Quantifying chaos: A tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This Letter presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.     

An equation for continuous chaos

A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable. The flow in state space allows for a “folded” Poincaré map (horseshoe map). Many more natural and artificial systems are governed by this type of equation.     

Chapman-Kolmogorov equation for discrete chaos

We derive an equation of the Chapman-Kolmogorov type for multi-dimensional discrete mappings under the impact of additive and multiplicative noise with arbitrary distribution.     

Large deviations for multiplicative chaos

The local singularities for a class of random measures, obtained by random iterated multiplications, are investigated using the thermodynamic formalism. This analysis can be interpreted as a rigorous study of the phase transition of a system with random interactions.Partially supported by SCIENCE grant CT000307UPR A014 du CNRS     

电阻电容分路的约瑟夫森结中阵发性混沌及混沌控制的计算机模拟

通过计算机模拟研究了电阻电容分路的约瑟夫森结中的混沌行为,给出了结电压随阻尼参数及偏置直流电流变化的分岔图,从而展示了混沌产生的方式及混沌出现的参数区间,并基于弱周期扰动理论提出了控制RCSJJ中混沌的方案,模拟结果证明了该方案的有效性.     

Quantifying Discriminating Strength for Gaussian States

The discriminating strength DS(ρAB) induced by local Gaussian unitary operators for any(n + m)-mode Gaussian state ρABis introduced in [Phys. Rev. A 83(2011) 042325]. In this paper, we further discuss the quantity by restricting to Hilbert-Schmidt norm. The analytic formulas of DS for two-mode squeezed thermal states and mixed thermal states are given. Then, the relationship between DS(ρAB) and DS((I ? Φ)(ρAB)) for some special Gaussian channels Φ is discussed. In addition, DS is compared with Gaussian entanglement for symmetric squeezed thermal states.     

Local Lyapunov exponents for spatiotemporal chaos

Local Lyapunov exponents are proposed for characterization of perturbations in distributed dynamical systems with chaotic behavior. Their relation to usual and velocity-dependent exponents is discussed. Local Lyapunov exponents are analytically calculated for coupled map lattices using random field approximation. Boundary Lyapunov exponents describing reflection of perturbations at boundaries are also introduced and calculated.     

The stoichiometry of FeSe

Tetragonal iron selenide, FeSe, the layered parent compound of the recently discovered superconducting arsenide family, has previously been shown to be non-magnetic and superconducting with a critical temperature near 8 K. There has, however, been a lack of consensus as to whether selenium vacancies present due to large deviations from ideal stoichiometry are required to give rise to the superconductivity. Here we describe the results of experiments that demonstrate simply that superconducting iron selenide can only be synthesized as a pure material when near stoichiometric (i.e. FeSe). Significant selenium deficiency or excess gives rise to secondary magnetic phases, and a suppression of the superconductivity.     

Asymptotic chaos

We provide in table I a list of normal forms of ordinary differential equations describing the dynamics of physical systems in conditions near to the simultaneous onset of up three instabilities. The first (quadratic) terms in the Taylor series for the nonlinear terms in these amplitude equations (as they are called in fluid dynamics) are given in each case. We focus on a particular example involving three instabilities and derive an asymptotic version of the corresponding normal form as a limit of small dissipation is approached. The numerical investigation of this asymptotic normal form strongly suggests that chaotic behavior occurs as close as one wants to the onset of the triple instability. This chaotic behavior is also exhibited by a return map constructed by direct numerical experiments on the amplitude equation. We also derive by analytic methods a model return map that qualitatively reproduces much of the dynamics observed numerically in the solutions of the asymptotic normal form in nearly homoclinic conditions. In the limit of strong contraction, this model map of the plane reduces to a unidimensional map that is valuable in understanding the dynamics of the original system.     

Quantum chaos

A quantum system which is allowed to interact with its boundary in a self-consistent way is shown to exhibit chaos. We conjecture that in general genuine wave chaos (decaying autocorrelation functions, exponential sensitivity of wavefunctions to initial wavefunction configurations) can be obtained whenever a wavefield is allowed to modify its confining boundaries in a self-consistent way. We suggest to test this conjecture in the acoustic regime.     

Quantum chaos

In this paper we present an overview of important recent results in the study of a very controversial topic, the so-called quantum chaos. The theoretical and numerical results are compared with real laboratory experiments with special emphasis on the problem of ionization of hydrogen atoms in external microwave fields. (c) 1996 American Institute of Physics.