Well-known Dynamical Systems Research Articles

Efficient detection of the quasi-periodic route to chaos in discrete maps by the three-state test

The three-state test (3ST) is a method based on ordinal pattern analysis for detecting chaos and determining the period in time series. For some well-known chaotic dynamical systems, we showed that the test behaves similar to Lyapunov exponents. However, the 3ST is detecting quasi-periodic motions both as regular and non-regular. In this paper, we propose to use the sensitivity of its chaos indicator $$\lambda $$ on time delay for clear discernment between quasi-periodic and chaotic dynamics. Simulation results obtained using the logistic map and the sine-circle map attest that the sensitivity of $$\lambda $$ on time delay is sufficient for the detection of the periodic and quasi-periodic route to chaos. A comparison with the permutation entropy confirms the effectiveness of the 3ST for the analysis of discrete time series data.

Protein Folding in the 2D Hydrophobic–Hydrophilic (HP) Square Lattice Model is Chaotic

Among the unsolved problems in computational biology, protein folding is one of the most interesting challenges. To study this folding, tools like neural networks and genetic algorithms have received a lot of attention, mainly due to the NP completeness of the folding process. The background idea that has given rise to the use of these algorithms is obviously that the folding process is predictable. However, this important assumption is disputable as chaotic properties of such a process have been recently highlighted. In this paper, which is an extension of a former work accepted to the 2011 International Joint Conference on Neural Networks (IJCNN11), the topological behavior of a well-known dynamical system used for protein folding prediction is evaluated. It is mathematically established that the folding dynamics in the 2D hydrophobic–hydrophilic (HP) square lattice model, simply called the 2D model in this document, is indeed a chaotic dynamical system as defined by Devaney. Furthermore, the chaotic behavior of this model is qualitatively and quantitatively deepened, by studying other mathematical properties of disorder, namely: the indecomposability, instability, strong transitivity, and constants of expansivity and sensitivity. Some consequences for both biological paradigms and structure prediction using this model are then discussed. In particular, it is shown that some neural networks seems to be unable to predict the evolution of this model with accuracy, due to its complex behavior.

AN EFFECTIVE METHOD FOR DETECTING CHAOS IN FRACTIONAL-ORDER SYSTEMS

This paper illustrates a reliable binary test for detecting the presence of chaos in nonlinear systems described by fractional-order differential equations. The method, which is inspired by the technique proposed in [Gottwald & Melbourne, 2004] for integer-order differential equations, does not require phase space reconstruction of the fractional system. It consists of obtaining the data series, constructing a random walk-type process and studying how the variance of the random walk scales with time. In order to show the capabilities of the approach, the test is successfully applied to three well-known dynamical systems, i.e. fractional Chua, Chen and Lorenz systems.

Two sided Sand Piles Model and unimodal sequences

We introduce natural generalizations of two well-known dynamical systems, the Sand Piles Model and the Brylawski's model. We describe their order structure, their reachable configuration's characterization, their fixed points and their maximal and minimal length's chains. Finally, we present an induced model generating the set of unimodal sequences which amongst other corollaries, implies that this set is equipped with a lattice structure.

A Centralized Feedback Control Model for Resource Management in Wireless Networks

In a wireless environment, guaranteeing QoS is challenging because applications at multiple devices share the same limited radio bandwidth. In this paper we introduce and study a resource management model for centralized wireless networks, using feedback control theory. Before applying in practice, the proposed model is evaluated using the well-known 20-sim dynamic system simulator. The experimental results show that flexible and efficient resource allocation can be achieved for a variety of system parameters and WLAN scenarios; however, care should be taken in setting the control parameters and coefficients.

Shifts of finite type and Fibonacci Harps

We make an explicit connection between the Fibonacci Harp (or Fibonacci String) and two well-known dynamical systems: subshifts of finite type and the baker’s map on the unit interval. In particular, we show that the boundary of the Fibonacci Harp is an embedding of a commonly studied shift of finite type in the unit interval. Moreover, every shift of finite type embeds as the boundary of a lattice harp.

The base-scale entropy analy-sis of short-term heart rate variability signal

The complexity of heart rate variability (HRV) signal can reflect physiological functions and healthy status of heart system. Detecting complexity of the short-term HRV signal has an important practical meaning. We introduce the base-scale entropy method to analyze the complexity of time series. The advantages of our method are its simplicity, extremely fast calculation for very short data and anti-noise characteristic. For the well-known chaotic dynamical system—logistic map, it is shown that our complexity behaves similarly to Lyapunov exponents, and is especially effective in the presence of random Gaussian noise. This paper addresses the use of base-scale entropy method to 3 low-dimensional nonlinear deterministic systems. At last, we apply this idea to short-term HRV signal, and the result shows the method could robustly identify patterns generated from healthy and pathologic states, as well as aging. The base-scale entropy can provide convenience in practically applications.

