Scaling of dynamical properties of the Fermi-Ulam accelerator

We investigate numerically the chaotic sea of the complete Fermi-Ulam model (FUM) and of its simplified version (SFUM). We perform a scaling analysis near the integrable to non-integrable transition to describe the average energy as function of time t and as function of iteration (or collision) number n. When t is employed as independent variable, the exponents of FUM and SFUM are different. However, when n is used, the exponents are the same for both FUM and SFUM. In the collision number analysis, we present analytical arguments supporting the values of the exponents related to the control paramenter and to the initial velocity. We describe also how the scaling exponents obtained by using t as independent variable are related to the ones obtained with n. In contrast to SFUM, the average energy in FUM saturates for long times. We discuss the origin of the observed differences and similarities between FUM and its simplified version.     

Fuzzy model based adaptive synchronization of uncertain chaotic systems: Robust tracking control approach

This Letter presents fuzzy model-based robust tracking control for the adaptive synchronization of uncertain chaotic systems. Fuzzy model and adaptive algorithm are employed to present the unknown chaotic systems. H^∞ and sliding mode control are combined to construct a robust tracking controller. The incorporated H^∞ controller can attenuate the external disturbance and approximation error to any prescribed level. The proposed scheme guarantees that all the variables are bounded and the tracking error is compensated.     

Stochastic optimization of spin-glasses on cellular neural/nonlinear network based processors

Nowadays, Cellular Neural/Nonlinear Networks (CNN) are practically implemented in parallel, analog computers, showing a fast developing trend. It is important also for physicists to be aware that such computers are appropriate for implementing in an elegant manner practically important algorithms, which are extremely slow on the classical digital architecture. Here, CNN is used for optimization of spin-glass systems. We prove, that a CNN in which the parameters of all cells can be separately controlled, is the analog correspondent of a two-dimensional Ising type spin-glass system. Using the properties of CNN we show that one single operation on the CNN chip would yield a local minimum of the spin-glass energy function. By using this property a fast optimization method, similar to simulated annealing, can be built. After estimating the simulation time needed for this algorithm on CNN based computers, and comparing it with the time needed on normal digital computers using the classical simulated annealing algorithm, the results look promising: a speed-up of the order 10¹² is expected already at 50×50 lattice sizes. Hardwares realized nowadays are of 128×128 size. Also, there seem to be no technical difficulties adapting CNN chips for such problems and the needed local control of the parameters could be fully developed in the near future.     

Forbidden ordinal patterns in higher dimensional dynamics

Forbidden ordinal patterns are ordinal patterns (or rank blocks) that cannot appear in the orbits generated by a map taking values on a linearly ordered space, in which case we say that the map has forbidden patterns. Once a map has a forbidden pattern of a given length L₀, it has forbidden patterns of any length L≥L₀ and their number grows superexponentially with L. Using recent results on topological permutation entropy, in this paper we study the existence and some basic properties of forbidden ordinal patterns for self-maps on n-dimensional intervals. Our most applicable conclusion is that expansive interval maps with finite topological entropy have necessarily forbidden patterns, although we conjecture that this is also the case under more general conditions. The theoretical results are nicely illustrated for n=2 both using the naive counting estimator for forbidden patterns and Chao’s estimator for the number of classes in a population. The robustness of forbidden ordinal patterns against observational white noise is also illustrated.     

Choosing effective controlled nodes for scale-free network synchronization

Previous studies concerning pinning control of complex-network synchronization have very often demonstrated that in an unweighted symmetrical scale-free network, controlling the high-degree nodes is more efficient than controlling randomly chosen ones; due to the heterogeneity of the node-degree or edge-connection distribution of the scale-free network, small-degree nodes have relatively high probabilities of being chosen at random but their control has less influence on the other nodes through the network. This raises the question of whether or not controlling the high-degree nodes is always better than controlling the small ones in scale-free networks. Our answer to this is yes and no. In this study, we carry out extensive numerical simulations to show that in an unweighted symmetrical Barabasi-Albert scale-free network, when the portion of controlled nodes is relatively large, controlling the small nodes becomes better than controlling the big nodes and controlling randomly chosen nodes has approximately the same effect as controlling the big ones. However, we also show that for normalized weighted scale-free networks, controlling the big nodes is in fact always better than controlling the small ones.     

