Detecting phase synchronization in noisy systems

A quantitative method for automatic detection of phase synchronization in noisy experimental bivariate time series is proposed, based on the fact that instantaneous phases of phase-synchronized (sub) systems are mutually dependent in a specific way irrespective of a relation between the original time series. The level of dependence between the instantaneous phases is quantified by a statistical dependence parameter, which also reflects the strength of the systems' phase synchronization. Ranges of the parameter values, for which the detection of the phase synchronization can be considered reliable, are estimated by using the technique of surrogate data. Possible applications of the proposed method are demonstrated by using both numerically generated and real experimental data, namely solutions of two coupled Rössler systems, mammalian cardio-respiratory data, and long-term recordings of surface atmospheric temperature and sunspot numbers.     

Globally enumerating unstable periodic orbits for observed data using symbolic dynamics

The unstable periodic orbits of a chaotic system provide an important skeleton of the dynamics in a chaotic system, but they can be difficult to find from an observed time series. We present a global method for finding periodic orbits based on their symbolic dynamics, which is made possible by several recent methods to find good partitions for symbolic dynamics from observed time series. The symbolic dynamics are approximated by a Markov chain estimated from the sequence using information-theoretical concepts. The chain has a probabilistic graph representation, and the cycles of the graph may be exhaustively enumerated with a classical deterministic algorithm, providing a global, comprehensive list of symbolic names for its periodic orbits. Once the symbolic codes of the periodic orbits are found, the partition is used to localize the orbits back in the original state space. Using the periodic orbits found, we can estimate several quantities of the attractor such as the Lyapunov exponent and topological entropy.     

Motion in a stochastic layer described by symbolic dynamics

The motion in the stochastic layer surrounding an island can be studied by using the standard map: This problem is of direct relevance to the diffusion of magnetic field lines in a tokamak. In a previous work it was shown that this process can be adequately modelled by a continuous time random walk (CTRW) describing transitions of the running point between three basins representing, respectively, trapped motion around the island, and passing motion above or below the island. The sticking property of the island deeply modifies the nature of the transport process, leading to subdiffusive behavior. In the present work it is shown that the motion can be analyzed in terms of a symbolic dynamics which leads to the possibility of an automatic measurement of the data necessary for the construction of the CTRW. The logical features of the procedure are described, and the method is applied to an analysis of long time series, thus completing the results of the previous work. (c) 1998 American Institute of Physics.     

Average dynamics of noisy maps

The average trajectories and fluctuations around them resulting from noisy, non-linear maps are analyzed. After describing their scaling properties they are discussed using a deterministic average equation of motion. A bifurcation gap is found and the existence of exceptional attractors for special initial conditions is predicted.     

One-dimensional three-body problem via symbolic dynamics

Symbolic dynamics is applied to the one-dimensional three-body problem with equal masses. The sequence of binary collisions along an orbit is expressed as a symbol sequence of two symbols. Based on the time reversibility of the problem and numerical data, inadmissible (i.e., unrealizable) sequences of collisions are systematically found. A graph for the transitions among various regions in the Poincare section is constructed. This graph is used to find an infinite number of periodic sequences, which implies an infinity of periodic orbits other than those accompanying a simple periodic orbit called the Schubart orbit. Finally, under reasonable assumptions on inadmissible sequences, we prove that the set of admissible symbol sequences forms a Cantor set. (c) 2000 American Institute of Physics.     

Randomizing nonlinear maps via symbolic dynamics

Pseudo Random Number Generators (PRNG) have attracted intense attention due to their obvious importance for many branches of science and technology. A randomizing technique is a procedure designed to improve the PRNG randomness degree according the specific requirements. It is obviously important to quantify its effectiveness. In order to classify randomizing techniques based on a symbolic dynamics’ approach, we advance a novel, physically motivated representation based on the statistical properties of chaotic systems. Recourse is made to a plane that has as coordinates (i) the Shannon entropy and (ii) a form of the statistical complexity measure. Each statistical quantifier incorporates a different probability distribution function, generating thus a representation that (i) sheds insight into just how each randomizing technique operates and also (ii) quantifies its effectiveness. Using the Logistic Map and the Three Way Bernoulli Map as typical examples of chaotic dynamics it is shown that our methodology allows for choosing the more convenient randomizing technique in each instance. Comparison with measures of complexity based on diagonal lines on the recurrence plots [N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Phys. Rep. 438 (2007) 237] support the main conclusions of this paper.     

