbaum), and (A.S. Pikovsky) 0.3truecm

Symbolic Dynamics Description of Chaotic Systems 0.01truecm

We study a number of different chaotic systems using the theory of symbolic dynamics. We show how symbolic dynamics can be used to describe -modal one-dimensional maps and two-dimensional maps of the Hénon type. Different billiard systems, such as the three disk system, hyperbola billiard, stadium billiard and the wedge billiard are described and we show how to construct a well ordered symbolic plane and find pruning fronts for these systems. These results will be applied to describe semiclassically quantum systems. -0.1truecm P. Cvitanovic and K.T. Hansen 0.3truecm

Sound Waves in Crystals 0.01truecm

The experimental study of eigenmodes of microwave cavities has received considerable credit in recent years. For an essentially two dimensional cavity, the corresponding electrodynamical wave equation coincides mathematically with the quantum mechanical Schrödinger equation. The cavities have to be cooled to a temperature of about 2K to achieve a high resolution in the spectra. It is investigated whether a comparably high resolution can be achieved by exciting the eigenmodes of a crystal with sound waves. The study of three dimensional geometries should also be possible. -0.1truecm C. Ellegaard, T. Guhr, and M. Oxborrow 0.3truecm

Correlation Holes in Microwave Spectra 0.01truecm

In order to achieve a high resolution in microwave cavity spectra, it is necessary to cool the cavity down to a temperature of about 2K. The correlation hole technique, frequently used in molecular physics, decouples to some extent the statistical influences of the widths and the positions of the resonances. We study if this method works for cavity spectra as well, to become less dependent on high resolution and cooling. -0.1truecm T. Guhr, (R. Hofferbert), and (A. Richter) 0.3truecm

Fourier Bessel Analysis in Graded Matrix Spaces 0.01truecm

The generating function of the correlation functions in theories involving random matrices can often be mapped onto theories, that show a so called hopping coupling and that are formulated in spaces of graded matrices. To evaluate the integrals an expansion is employed similar to the well known plane wave expansion. The angular degrees of freedom appear in generalized spherical harmonics and can thus be integrated by using results from representation theory. -0.1truecm T. Guhr 0.3truecm

Mixing of a Fixed Matrix to a Gaussian Unitary Ensemble 0.01truecm

A given matrix is, with a strength parameter, mixed to a Gaussian Unitary Ensemble. The correlation functions for arbitrary level number are evaluated by using the graded eigenvalue method. The results involve the graded symmetric functions which are related to the representations of supergroups. -0.1truecm T. Guhr 0.3truecm

Thermoelectricity in Bimetallic Film 0.01truecm

In 1821, Thomas Seebeck discovered that an D.C. electromotive force can be generated by heating a junction between two different conductors. This phenomenon, the so-called Seebeck effect, is reversible; it can be used to convert thermal energy into electrical energy with Carnot efficiency. Because irreversible phenomena (Joule heating and thermal conduction) inevitably accompany the Seebeck effect, however, no device based on the Seebeck effect, that can generate a sizeable amount of electricity at a reasonable efficiency, has ever been built. In 1993, I am attempting to generate an A.C.thermoelectric current, within a sputtered Al-Ni bilayer that lies on a piezoelectric PVDF substrate, by applying an ultrasonic (2MHz) wave normal to the substrate's surface. So far, I have generated only heat. -0.1truecm M. Oxborrow 0.3truecm

Embedded Gaussian Unitary Ensemble 0.01truecm

The Embedded Gaussian Ensembles model the forces in physical systems more realistically than the plain Gaussian Ensembles. The graded eigenvalue method and the graded Fourier Bessel Analysis are applied in order to study the Embedded Gaussian Unitary Ensemble. -0.1truecm T. Guhr 0.3truecm

