Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps

Ensemble averages of the sensitivity to initial conditions xi(t) and the entropy production per unit of time of a new family of one-dimensional dissipative maps, x(t+1)=1-ae(-1/|x(t)|(z))(z>0), and of the known logisticlike maps, x(t+1)=1-a|x(t)|(z)(z>1), are numerically studied, both for strong (Lyapunov exponent lambda(1)>0) and weak (chaos threshold, i.e., lambda(1)=0) chaotic cases. In all cases we verify the following: (i) both [ln((q)x triple bond (x(1-q)-1)/(1-q); ln((1)x=ln(x] and [S(q) triple bond (1- sigma p(q)(i))/(q-1); S(1)=- sigma p(i)ln(p(i)] linearly increase with time for (and only for) a special value of q, q(av)(sen), and (ii) the slope of and that of coincide, thus interestingly extending the well known Pesin theorem. For strong chaos, q(av)(sen)=1, whereas at the edge of chaos q(av)(sen)(z)<1.     

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.     

Synchronization and suppression of chaos in non-locally coupled map lattices

We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a power-law fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.     

基于时空混沌系统的单向Hash函数构造

提出了一种基于时空混沌系统的Hash函数构造方法.以线性变换后的消息数作为一组初值来驱动单向耦合映像格子的时空混沌系统，产生时空混沌序列，取其空间最后一组混沌序列的适当项，线性映射为Hash值要求的128bit值.研究结果表明，这种基于时空混沌系统的Hash函数具有很好的单向性、弱碰撞性、初值敏感性，较基于低维混沌映射的Hash函数具有更强的保密性能，且实现简单. 关键词： 时空混沌 Hash函数 单向耦合映像格子     

Inertial mass of the Abrikosov vortex

We show that a large contribution to the inertial mass of the Abrikosov vortex comes from transversal displacements of the crystal lattice. The corresponding part of the mass per unit length of the vortex line is M(l)=(m(2)(e)c(2)/64 pi alpha(2)mu lambda(4)(L))ln((lambda(L)/xi), where m(e) is the bare electron mass, c is the speed of light, alpha=e(2)/Planck's over 2 pi c approximately 1/137 is the fine structure constant, mu is the shear modulus of the solid, lambda(L) is the London penetration length, and xi is the coherence length. In conventional superconductors, this mass can be comparable to or even greater than the vortex core mass computed by Suhl [Phys. Rev. Lett. 14, 226 (1965)]].     

Breakdown of dynamic scaling in surface growth under shadowing

Using Monte Carlo simulations and experimental results, we show that for common thin film deposition techniques, such as sputter deposition and chemical vapor deposition, a mound structure can be formed with a characteristic length scale, or "wavelength" lambda, that describes the separation of the mounds. We show that the temporal evolution of lambda is distinctly different from that of the mound size, or lateral correlation length xi. The formation of a mound structure is due to nonlocal growth effects, such as shadowing, that lead to the breakdown of the self-affinity of the morphology described by the well-established dynamic scaling theory. We show that the wavelength grows as a function of time in a power law form, lambda approximately t(p), where p approximately equals 0.5 for a wide range of growth conditions, while the mound size grows as xi approximately t(1-z), where 1/z varies depending on growth conditions.     

Controlling chaos: Stabilization of unstable orbits using exponential control for maps and flows

We demonstrate that chaos can be controlled using multiplicative exponential feedback control. Unstable fixed points, unstable limit cycles and unstable chaotic trajectories can all be stabilized using such control which is effective both for maps and flows. The control is of particular significance for systems with several degrees of freedom, as knowledge of only one variable on the desired unstable orbit is sufficient to settle the system onto that orbit. We find in all cases that the transient time is a decreasing function of the stiffness of control. But increasing the stiffness beyond an optimum value can increase the transient time. We have also used such a mechanism to control spatiotemporal chaos is a well-known coupled map lattice model.     

Dynamical behavior of the multiplicative diffusion coupled map lattices

We report a dynamical study of multiplicative diffusion coupled map lattices with the coupling between the elements only through the bifurcation parameter of the mapping function. We discuss the diffusive process of the lattice from an initially random distribution state to a homogeneous one as well as the stable range of the diffusive homogeneous attractor. For various coupling strengths we find that there are several types of spatiotemporal structures. In addition, the evolution of the lattice into chaos is studied. A largest Lyapunov exponent and a spatial correlation function have been used to characterize the dynamical behavior. (c) 1996 American Institute of Physics.     