Entanglement dynamics of three-qubit states in noisy channels

We study entanglement dynamics of the three-qubit system which is initially prepared in pure Greenberger-Horne-Zeilinger (GHZ) or W state and transmitted through one of the Pauli channels s_z\sigma_z, s_x\sigma_x, s_y\sigma_y or the depolarizing channel. With the help of the lower bound for three-qubit concurrence we show that the W state preserves more entanglement than the GHZ state in transmission through the Pauli channel s_z\sigma_z. For the Pauli channels s_x\sigma_x, s_y\sigma_y and the depolarizing channel, however, the entanglement of the GHZ state is more resistant against decoherence than the W-type entanglement. We also briefly discuss how the accuracy of the lower bound approximation depends on the rank of the density matrix under consideration.     

Mean discharge frequency locking in the response of a noisy neuron model to subthreshold periodic stimulation

Leaky integrate-and-fire neuron models display stochastic resonance-like behavior when stimulated by subthreshold periodic signal and noise. Previous works have shown that matching between the time scales of the noise induced discharges and the modulation period can account for this phenomenon at low modulation amplitudes, but not large subthreshold modulation amplitude. In order to examine the discharge patterns of the model in this regime, we introduce a method for the computation of the power spectral density of the discharge train. Using this method, we clarify the role of the distribution of the input phase at discharge times. Finally, we argue that for large subthreshold inputs, mean discharge frequency locking accounts for the enhanced response.     

Quantifying heart rate dynamics using different approaches of symbolic dynamics

The analysis of symbolic dynamics applied to physiological time series is able to retrieve information about dynamical properties of the underlying system that cannot be gained with standard methods like e.g. spectral analysis. Different approaches for the transformation of the original time series to the symbolic time series have been proposed. Yet the differences between the approaches are unknown. In this study three different transformation methods are investigated: (1) symbolization according to the deviation from the average time series, (2) symbolization according to several equidistant levels between the minimum and maximum of the time series, (3) binary symbolization of the first derivative of the time series. Furthermore, permutation entropy was used to quantify the symbolic series. Each method was applied to the cardiac interbeat interval series RR _i and its difference ΔRR _I of 17 healthy subjects obtained during head-up tilt testing. The symbolic dynamics of each method is analyzed by means of the occurrence of short sequences (“words”) of length 3. The occurrence of words is grouped according to words without variations of the symbols (0V%), words with one variation (1V%), two like variations (2LV%) and two unlike variations (2UV%). Linear regression analysis showed that for method 1 0V%, 1V%, 2LV% and 2UV% changed with increasing tilt angle. For method 2 0V%, 2LV% and 2UV% changed with increasing tilt angle and method 3 showed changes for 0V% and 1V%. Furthermore, also the permutation entropy decreased with increasing tilt angle. In conclusion, all methods are capable of reflecting changes of the cardiac autonomic nervous system during head-up tilt. All methods show that even the analysis of very short symbolic sequences is capable of tracking changes of the cardiac autonomic regulation during head-up tilt testing.     

Constructing dynamical systems with specified symbolic dynamics

In this paper we demonstrate how to construct signals (time series) of continuous-time dynamical systems that exhibit a given symbolic dynamics. This is achieved without construction of the ordinary differential equations that generate the flow. This construction is of theoretical interest and is useful as a source of dynamical data that can be used to test various data analysis algorithms.     

The noisy Hegselmann-Krause model for opinion dynamics

In the model for continuous opinion dynamics introduced by Hegselmann and Krause, each individual moves to the average opinion of all individuals within an area of confidence. In this work we study the effects of noise in this system. With certain probability, individuals are given the opportunity to change spontaneously their opinion to another one selected randomly inside the opinion space with different rules. If the random jump does not occur, individuals interact through the Hegselmann-Krause’s rule. We analyze two cases, one where individuals can carry out opinion random jumps inside the whole opinion space, and other where they are allowed to perform jumps just inside a small interval centered around the current opinion. We found that these opinion random jumps change the model behavior inducing interesting phenomena. Using pattern formation techniques, we obtain approximate analytical results for critical conditions of opinion cluster formation. Finally, we compare the results of this work with the noisy version of the Deffuant et al. model [G. Deffuant, D. Neu, F. Amblard, G. Weisbuch, Adv. Compl. Syst. 3, 87 (2000)] for continuous-opinion dynamics.     