Four Spheres Scattering Problem 0.01truecm

The scattering of a particle in a geometry of three discs in two dimensions was studied classically and quantum mechanically by several authors. Particularly, it was one of the first problems investigated with the periodic orbit theory. We study the analogous problem in three dimensions, the tetrahedron of four spheres. We expect new information about the dependence of (semi)classical methods on the dimensionality. The model is also more realistic for applications in condensed matter physics. -0.1truecm (A. Wirzba) and T. Guhr 0.3truecm

Periodic Orbits of Nonscaling Hamiltonian Systems from Quantum Mechanics. 0.01truecm

The spectral formfactor is utilized to define a quantum equivalent of the classical -plot of the periodic orbits of a Hamiltonian system. For scaling systems where the classical dynamics is independent of the energy (except for a trivial scaling) the quantal -plot may be reduced to the Fourier transform of the energy spectrum. For nonscaling systems on the other hand, the quantal -plot reveals the richness of the classical dynamics with its bifurcations of periodic orbits. The analysis may be extended into the regime where the diagonal form of the semiclassical formfactor is not expected to hold and where the connection to Random Matrix Theories is largely unknown. -0.1truecm (M. Baranger), (M. Haggarty), B. Lauritzen, (D. Meredith), and (D. Provost) 0.3truecm

Bound States in Twisting Tubes 0.01truecm

It is known that an infinite tube with constant cross-section has propagating modes if the frequency is above a certain cutoff frequency. What is not so widely known is that if the tube is bent there can also be localized states around the bend. These states exist as solutions of both Schrodinger's equation and also of Maxwell's equations when the tube is very narrow in one dimension so as to be quasi two-dimensional. This contrasts with the corresponding classical motion which is not localized. I am interested in understanding the connection between the classical and quantum behaviour by using semiclassical approximations to the quantum Green function. Hopefully this project will result in predictions for microwave experiments. -0.1truecm N.D. Whelan 0.3truecm

Classical Transport in Near-Integrable Chaotic Systems 0.01truecm

The KAM theorem of classical mechanics asserts that if an integrable Hamiltonian is subject to a small perturbation then only a small fraction of the tori are destroyed. However, this theorem has certain smoothness conditions for the perturbation. These conditions are violated for a number of systems of common interest such as the Bunimovich stadium, the Sinai billiard and the wedge in a constant gravitational field. Because of this violation these three problems share the property of being integrable for one choice of parameter and completely chaotic for arbitrarily close parameter values. This non-KAM behaviour is not very well understood. I am interested in understanding the nature of the classical transport of these chaotic systems when their hamiltonians are close to the integrable limit. Results obtained so far indicate that a trajectory remains trapped for a long time in a small region of phase space before making a sudden jump to a different region. It would be good to understand the mechanism for this behaviour and the extent to which understanding the nearby integrable Hamiltonian helps in understanding the chaotic Hamiltonian. In the longer term, it would also be interesting to see how this is reflected in the corresponding quantum dynamics. -0.1truecm N.D. Whelan 0.3truecm

Semiclassical Analysis of a Hyperbolic System 0.01truecm

Much is now understood about the connection between classical periodic orbits and the quantum Green function. We want to use this understanding to study an apparently simple problem. We consider free motion between confocal hyperbolae. This problem is integrable both classically and quantum mechanically. The quantum eigenvalue problem can be solved in terms of Mathieu functions and the classical problem has two independent integrals of motion in involution. What makes this problem interesting is that there is only one classical periodic orbit, which is unstable. This should make the connection between the quantum and classical problems particularly clear. We hope to be able to describe scattering resonances in terms of the one unstable periodic orbit. -0.1truecm G. Vattay and N.D. Whelan 0.3truecm

