The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.Research in part supported by AFOSR and by the Department of Energy (Scientific Computing Staff Office of Energy Research)     

Multifractal Analysis of Inhomogeneous Bernoulli Products

We are interested to the multifractal analysis of inhomogeneous Bernoulli products which are also known as coin tossing measures. We give conditions ensuring the validity of the multifractal formalism for such measures. On another hand, we show that these measures can have a dense set of phase transitions.     

Pseudo Random Coins Show More Heads Than Tails

Tossing a coin is the most elementary Monte-Carlo experiment. In a computer the coin is replaced by a pseudo random number generator. It can be shown analytically and by exact enumerations that popular random number generators are not capable of imitating a fair coin: pseudo random coins show more “heads” than “tails.” This bias explains the empirically observed failure of some random number generators in random walk experiments. It can be traced down to the special role of the value zero in the algebra of finite fields.     

Brownian ratchets and Parrondo's games

Parrondo's games present an apparently paradoxical situation where individually losing games can be combined to win. In this article we analyze the case of two coin tossing games. Game B is played with two biased coins and has state-dependent rules based on the player's current capital. Game B can exhibit detailed balance or even negative drift (i.e., loss), depending on the chosen parameters. Game A is played with a single biased coin that produces a loss or negative drift in capital. However, a winning expectation is achieved by randomly mixing A and B. One possible interpretation pictures game A as a source of "noise" that is rectified by game B to produce overall positive drift-as in a Brownian ratchet. Game B has a state-dependent rule that favors a losing coin, but when this state dependence is broken up by the noise introduced by game A, a winning coin is favored. In this article we find the parameter space in which the paradoxical effect occurs and carry out a winning rate analysis. The significance of Parrondo's games is that they are physically motivated and were originally derived by considering a Brownian ratchet-the combination of the games can be therefore considered as a discrete-time Brownian ratchet. We postulate the use of games of this type as a toy model for a number of physical and biological processes and raise a number of open questions for future research. (c) 2001 American Institute of Physics.     

Provably secure experimental quantum bit-string generation

Coin tossing is a cryptographic task in which two parties who do not trust each other aim to generate a common random bit. Using classical communication this is impossible, but nontrivial coin tossing is possible using quantum communication. Here we consider the case when the parties do not want to toss a single coin, but many. This is called bit-string generation. We report the experimental generation of strings of coins which are provably more random than achievable using classical communication. The experiment is based on the "plug and play" scheme developed for quantum cryptography, and therefore well suited for long distance quantum communication.     

Nonstandard (non-σ-additive) probabilities in algebraic quantum field theory

Traditionally, physicists deduce the observational (physical) meaning of probabilistic predictions from the implicit assumption that thewell-defined events whose probabilities are 0 never occur. For example, the conclusion that in a potentially infinite sequence of identical experiments with probability 0.5 (like coin tossing) the frequency of heads tends to 0.5 follows from the theorem that sequences for which the frequencies do not tend to 0.5 occur with probability 0. Similarly, the conclusion that in quantum mechanics, measuring a quantity always results in a number from its spectrum is justified by the fact that the probability of getting a number outside the spectrum is 0. In the mid-60s, a consistent formalization of this assumption was proposed by Kolmogorov and Martin-Löf, who defined arandom element of a probability space as an element that does not belong to any definable set of probability 0 (definable in some reasonable sense). This formalization is based on the fact that traditional probability measures are σ-additive, i.e., that the union of countably many sets of probability 0 has measure 0. In quantum mechanics with infinitely many degrees of freedom (e.g., in quantum field theory) and in statistical physics one must often consider non-σ-additive measures, for which the Martin-Löf's definition does not apply. Many such measures can be defined as “limits” of standard probability distributions. In this paper, we formalize the notion of a random element for such finitely-additive probability measures, and thus explain the observational (physical) meaning of such probabilities.     

Multifarious intertwined basin boundaries of strange nonchaotic attractors in a quasiperiodically forced system

A variety of different dynamical regimes involving strange nonchaotic attractors (SNAs) can be observed in a quasiperiodically forced delayed system. We describe some numerical experiments giving evidences of intertwined basin boundaries (smooth, non-Wada fractal and Wada property) for SNAs. In particular, we show that Wada property, fractality and smoothness can be intertwined on arbitrarily fine scales. This suggests that SNAs can exhibit the final state sensitivity and unpredictable behaviors. An interesting dynamical transition of SNAs together with associated mechanisms from non-Wada fractal to Wada intertwined basin boundaries is examined. A scaling exponent is used to characterize the intertwined basin boundaries.     

