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Classical chaos


Deterministic Approach to Die Tossing truecm

By means of a recursive transformation involving the construction of flight and collision maps we have investigated the structure of the basins of attraction for a die toss. 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. A similar analysis has been performed for the Galton Apparatus (or so-called Probability machine). We have shown that the outcome distribution for this machine in general is not binominal. It may be approximated by a Gaussian distribution but the standard deviation in this distribution will depend on the coefficient of restitution in the ball-pin collisions. For certain parameters the outcome distribution may be double humped. truecm

M. Christensen, L.-U. W. Hansen and E. Mosekilde truecm

Interference Effects in Semiclassical Orbits truecm

We investigate the relationship between the power spectrum of a wave field and that of a spatially uncorrelated source exciting it. Correlation functions are also examined. It is found that interference effects can significantly alter the structure of the power spectrum, leading to oscillations even when the power spectrum of the source is smooth. We derive a semiclassical approximation in which these oscillations are related to orbits of the geometrical limit of the wave system. We also derive a trace formula that approximates a spatial average of the wave power spectrum as a sum over periodic orbits. These calculations explain the structure of a measured power spectrum of the fluctuations in the height of a fluid surface generated by the circular hydraulic jump, which provided the motivation for the study. truecm

S.C. Creagh and P. Dimon truecm

Steady Fast Kinematic Dynamos: The Topological Entropy Bound and the Effect of Magnetic Field Diffusion truecm

We utilize recently developed periodic orbit techniques to establish that the topological entropy bounds the dynamo growth rate, estimate the magnetic field diffusion induced suppression of the high frequency eigenmodes, and give periodic orbit formulas for the cancellation exponent and the generalized dimension , quantities characterizing the singular nature of dynamo eigenfunctions. truecm

P. Cvitanovic and (E. Ott) truecm

Sum Rules for Periodic Orbit Theory truecm

A family of exact sum rules for stabilities of periodic points of d-dimensional polynomial mappings is derived and checked numerically. Associated Fredholm determinants are of particularly simple polynomial form. Possible applications to periodic orbit calculations for continuous time flows are discussed. truecm

P. Cvitanovic and G. Vattay truecm

Dynamical Averaging in Terms of Periodic Orbits truecm

Periodic orbit theory methods for evaluation of average values of observables for chaotic dynamical systems are reviewed and illustrated by several examples, such as evaluation of the Lyapunov exponents and the diffusion constants. truecm

P. Cvitanovic truecm

Trace Formula for Dynamic Systems in the Presence of Noise truecm

Evolution operator methods for evaluation of averages in chaotic 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. truecm

P. Cvitanovic, (R. Mainieri) and G. Vattay truecm

Symbolic Dynamics and Markov Partitions for the Stadium Billiard truecm

We investigate the Bunimovich stadium dynamics and show that in the limit of infinitely long stadium the symbolic dynamics is a subshift of finite type. For a stadium of finite length the Markov partitions are infinite, but the inadmissible symbol sequences can be determined exactly by means of the appropriate pruning front. We outline a construction of a sequence of finite Markov graph approximations by means of approximate pruning fronts with finite numbers of steps. truecm

P. Cvitanovic and (K.T. Hansen) truecm

Topology and Bifurcation Structure of Maps of Hénon type truecm

We construct a series of n-modal approximations to maps of the Hénon type and apply the corresponding symbolic dynamics to description of the admissible orbits and possible bifurcation structures for such maps. truecm

P. Cvitanovic and (K.T. Hansen) truecm

Periodic Orbit Expansions for Power Spectra of Chaotic Systems truecm

We develop the periodic orbit theory of power spectra for chaotic dynamical systems. The theory is tested numerically on several 1-dimensional mappings. truecm

P. Cvitanovic, (M.J. Feigenbaum) and (A.S. Pikovsky) truecm

Bifurcation Analyses truecm

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. We have also investigated the solutions to Duffing's equation for a forced nonlinear oscillator. This equation 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. Finally, we have studied an extension of 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. truecm

C. Knudsen, E. Mosekilde, (J. Sturis) and J.S. Thomsen truecm

Phase Locking in Long Josephson Diodes truecm

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. 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. truecm

(N. Grønbech-Jensen), (Y.S. Kivskar) and M. Samuelsen truecm

Vehicle Dynamics truecm

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. Poincare sections and return maps have been applied to describe the structure of the periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behavior can be fully understood as a nonlinear iterative process. truecm

R. Feldberg, C. Knudsen and (H. True)

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Mon Mar 6 19:42:06 MET 1995