Determination of the Noise Level of Chaotic Time Series 0.01truecm

We analyze analytically the influence of gaussian measurement noise on the shape of the estimated corelation integral of an otherwise low dimensional signal. If the data is embedded in a space of higher dimension than strictly required to reconstruct the dynamics, the extra dimensions are dominated by the noise which results in a certain shape of the correlation integral. For the case that only Gaussian noise is present, this shape can be calculated analytically as a function of the noise level. This leads to a method to determine the noise level present in the data. Furthermore, the analytical result shows that a noise level of more than 2% will obscure any possible scaling of the correlation integral and thus makes it impossible to estimate the correlation dimension. -0.1truecm T. Schreiber 0.3truecm

Tunnelling in Multidimensional Integrable Systems 0.01truecm

We examine tunnelling from a point of view that makes it possible to treat intrinsically multidimensional systems. The semiclassical theory of tunnelling in dimension is very well understood, but much less so in more dimensions. The -dimensional theory is interpreted in terms of the construction of wavefunctions from complex Lagrangian manifolds in complexified phase space. With this interpretation it is possible to see how to generalize the calculation to more dimensions in cases where a quantum state can be identified with a Lagrangian manifold. The most obvious case is that of classically integrable systems where the quantum states are derived from invariant tori. We illustrate the theory by calculating the splittings in a -dimensional double well potential for which the Hamiltonian is integrable, but nonseparable. Because the system is nonseparable, it is not possible to reduce it to -dimensional problems - which has been the only way to solve tunnelling in integrable systems before. -0.1truecm S. Creagh 0.3truecm

Anomalous Tunnelling Splittings in Near-integrable Systems 0.01truecm

We compute the tunnelling rates in a near-integrable system under the assumption that the states can be associated with a complex Lagrangian manifold. The energy levels, without tunnelling corrections, have been well predicted by applying EBK quantization to KAM tori. The tunnelling splittings are computed using the technique previously used for exactly integrable systems. The splittings so obtained are found to be significantly smaller that the actual splittings. This indicates that the mechanism for tunnelling in even near-integrable systems is significantly different from that in exactly integrable systems. This is in line with previous calculations that have shown that the introduction of nonintegrability significantly enhances tunnelling rates. -0.1truecm S. Creagh 0.3truecm

Long Time Behaviour in Quantum Maps 0.01truecm

I have been using quantum maps to investigate the long-time behaviour of semiclassical approximations. The recent advances of reorganization of the trace formula into Riemmann-zeta-function-like determinants has the potential problem that they require pushing the semiclassical approximations to times that are very large on classical scales, and it is not obvious that the approximations are always valid for such long times. I investigated the validity of the formalism by testing it for quantizations of a perturbed Arnol'd cat map. For sufficiently small perturbations, the map is hyperbolic and semiclassical approximations hold well for long times. It is found that the zeta-function formulation gives accurate results in this case, including an explicit and simple realization of the `resurgence' predicted by Berry, (Les Houches Lecture Series 52 (North-Holland, 1990)). As the perturbation is increased, nonhyperbolic structures appear and the twists and folds associated with these should make the semiclassical behaviour much worse. It is indeed found that the zeta function deteriorates after the transition to nonhyperbolic behaviour. -0.1truecm S. Creagh 0.3truecm

Chaos in Periodically Driven Gunn Diodes 0.01truecm

We have applied numerical simulation to explore the highly nonlinear dynamic phenomena that can arise in Gunn diodes by interaction between the internally generated domain mode and an external microwave signal. By adjusting the time of domain formation and the speed of propagation, the internal oscillation entrains with the external signal. This produces a devil's staircase of frequency-locked solutions. At higher microwave amplitudes, period-doubling and other forms of mode-converting bifurcations can be seen. In this interval the diode also exhibits spatio-temporal chaos. At still higher microwave amplitudes, transitions to delayed, quenched and limited space-charge accumulation modes take place. -0.1truecm R. Feldberg, C. Knudsen, E. Mosekilde, and J.S. Thomsen 0.3truecm

