Publications with Abstracts by Alfred Hübler (2007)
* Publications in journals with a strict referee process
- A. Hübler and K. C. Phelps, Guiding a self-adjusting system through chaos, paper presented at Symposium on Complex Systems Engineering, RAND Corporation, Santa Monica, CA, January 11-12, 2007, accepted by Complexity
Abstract: We study the parametric controls of self-adjusting systems with numerical models. We investigate the situation where the target dynamics changes slowly and passes through a chaotic region. We find that feedback destabilizes controls if the target is chaotic. If the control is unstable the system migrates to the closest non-chaotic target, i.e. it adapts to the edge of chaos. For weak controls the deviation between system dynamics and target is larger, but the system dynamics is less chaotic and therefore more predictable.
Preprint
- B. Chase and A. Hübler, Inverse Energy-Uncertainty Relation for a Simple Information Engine, submitted to Physica D
Abstract: We examine the relationship between uncertainty in initial conditions and subsequent energy
gain for a simple information engine in the likeness of Maxwell’s Demon. Our engine consists of two
noninteracting idealized classical particles in a two-compartment box. Using information about the
initial states of the particles and our uncertainty in those states, we desire to capture both particles
to perform work. We find that in certain cases the energy extracted is inversely proportional to
the initial uncertainty. We also examine properties of a nonlinear container. We also note that the
average energy over all initial condition cases is approximately the fraction of particles which we
can predict long enough for the possibility of capture to occur.
Preprint
- *M. Singleton and A. Hübler, Learning rate and attractor size of the single-layer perceptron, Phys. Rev. E 75, 026704 (2007).
Abstract: We study the simplest possible order one single-layer perceptron with two inputs, using the delta rule with
online learning, in order to derive closed form expressions for the mean convergence rates. We investigate the
rate of convergence in weight space of the weight vectors corresponding to each of the 14 out of 16 linearly
separable rules. These vectors follow zigzagging lines through the piecewise constant vector field to their
respective attractors. Based on our studies, we conclude that a single-layer perceptron with N inputs will
converge in an average number of steps given by an Nth order polynomial in t
l , where t is the threshold, and
l is the size of the initial weight distribution. Exact values for these averages are provided for the five linearly
separable classes with N=2. We also demonstrate that the learning rate is determined by the attractor size, and
that the attractors of a single-layer perceptron with N inputs partition RN x RN.
Preprint
- A. Hübler, G. Foster and K. C. Phelps, Managing chaos: Thinking out of the box, Complexity 12(3), 10-13 (2007).
Preprint
- D. Farrell and A. Hübler, Acceleration Beyond the Wave Speed in Dissipative
Wave-Particle Systems, submitted to Physics Letters A
Abstract:Quantitative reasoning skills are a fundamental tool in many fields, ranging from Mathematics to Engineering and from Business to Rhetoric. However, quantitative reasoning is almost never taught as a course, except in the context of other disciplines, such as Mathematics or Physics. We introduce basic elements of a course in reasoning, such as the definition of a concept and the definition of a strategy and study the response of the students. We apply the definitions to algebraic proofing. We conceptualize algebraic concepts, i.e. we name each concept, define its range of applicability and illustrate each concept with typical examples. We also conceptualize strategies for proofing. We find that a diverse population of female middle school students readily accepts this approach and achieves proofing skills on a level which is comparable to university freshmen.
Preprint
- *G. Foster, A. Hübler, K. Dahmen, Resonant forcing of multi-dimensional chaotic map dynamics, Phys. Rev. E 75, 036212 (2007).
Abstract: We study resonances of chaotic map dynamics. We use the calculus of variations to determine the additive forcing function that induces the largest response. We find that resonant forcing functions complement the separation of nearby trajectories, in that the product of the displacement of nearby trajectories and the resonant forcing is a conserved quantity. As a consequence, the resonant function will have the same periodicity as the displacement dynamics and if the displacement dynamics are irregular, then the resonant forcing function will be irregular as well. Furthermore we show that resonant forcing functions of chaotic systems decrease exponentially, where the rate equals the negative of the largest Lyapunov exponent of the unperturbed system. We compare the response to optimal forcing with random forcing, and find that the optimal forcing is particularly effective if the largest Lyapunov exponent is significantly larger than the other Lyapunov exponents. However, if the largest Lyapunov exponent is much larger than unity, then the optimal forcing decreases rapidly and is only as effective as a single push forcing.