Can a polynomial interpolation improve on the Kaplan-Yorke dimension?

The Kaplan-Yorke dimension can be derived using a linear interpolation between an h-dimensional Lyapunov exponent λ(h)>0 and an h+1-dimensional Lyapunov exponent λ(h+1)<0. In this Letter, we use a polynomial interpolation to obtain generalized Lyapunov dimensions and study the relationships among them for higher-dimensional systems.     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

A three-scroll chaotic attractor

This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.     

Character of the map for switched dynamical systems for observations on the switching manifold

It has been proposed to obtain the discrete-time models of switching dynamical systems by observing the states at the switching instants. Apart from the lowering of dimension, such switching maps or impact maps offer advantage in modeling systems that exhibit chattering. In this Letter we derive the nature of the switching map for the special case of grazing orbits. We show that the map is discontinuous in the neighborhood of a grazing orbit, and that it has a square root slope singularity on one side of the discontinuity. We illustrate the above by obtaining the switching maps for two example systems: the Colpitt's oscillator in the electrical domain and the soft impact oscillator in the mechanical domain.     

Initial growth of Boltzmann entropy and chaos in a large assembly of weakly interacting systems

We introduce a high-dimensional symplectic map, modeling a large system, to analyze the interplay between single-particle chaotic dynamics and particles interactions in thermodynamic systems. We study the initial growth of the Boltzmann entropy, S_B, as a function of the coarse-graining resolution (the late stage of the evolution is trivial, as the system is subjected to no external drivings). We show that a characteristic scale emerges, and that the behavior of S_B vs t, at variance with the Gibbs entropy, does not depend on the resolution, as far as it is finer than this scale. The interaction among particles is crucial to achieve this result, while the rate of entropy growth, in its early stage, depends essentially on the single-particle chaotic dynamics. It is possible to interpret the basic features of the dynamics in terms of a suitable Markov approximation.     

混沌神经网络的动态阈值控制

提出了混沌神经网络的动态阈值控制方法, 将大脑脑电波的主要成分, 正弦信号作为控制变量实现对混沌神经网络内部状态的阈值动态改变, 从而达到了控制混沌神经网络混沌的目的. 利用该方法可以将混沌神经网络的输出稳定在一个与网络初始模式相关的存储模式和其反相模式上, 从而使混沌神经网络在模式识别、信息搜索等信息处理功能得以实现. 该控制方法不需要事先指定阈值, 是一种自适应方法, 符合实际人脑的思维运动. 关键词： 混沌控制 混沌神经网络 动态阈值控制 信息处理     

Dynamical role of the degree of intraspecific cooperation: A simple model for prebiotic replicators and ecosystems

We present a simple mean field model to analyze the dynamics of competition between two populations of replicators in terms of the degree of intraspecific cooperation (i.e., autocatalysis) in one of these populations. The first population can only replicate with Malthusian kinetics while the second one can reproduce with Malthusian or autocatalytic replication or with a combination of both reproducing strategies. The model consists of two coupled, nonlinear, autonomous ordinary differential equations. We investigate analytically and numerically the phase plane dynamics and the bifurcation scenarios of this ecologically coupled system, focusing on the outcome of competition for several degrees of intraspecific cooperation, σ, in the second population of replicators. We demonstrate that the dynamics of both populations can not be governed by a limit cycle, and also that once cooperation is considered, the topology of phase space does not allow for coexistence. Even for low values of the degree of intraspecific cooperation, for large enough autocatalytic replication rates, the second population of replicators is able to outcompete the first one, having a wide basin of attraction in state space. We characterize the same power law dependence between the outcompetition extinction times, τ, and the degree of intraspecific cooperation for both populations, given by τ∼c_iσ⁻¹. Our results suggest that, under some kinetic conditions, the appearance of autocatalysis might be favorable in a population of replicators growing with Malthusian kinetics competing with another population also reproducing exponentially.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation

This Letter is concerned with bifurcation and chaos control in scalar delayed differential equations with delay parameter τ. By linear stability analysis, the conditions under which a sequence of Hopf bifurcation occurs at the equilibrium points are obtained. The delayed feedback controller is used to stabilize unstable periodic orbits. To find the controller delay, it is chosen such that the Hopf bifurcation remains unchanged. Also, the controller feedback gain is determined such that the corresponding unstable periodic orbit becomes stable. Numerical simulations are used to verify the analytical results.     

Fermi acceleration with memory-dependent excitation

Some scaling properties for a classical particle confined to bounce between two walls, where one wall is fixed and the other one moves in time according to a random signal with a memory length are studied. We have considered two different kinds of collisions of the particle with the moving wall namely: (i) elastic and (ii) inelastic. The dynamics of the model is described in terms of a two-dimensional nonlinear mapping. For the case of elastic collisions, we show that the memory of the stochastic signal affects directly the behaviour of the average velocity of the particle. It then exhibits different slopes for the average velocity at different stages of the series with β≅3/4 for a short time, β≅1 for the average stage and β≅1/2 for a long time, as predicted by the Central Limit Theorem, therefore leading to the Fermi acceleration. The situation where inelastic collisions are taken into account yields a more drastic change, particularly suppressing the Fermi acceleration.     

Vehicular traffic through a self-similar sequence of traffic lights

We study the dynamical behavior of vehicular traffic through a sequence of traffic lights positioned self-similarly on a highway, where all traffic lights turn on and off simultaneously with cycle time T_s. The signals are positioned self-similarly by Cantor set. The nonlinear-map model of vehicular traffic controlled by self-similar signals is presented. The vehicle exhibits the complex behavior with varying cycle time. The tour time is much lower such that signals are positioned periodically with the same interval. The arrival time T(x) at position x scales as (T(x)-x)∝x^d_f, where d_f is the fractal dimension of Cantor set. The landscape in the plot of T(x)−x against cycle time T_s shows a self-affine fractal with roughness exponent α=1−d_f.     

Degree distributions of the visibility graphs mapped from fractional Brownian motions and multifractal random walks

The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

Formal analytical solutions for the Gross-Pitaevskii equation

Considering the Gross-Pitaevskii integral equation we are able to formally obtain an analytical solution for the order parameter Φ(x) and for the chemical potential μ as a function of a unique dimensionless non-linear parameter Λ. We report solutions for different ranges of values for the repulsive and the attractive non-linear interactions in the condensate. Also, we study a bright soliton-like variational solution for the order parameter for positive and negative values of Λ. Introducing an accumulated error function we have performed a quantitative analysis with respect to other well-established methods as: the perturbation theory, the Thomas-Fermi approximation, and the numerical solution. This study gives a very useful result establishing the universal range of the Λ-values where each solution can be easily implemented. In particular, we showed that for Λ<−9, the bright soliton function reproduces the exact solution of GPE wave function.     

Vehicular motion through a sequence of traffic lights controlled by logistic map

We study the dynamical behavior of a single vehicle through the sequence of traffic lights controlled by the logistic map. The phase shift of traffic lights is determined by the logistic map and varies from signal to signal. The nonlinear dynamic model of the vehicular motion is presented by the nonlinear map including the logistic map. The vehicle exhibits the very complex behavior with varying both cycle time and logistic-map parameter a. For a>3, the dependence of arrival time on the cycle time becomes smoother and smoother with increasing a. The dependence of vehicular motion on parameter a is clarified.