Co-existence of multiple attractors in the PWM controlled DC drives

DC shunt and series drives are extensively used in the industry. The occurrence of bifurcation and chaos in dc shunt and permanent magnet drives are well known. It is observed that the behavior of the drives not only depends on the value of system parameters but also on the value of initial conditions. Multiple attractors can exist for same parameter value. Different choice of initial conditions gives different periodic behavior of the system. The drive is intended to operate in a parameter range to give period-1 behavior. We report the existence of sub- harmonic oscillations in the period-1 region of the bifurcation diagram along with co-existing attractor with fractal basin boundaries in PWM controlled dc series drives. The series drive is extensively used in electric traction and other applications. The dc drives are run with dc input voltage. This dc voltage may be derived from a dc source or an ac source with a rectifier. The dc series drive shows different bifurcation behavior when different types of input voltage and switching elements are used. The existence of period-1, period-2 and period-4 orbits are observed with different initial conditions in the desired period-1 region of the bifurcation diagram. The dependence of system’s behavior on initial condition may render the system’s behavior unpredictable. These phenomena may have serious implication in performance.     

Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Exact solutions and symmetry analysis for the limiting probability distribution of quantum walks

In the literature, there are numerous studies of one-dimensional discrete-time quantum walks (DTQWs) using a moving shift operator. However, there is no exact solution for the limiting probability distributions of DTQWs on cycles using a general coin or swapping shift operator. In this paper, we derive exact solutions for the limiting probability distribution of quantum walks using a general coin and swapping shift operator on cycles for the first time. Based on the exact solutions, we show how to generate symmetric quantum walks and determine the condition under which a symmetric quantum walk appears. Our results suggest that choosing various coin and initial state parameters can achieve a symmetric quantum walk. By defining a quantity to measure the variation of symmetry, deviation and mixing time of symmetric quantum walks are also investigated.     

Multiple coexisting attractors,Basin boundaries and basic sets

Orbits initialized exactly on a basin boundary remain on that boundary and tend to a subset on the boundary. The largest ergodic such sets are called basic sets. In this paper we develop a numerical technique which restricts orbits to the boundary. We call these numerically obtained orbits “straddle orbits”. By following straddle orbits we can obtain all the basic sets on a basin boundary. Furthermore, we show that knowledge of the basic sets provides essential information on the structure of the boundaries. The straddle orbit method is illustrated by two systems as examples. The first system is a damped driven pendulum which has two basins of attraction separated by a fractal basin boundary. In this case the basic set is chaotic and appears to resemble the product of two Cantor sets. The second system is a high-dimensional system (five phase space dimensions), namely, two coupled driven Van der Pol oscillators. Two parameter sets are examined for this system. In one of these cases the basin boundaries are not fractal, but there are several attractors and the basins are tangled in a complicated way. In this case all the basic sets are found to be unstable periodic orbits. It is then shown that using the numerically obtained knowledge of the basic sets, one can untangle the topology of the basin boundaries in the five-dimensional phase space. In the case of the other parameter set, we find that the basin boundary is fractal and contains at least two basic sets one of which is chaotic and the other quasiperiodic.     

Global isochrons of a planar system near a phaseless set with saddle equilibria

Given an attracting periodic orbit of a system of ordinary differential equations, one can assign an asymptotic phase to any initial condition that approaches such a periodic orbit. All initial conditions with the same asymptotic phase lie on what is known as an isochron. Isochrons foliate the basin of attraction, and may have intriguing geometric properties. We present here two cases of a planar vector field for which the basin boundary — also referred to as the phaseless set — contains saddle equilibria and their stable manifolds. A continuation-based approach, in combination with Poincaré compactification when the basin is unbounded, allows us to compute isochrons accurately and visualise them as smooth curves to clarify their overall geometry.     