Bifurcation Analyses for a Forced Chemical Oscillator 0.01truecm

We have performed a detailed numerical bifurcation analysis of the forced Brusselator model, exposing local and global bifurcation curves that constitute the internal structure of the dominant Arnol'd tongues. The results are presented as phase diagrams and one-parameter bifurcation diagrams. Two theorems concerning the existence of global bifurcations near generic codimension-2 bifurcation points are stated and proved. It is argued that the results are generic to a class of periodically forced self-oscillating systems. -0.1truecm C. Knudsen, (J. Sturis), and J. S. Thomsen 0.3truecm

Duffing's equation for a forced nonlinear oscillator 0.01truecm

In collaboration with Y. Ueda, University of Kyoto, we have investigated the solutions to Duffing's equation for a forced nonlinear oscillator. This equation, which in many ways is prototypical to modern nonlinear dynamics, is known to exhibit a wealth of complex phenomena, including deterministic chaos, coexisting regular and chaotic solutions, fractal basin boundaries, cascades of period-doubling bifurcations, as well as blue sky catastrophes. We have determined the phase diagram for forcing amplitudes exceeding those hitherto studied. New regions of chaotic behavior have been found. The basins of attraction for two coexisting chaotic attractors have been determined, and the chaotic solutions have been characterized in terms of their Lyapunov exponents and fractal dimensions. We have also studied an extension to Duffing's equation to three variables with external forcing. Starting from a phase-space conserving chaos, three prototypes of chaotic attractors with a dimension larger than three have been obtained. -0.1truecm E. Mosekilde and J.S. Thomsen 0.3truecm

Phase Locking in Long Josephson Diodes 0.01truecm

Phase-locking of fluxon motion to an external signal has been extensively studied using a simple particle model that captures many of the characteristics observed experimentally in long Josephson junctions. When the influence of the external ac drive manifests itself only through the boundary conditions, and when there is exactly one fluxon present in the junction at all times, the simple model reduces the perturbed sine-Gordon equation into an implicit two-dimensional map. Hence, it is possible to obtain analytical expressions for the existence of fixed points and their stability. When the fluxon is in a phase-locked state, it has been observed that the stability of the fixed point changes under parameter variations. This can result in period-doubling bifurcations and chaotic oscillations in the soliton propagation.

We have investigated the effect of thermal noise on the phase-locked state and shown that the noise measurements can be used to detect internal bifurcations in the system. This is particularly relevant to the prediction of the line width of the emitted signal from a phase-locked Josephson junction, since the line width increases with the variance of the time of flight for the fluxon.

The project has also led to the development of a numerical scheme for solving a nonlinear partial integro-differential equation with nonlocal time dependence. In particular we have considered the dynamics of long Josephson diodes modeled by use of the microscopic theory for tunneling between superconductors. It is demonstrated that the detailed behavior of a solitonic mode is different from the results of the conventional sine-Gordon equation. -0.1truecm M. Samuelsen, (N. Grønbech-Jensen), and (Y.S. Kivskar) 0.3truecm

Deterministic Approach to Die Tossing 0.01truecm

Usually, the motion of a die is so rapid that the human eye is unable to follow the trajectory in detail. Combined with the actual complexity of the motion, which appears to have defied mathematical treatment for centuries, this may have contributed to the widespread notion that a die toss is random. However, by virtue of the dissipation of energy, a die must always come to rest within a finite time. Therefore, a die toss cannot show the sensitive dependence on initial conditions found in chaotic systems or in systems with fractal basin boundaries. Nonetheless, die tossing has certain properties in common with these systems. In particular, the basins of attraction in the space of initial conditions show a certain self-similarity in structure.