Preprint
- *C. Strelioff, J. Crutchfield, A. Hübler, k-th Order Markov Chains: Bayesian Inference,
Model Comparison, Entropy Rate, and Out-of-class Modeling
, submitted to Physical Review E in March 2007
Abstract: Markov chains are a natural and well understood tool for describing one-dimensional patterns in
time or space. We show how to infer k-th order Markov chains, for arbitrary k, from finite data.
In this process we employ Bayesian methods applied to both parameter estimation and model-
order selection. Extending existing results for multinomial models of discrete data, we present
new results which connect inference to statistical mechanics through information-theoretic (type
theory) techniques. We establish a direct relationship between Bayesian evidence and the partition
function which allows for direct calculation of the expectation and variance of the conditional relative
entropy and the source entropy rate. Finally, we explore a novel method for understanding out-of-
class modeling. We demonstrate the use of finite-date-size scaling with model order comparison to
infer the structure of out-of-class processes.
Preprint
- Gintautas, V. and Hubler, A. Mixed Reality States in a Bidirectionally Coupled Interreality System. Will appear in May 2007 issue of PRE.
Abstract:: We present experimental data on the limiting behavior of an interreality system comprising a
virtual horizontally driven pendulum coupled to its real-world counterpart, where the interaction
time scale is much shorter than the time scale of the dynamical system. We present experimental
evidence that if the physical parameters of the simplified virtual system match those of the real
system within a certain tolerance, there is a transition from an uncorrelated dual reality state to
a mixed reality state of the system in which the motion of the two pendula is highly correlated.
The region in parameter space for stable solutions has an Arnold tongue structure for both the
experimental data and for a numerical simulation. As virtual systems better approximate real ones,
even weak coupling in other interreality systems may produce sudden changes to mixed reality
states.
Preprint
- A. Hübler, Understanding Complex Systems: Defining an abstract concept
, Complexity 12(5), 5-9(2007).
Preprint
- Austin Gerig and Alfred Hüubler, Chaos in a one-dimensional compressible flow, to appear in Phys. Rev. E
We study the dynamics of a one-dimensional discrete flow with open boundaries - a series of moving
point particles connected by ideal springs. These particles flow towards an inlet at constant velocity,
pass into a region where they are free to move according to their nearest neighbor interactions, and
then pass an outlet where they travel with a sinusoidally varying velocity. As the amplitude of the
outlet oscillations is increased, we find that the resident time of particles in the chamber follows a
bifurcating (Feigenbaum) route to chaos. This irregular dynamics may be related to the complex
behavior of many particle discrete flows or is possibly a low-dimensional analogue of non-stationary
flow in continuous systems.
Preprint
- Vadas Gintautas, Glenn Foster, and Alfred W. Hubler, Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics, preprint submitted to J. Stat. Phys.
We study resonances of multidimensional chaotic map
dynamics. We use the calculus of variations to determine
the additive forcing function that induces the largest
response, that is, the greatest deviation from the unperturbed
dynamics. We include the additional constraint
that only select degrees of freedombe forced, corresponding
to a very general class of problems in which not all of
the degrees of freedom in an experimental system are accessible
to forcing. We find that certain Lagrange multipliers
take on a fundamental physical role as the efficiency
of the forcing function and the effective forcing experienced
by the degrees of freedom which are not forced
directly. Furthermore, we find that the product of the
displacement of nearby trajectories and the effective total
forcing function is a conserved quantity. We demonstrate
the efficacy of this methodology with several examples.
Preprint
- Jian Xu and Alfred W. Hubler, Detecting quasiperiodic structures with double diffraction, preprint submitted to Europhysics Letters.
We study interference patterns of double di
raction systems with quasiperiodic
structures. A quasiperiodic linear array of scatterers converts single delta pulses into a sequence
of quasiperiodic pulses. This pulse train is di
racted from a second set of scatterers. We find that
the interference pattern after the second di
raction has a pronounced peak if both sets of scatterers
have similar quasiperiodic structures. We show that this method can be used for identifying the
Fibonacci chain and related quasiperiodic sequences, if the number of scatterers in the first set is
at least twice as large as the number of scatterers in the second set, and if the distances among the
two sets of scatterers and the detector are all large compared to the size of the sets. This method
may provide a methodology for identifying the structure of quasicrystals and quasiperiodic layered
materials with a large signal to noise ratio.
Preprint