Oscillatory behavior of critical amplitudes of the Gaussian model on a hierarchical structure

We studied oscillatory behavior of critical amplitudes for the Gaussian model on a hierarchical structure presented by a modified Sierpinski gasket lattice. This model is known to display nonstandard critical behavior on the lattice under study. The leading singular behavior of the correlation length xi near the critical coupling K=K(c) is modulated by a function which is periodic in ln/ln(K(c)-K)/. We have also shown that the common finite-size scaling hypothesis, according to which for a finite system at criticality xi should be of the order of the size of the system, is not applicable in this case. As a consequence of this, the exact form of the leading singular behavior of xi differs from the one described earlier (which was based on the finite-size scaling assumption).     

Finite-size scaling and universality of the thermal resistivity of liquid 4He near Tlambda

We present measurements of the thermal resistivity rho(t,P,L) near the superfluid transition of 4He at saturated vapor pressure and confined in cylindrical geometries with radii L=0.5 and 1.0 microm [t identical with T/T(lambda)(P)-1]. For L=1.0 microm measurements at six pressures P are presented. At and above T(lambda) the data are consistent with a universal scaling function F(X)=(L/xi(0))(x/nu)(rho/rho(0)), X=(L/xi(0))(1/nu)t valid for all P (rho(0) and x are the pressure-dependent amplitude and effective exponent of the bulk resistivity rho, and xi=xi(0)t(-nu) is the correlation length). Indications of breakdown of scaling and universality are observed below T(lambda).     

Temperature-dependent anisotropy of the penetration depth and coherence length of MgB2

We report measurements of the temperature-dependent anisotropies (gamma(lambda) and gamma(xi)) of both the London penetration depth lambda and the upper critical field of MgB2. Data for gamma(lambda)=lambda(c)/lambda(a) was obtained from measurements of lambda(a) and lambda(c) on a single crystal sample using a tunnel diode oscillator technique. gamma(xi)=H(perp)c(c2)/H(||c)(c2) was deduced from field-dependent specific heat measurements on the same sample. Gamma(lambda) and gamma(xi) have opposite temperature dependencies, but close to T(c) tend to a common value (gamma(lambda) similar or equal to gamma(xi)=1.75 +/- 0.05). These results are in good agreement with theories accounting for the two-gap nature of MgB2.     

Does synchronization of networks of chaotic maps lead to control?

We consider networks of chaotic maps with different network topologies. In each case, they are coupled in such a way as to generate synchronized chaotic solutions. By using the methods of control of chaos we are controlling a single map into a predetermined trajectory. We analyze the reaction of the network to such a control. Specifically we show that a line of one-dimensional logistic maps that are unidirectionally coupled can be controlled from the first oscillator whereas a ring of diffusively coupled maps cannot be controlled for more than 5 maps. We show that rings with more elements can be controlled if every third map is controlled. The dependence of unidirectionally coupled maps on noise is studied. The noise level leads to a finite synchronization lengths for which maps can be controlled by a single location. A two-dimensional lattice is also studied.     

Spatiotemporal dynamics of Bose-Einstein condensates in moving optical lattices

Spatiotemporal dynamics of Bose-Einstein condensates in moving optical lattices have been studied. For a weak lattice potential, the perturbed correction to the heteroclinic orbit in a repulsive system is constructed. We find the boundedness conditions of the perturbed correction contain the Melnikov chaotic criterion predicting the onset of Smale-horseshoe chaos. The effect of the chemical potential on the spatiotemporal dynamics is numerically investigated. It is revealed that the variance of the chemical potential can lead the systems into chaos. Regulating the intensity of the lattice potential can efficiently suppress the chaos resulting from the variance of the chemical potential. And then the effect of the phenomenological dissipation is considered. Numerical calculation reveals that the chaos in the dissipative system can be suppressed by adjusting the chemical potential and the intensity of the lattice potential.     

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The dynamics of cold atoms in conservative optical lattices obviously depends on the geometry of the lattice. But very similar lattices may lead to deeply different dynamics. In a 2D optical lattice with a square mesh, it is expected that the coupling between the degrees of freedom leads to chaotic motions. However, in some conditions, chaos remains marginal. The aim of this paper is to understand the dynamical mechanisms inhibiting the appearance of chaos in such a case. As the quantum dynamics of a system is defined as a function of its classical dynamics – e.g. quantum chaos is defined as the quantum regime of a system whose classical dynamics is chaotic – we focus here on the dynamical regimes of classical atoms inside a well. We show that when chaos is inhibited, the motions in the two directions of space are frequency locked in most of the phase space, for most of the parameters of the lattice and atoms. This synchronization, not as strict as that of a dissipative system, is nevertheless a mechanism powerful enough to explain that chaos cannot appear in such conditions.     