In the present project we have investigated the structure of these basins of attraction by means of a recursive transformation involving the construction of flight and collision maps. We have proposed quantitative measures for the average size of the basins of attraction and for the mixing of these basins, and we have applied these measures to determine how the sensitivity to the initial conditions varies with the energy and orientation of the die. -0.1truecm R. Feldberg, C. Knudsen, and E. Mosekilde 0.3truecm

Vehicle Dynamics 0.01truecm

The dynamical behavior of railway vehicles is controlled by two nonlinear influences arising from the specific form of the wheel and rail profiles and from the dry friction in the contact between wheel and rail, respectively. Because of these nonlinearities, bifurcations, self-sustained oscillations and various forms of deterministic chaos may arise in the wheel/rail interaction. These phenomena can be observed in theoretical investigations and in experiments under laboratory conditions. In practical railway operation the deterministic response tends to be swamped by stochastic phenomena arising from various irregularities. However, the deterministic response may influence the wear of the wheels and the suspense system.

In the present project we have studied a dicone moving on a pair of cylindrical rails. Accounting for nonlinear friction, the equations of motion for this mechanical system consist of a set of nonlinear differential-algebraic equations. We have investigated the nonlinear dynamic behavior arising from these equations in detail with the aid of special in-house software and path-following algorithms. -0.1truecm C. Knudsen, R. Feldberg, (E. Slivsgaard), (H. True), and (M. Rose) 0.3truecm

Chaos in Economic Systems 0.01truecm

In collaboration with the Sloan School of Management, MIT, we have performed a series of experiments with human decision making behavior in simulated corporate environments. Participants were asked to operate a simplified production-distribution chain to minimize costs. Performance was systematically suboptimal, however, and in many cases the subjects were unable to secure the stable operation of the system. As a result, large-scale oscillations and various forms of highly nonlinear dynamic phenomena were observed.

A model of the applied ordering policy has been proposed. Econometric estimates show that the model is an excellent representation of the actual decisions. With different parameters, computer simulations of the estimated order policy produce a great variety of complex dynamic behaviors. Analyses of the parameter space reveal an extremely complex structure having a fractal boundary between the stable and unstable solutions, and with fingers of periodic solutions penetrating deeply intro regions representing quasiperiodic and chaotic solutions. In certain parts of the parameter space, any neighborhood of a given solution contains a qualitatively different solution. Thus, changes on the margin can produce a completely different system behavior.

Our results provide direct experimental evidence that chaos can be produced by the decision making behavior of real people in simple managerial systems. The consequent implications for the ability of human subjects to cope with complex dynamical systems are explored.

We have also studied how mode-locking and other nonlinear dynamic phenomena arise through the interaction of two capital-producing sectors in a disaggregated economic long-wave model. One sector might represent the construction of buildings and infrastructure capital with long lifetimes while the other represents production of machinery, transport means, etc., with much shorter lifetimes. -0.1truecm (E.R. Larsen), Chr. Haxholdt, Chr. Kampmann, E. Mosekilde, and J.S. Thomsen 0.3truecm

Experimental Observation of the Lorentz Group in Nonlinear Circuits 0.01truecm

We have measured Berry's geometrical phase of the Lorentz group in a pair of connected driven dissipative electronic oscillators in the vicinity of a period-doubling bifurcation. Surprisingly, under certain conditions, a small signal transforms just as the Lorentz transformation of a spinor, apart from some factors. By measuring along what corresponds to a closed loop on a hyperboloid (the invariant surface of the Lorentz group), we can determine Berry's phase of the Lorentz group. -0.1truecm H. Svensmark and P. Dimon 0.3truecm

Recursive definition of global cellular-automata mappings 0.01truecm

We have proposed a recursive definition of the global mappings for deterministic cellular automatas. The method is based on a graphical representation of global cellular-automata mappings. The recursive definition defines the change of the global mapping as the size of the automata is increased. A proof of lattice size invariance of the global mappings is derived from an approximation to the global mappings. The recursive definitions are applied to calculate the fractal dimension of the set of reachable states and of the set of fixed points of cellular automata on an infinite lattice. -0.1truecm R. Feldberg, C. Knudsen, and (S. Rasmussen)

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Nikolaj Berntsen
Mon Feb 6 10:24:27 MET 1995
14. Feb. 1995
Nikolaj Berntsen,