On the Fluctuation Relation for Nosé-Hoover Boundary Thermostated Systems

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nosé-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is verified at all times, as expected. Times longer than the mesoscopic scale, needed for local equilibrium to be settled, are required if a different initial ensemble is considered. This shows how the transient fluctuation relation asymptotically leads to the steady state relation when, as explicitly checked in our systems, the condition found in (D.J. Searles, et al., J. Stat. Phys. 128:1337, 2007), for the validity of the steady state fluctuation relation, is verified. For the steady state fluctuations of the phase space contraction rate Λ and of the dissipation function Ω, a similar relaxation regime at shorter averaging times is found. The quantity Ω satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale; the quantity Λ appears to begin a monotonic convergence after such times. This is consistent with the fact that Ω and Λ differ by a total time derivative, and that the tails of the probability distribution function of Λ are Gaussian.     

Fractal basin boundaries

Basin boundaries for dynamical systems can be either smooth or fractal. This paper investigates fractal basin boundaries. One practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes. The structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected. Examples and discussion of both types of structures are given, and it appears that fractal basin boundaries should be common in typical dynamical systems. Lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed.     

Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction–diffusion systems

In bistable systems, the long-term behavior of solutions depends on the location of the initial conditions. In a deterministic setting, where the initial condition is kept fixed in one particular basin of attraction, repeated numerical simulations will always lead to the same long-term behavior. The other possible asymptotic solution type will never be observed. This clear distinction does not hold anymore if the system is forced by random fluctuations. In this case, both asymptotic solutions can be attained, and the relative frequency of different long-term behaviors observed in many repeated simulation runs will follow a certain probability distribution. We present a simple reaction–diffusion model of spatial predator–prey interaction, where depending on the initial spatial distribution of the two populations either spatially homogeneous or spatiotemporal irregular oscillations may be observed. We show by repeated stochastic simulations that, when starting in the basin of attraction of the spatiotemporal irregular solution, in the randomly forced system the probability to observe spatially homogeneous oscillations instead of spatiotemporally irregular oscillations follows a non-trivial bimodal distribution.     

Stability of Atomic Simulations with Matching Boundary Conditions

We explore the stability of matching boundary conditions in one space dimension, which was proposed recently for atomic simulations (Wang and Tang, Int. J. Numer. Mech. Eng., 93 (2013), pp. 1255-1285). For a finite segment of the linear harmonic chain, we construct explicit energy functionals that decay along with time. For a nonlinear atomic chain with its nonlinearity vanished around the boundaries, an energy functional is constructed for the first order matching boundary condition. Numerical verifications are also presented.     

Symmetry boundary conditions

A simple approach to energy conserving boundary conditions using exact symmetries is described which is especially useful for numerical simulations using the finite difference method. Each field in the simulation is normally either symmetric (even) or antisymmetric (odd) with respect to the simulation boundary. Another possible boundary condition is an antisymmetric perturbation about a nonzero value. One of the most powerful aspects of this approach is that it can be easily implemented in curvilinear coordinates by making the scale factors of the coordinate transformation symmetric about the boundaries. The method is demonstrated for magnetohydrodynamics (MHD), reduced MHD, and a hybrid code with particle ions and fluid electrons. These boundary conditions yield exact energy conservation in the limit of infinite time and space resolution. Also discussed is the interpretation that the particle charge reverses sign at a conducting boundary with boundary normal perpendicular to the background magnetic field.     

Relativistic quantum coin tossing

A relativistic quantum exchange protocol enabling “coin tossing at a distance” between two participants is proposed. The exchange protocol is based on the fact that to distinguish a pair of orthogonal states with certainty in relativistic quantum mechanics requires a finite time that depends on the structure of the states themselves. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 10, 684–689 (25 November 1999)     

Finite temperature Casimir effect in Kaluza–Klein spacetime

In this article, we consider the finite temperature Casimir effect in Kaluza–Klein spacetime due the vacuum fluctuation of massless scalar field with Dirichlet boundary conditions. We consider the general case where the extra dimensions (internal space) can be any compact connected manifold or orbifold without boundaries. Using piston analysis, we show that the Casimir force is always attractive at any temperature, regardless of the geometry of the internal space. Moreover, the magnitude of the Casimir force increases as the size of the internal space increases and it reduces to the Casimir force in (3+1)$(3+1)$-dimensional Minkowski spacetime when the size of the internal space shrinks to zero. In the other extreme where the internal space is large, the Casimir force can increase beyond all bound. Asymptotic behaviors of the Casimir force in the low and high temperature regimes are derived and it is observed that the magnitude of the Casimir force grows linearly with temperature in the high temperature regime.