耦合映象格子中时空混沌的控制

采用相同的相空间压缩方法,有效地控制了均匀及非均匀耦合映象格子中的时空混沌.数值模拟结果表明,在一定的相空间压缩参数区域内,控制时空混沌到均匀稳定状态时,控制结果与控制参数之间存在确定的函数关系.利用这个控制方程,选择不同的相空间压缩参数控制耦合映象格子中的时空混沌,获得了各种所需要的稳定斑图 关键词： 时空混沌 耦合映象格子     

Deterministic chaos in one-dimensional maps—the period doubling and intermittency routes

This paper is a review of the present status of studies relating to occurrence of deterministic chaos and its characterization in one-dimensional maps. As our primary aim is to introduce the nonspecialists into this fascinating world of chaos we start from very elementary concepts and give sufficient arguments for clarity of ideas. The two main scenarios during onset of chaos viz. the period doubling and intermittency are dealt with in detail. Although the logistic map is often discussed by way of illustration, a few more interesting maps are mentioned towards the end.     

The fuzzy logic of chaos and probabilistic inference

The logic of a physical system consists of the elementary observables of the system. We show that for chaotic systems the logic is not any more the classical Boolean lattice but a kind of fuzzy logic which we characterize for a class of chaotic maps. Among other interesting properties the fuzzy logic of chaos does not allow for infinite combinations of propositions. This fact reflects the instability of dynamics and it is shared also by quantum systems with diagonal singularity. We also generalize the fuzzy implication to a probabilistic implication following the hint of von Neumann. In this way we can evaluate the probability of the validity of the logical inference.     

Emergence of Fermi-Dirac thermalization in the quantum computer core

We model an isolated quantum computer as a two-dimensional lattice of qubits (spin halves) with fluctuations in individual qubit energies and residual short-range inter-qubit couplings. In the limit when fluctuations and couplings are small compared to the one-qubit energy spacing, the spectrum has a band structure and we study the quantum computer core (central band) with the highest density of states. Above a critical inter-qubit coupling strength, quantum chaos sets in, leading to quantum ergodicity of eigenstates in an isolated quantum computer. The onset of chaos results in the interaction induced dynamical thermalization and the occupation numbers well described by the Fermi-Dirac distribution. This thermalization destroys the noninteracting qubit structure and sets serious requirements for the quantum computer operability. Received 3 July 2001 and Received in final form 9 September 2001     

Flux line lattice reorientation in the borocarbide superconductors with H parallel a

Small angle neutron scattering studies of the flux line lattice in LuNi2B2C and ErNi2B2C induced by a field parallel to the a axis reveal a first order flux line lattice reorientation transition. Below the transition the flux line lattice nearest neighbor direction is parallel to the b axis, and above the transition it is parallel to the c axis. This transition cannot be explained using nonlocal corrections to the London model. In addition, the anisotropy of the penetration depth lambda and the coherence length xi change at the transition.     

Multiple period-doubling bifurcation route to chaos in periodically pulsed Murali-Lakshmanan-Chua circuit-controlling and synchronization of chaos

We consider a simple nonautonomous dissipative nonlinear electronic circuit consisting of Chua's diode as the only nonlinear element, which exhibit a typical period doubling bifurcation route to chaotic oscillations. In this paper, we show that the effect of additional periodic pulses in this Murali-Lakshmanan-Chua (MLC) circuit results in novel multiple-period-doubling bifurcation behavior, prior to the onset of chaos, by using both numerical and some experimental simulations. In the chaotic regime, this circuit exhibits a rich variety of dynamical behavior including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures, and so on. For certain types of periodic pulses, this circuit also admits transcritical bifurcations preceding the onset of multiple-period-doubling bifurcations. We have characterized our numerical simulation results by using Lyapunov exponents, correlation dimension, and power spectrum, which are found to be in good agreement with the experimental observations. Further controlling and synchronization of chaos in this periodically pulsed MLC circuit have been achieved by using suitable methods. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses. The above study also indicates that one can generate any desired n-period-doubling bifurcation behavior by applying n-periodic pulses to a chaotic system.