Modulated noisy biological dynamics: Three examples

Three examples of noisy biological dynamics modulated by a periodic signal are discussed. A minimal neuron model driven by stochastic noise and small periodic force show a firing statistic comparable with stochastic resonance as demonstrated in bistable systems. Similar results are obtained from responses to periodic vibrotactile stimulation on higher-order neuronal units of the somatosensory pathway. Finally, results from a bistable visual perception task exhibiting stochastic resonance are reported.     

Deconstructing spatiotemporal chaos using local symbolic dynamics

We find that the global symbolic dynamics of a diffusively coupled map lattice is well approximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis, we give rigorous results for a range of coupling strengths and different local model widths. Examples are presented of extracting a local symbolic model from data and of controlling spatiotemporal chaos.     

Reconstruction of chaotic signals using symbolic data

We discuss the reconstruction of dynamical systems from noisy time-series. In particular, we consider the use of the symbol statistics (coarse-grained signal data) as the target for reconstruction. The statistics of symbol sequences is relatively insensitive to moderate amounts of measurement noise (σ(noise)/σ(signal) ≈ 10–20%), while larger amounts produce a substantial bias. We show that it is possible to produce robust reconstructions even when σ(noise)/σ(signal) ≈ O(1). Our study shows that even at such high noise levels the procedure is convergent: i.e. the accuracy of parameter estimates is limited only by the amount of data and computer time available.     

累积放电模型及其符号动力学研究

基于累积释放模型提出了一种累积放电模型.相比于累积释放模型, 累积放电模型无须变化的阈值调制, 即可出现多种状态, 例如混沌态、锁频等. 利用符号动力学对其进行研究, 发现在一定的参数条件下, 模型的输出符号序列可以被用于监测模型参数的变化, 而且与神经系统的测量相似, 都具有很高的分辨率. 计算机仿真和电路实验得到的结果也验证了上述说法. 电路实验结果显示模型的输出符号序列对输入频率的分辨率最高可以达到0.05 Hz, 对电流幅值的分辨率可达到1 μA, 并且都具有很大的动态范围. 关键词： 符号动力学 混沌 累积释放模型 非线性电路     

Testing heteroskedasticity of unknown form using symbolic dynamics

This paper aims to provide a new step towards statistical inference based on symbolic dynamics. Relevant null hypotheses can be expressed in probabilities associated with a given set of symbols, and therefore new nonparametric tests can be developed. Symbolic analysis of the dynamics of the second moment of a stochastic process has lead us to present a new semi-parametric test of heteroskedasticity. The test is fairly general as minimal assumptions are required to be used. Monte Carlo experiments mainly show the great power of the test and the fact that is very competitive test as compared with other powerful tests currently available in the relevant literature.     

Looking at Fokker-Planck dynamics with a noisy instrument

We consider the class of experiments which can be characterized by a Fokker-PIanck dynamics corresponding to the overdamped motion of a state point in a suitable stochastic potential. We assume that the general form of the potential is known (or can be guessed with reasonable accuracy), but that its parameters are to be determined experimentally by measurements made with a noisy instrument. This possible method for determining the potential parameters, which exploits the system's own internal stochastic motion in order to explore rapidly its available parameter space, is substantially more efficient than traditional methods involving time averages of single point measurements, and yet does not appear to have been previously considered. The method could be important when, for example, the experiment must be completed in a limited time owing either to the expense of the experimental materials or to the temporary stationarity of the preparation, situations which are commonly encountered in experimental biochemistry and biology.     

Quasipotentials for simple noisy maps with complicated dynamics

The theory of nonequilibrium potentials or quasipotentials is a physically motivated approach to small random perturbations of dynamical systems, leading to exponential estimates of invariant probabilities and mean first exit times. In the present article we develop the mathematical foundation of this theory for discrete-time systems, following and extending the work of Freidlin and Wentzell, and Kifer. We discuss strategies for calculating and estimating quasipotentials and show their application to one-dimensionalS-unimodal maps. The method proves to be especially suited for describing the noise scaling behavior of invariant probabilities, e.g., for the map occurring as the limit of the Feigenbaum period-doubling sequence. We show that the method allows statements about the scaling behavior in the case of localized noise, too, which does not originally lie within the scope of the quasipotential formalism.