Dynamics on Leibniz manifolds

This paper shows that various well-known dynamical systems can be described as vector fields associated to smooth functions via a bracket that defines what we call a Leibniz structure. We show that gradient flows, some control and dissipative systems, and non-holonomically constrained simple mechanical systems, among other dynamical behaviors, can be described using this mathematical construction that generalizes the standard Poisson bracket currently used in Hamiltonian mechanics. The symmetries of these systems and the associated reduction procedures are described in detail. A number of examples illustrate the theoretical developments in the paper.

Combining global and multi-scale features in a description of the solar wind-magnetosphere coupling

Abstract. The solar wind-magnetosphere coupling during substorms exhibits dynamical features in a wide range of spatial and temporal scales. The goal of our work is to combine the global and multi-scale description of magnetospheric dynamics in a unified data-derived model. For this purpose we use deterministic methods of nonlinear dynamics, together with a probabilistic approach of statistical physics. In this paper we discuss the mathematical aspects of such a combined analysis. In particular we introduce a new method of embedding analysis based on the notion of a mean-field dimension. For a given level of averaging in the system the mean-filed dimension determines the minimum dimension of the embedding space in which the averaged dynamical system approximates the actual dynamics with the given accuracy. This new technique is first tested on a number of well-known autonomous and open dynamical systems with and without noise contamination. Then, the dimension analysis is carried out for the correlated solar wind-magnetosphere database using vBS time series as the input and AL index as the output of the system. It is found that the minimum embedding dimension of vBS - AL time series is a function of the level of ensemble averaging and the specified accuracy of the method. To extract the global component from the observed time series the ensemble averaging is carried out over the range of scales populated by a high dimensional multi-scale constituent. The wider the range of scales which are smoothed away, the smaller the mean-field dimension of the system. The method also yields a probability density function in the reconstructed phase space which provides the basis for the probabilistic modeling of the multi-scale dynamical features, and is also used to visualize the global portion of the solar wind-magnetosphere coupling. The structure of its input-output phase portrait reveals the existence of two energy levels in the system with non-equilibrium dynamical features such as hysteresis which are typical for non-equilibrium phase transitions. Further improvements in space weather forecasting tools may be achieved by a combination of the dynamical description for the global component and a statistical approach for the multi-scale component.Key words. Magnetospheric physics (solar wind– magnetosphere interactions; storms and substorms) – Space plasma physics (nonlinear phenomena)

Permutation entropy: a natural complexity measure for time series.

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.

Wavelet filtering of chaotic data

Abstract. Satisfactory method of removing noise from experimental chaotic data is still an open problem. Normally it is necessary to assume certain properties of the noise and dynamics, which one wants to extract, from time series. The wavelet based method of denoising of time series originating from low-dimensional dynamical systems and polluted by the Gaussian white noise is considered. Its efficiency is investigated by comparing the correlation dimension of clean and noisy data generated for some well-known dynamical systems. The wavelet method is contrasted with the singular value decomposition (SVD) and finite impulse response (FIR) filter methods.

Dependability modelling and evaluation by using stochastic Petri nets: application to two test cases

The systems modelling and the assessment of their performances are the two key phases of any reliability or risk analysis study. However, it is well known and admitted that most of the methods devoted to this goal are unsuitable if the physical behaviour of the systems cannot be made independent from the probabilistic behaviour, as is the case for dynamic process systems. To overcome this inefficiency, some alternative methods have recently appeared. But they are not very approachable and could be, for this reason, ignored by most of the practitioners. Among these methods, however, simulation methods should be confirmed. From this point of view, the authors propose a straightforward approach to this problem by using stochastic Petri nets on a simple and well-known dynamic system from the literature. In addition, for testing this approach, the authors have carried it out on a system with periodically tested components. The results obtained are compared to those already published, the limits of the method are underlined and its efficiency in solving this kind of problem is being examined.

Spatial extension of brain activity fools the single-channel reconstruction of EEG dynamics

We report here on a first attempt to settle the methodological controversy between advocates of two alternative reconstruction approaches for temporal dynamics in brain signals: the single-channel method (using data from one recording site and reconstructing by time-lags), and the multiple-channel method (using data from a spatially distributed set of recordings sites and reconstructing by means of spatial position). For the purpose of a proper comparison of these two techniques, we computed a series of EEG-like measures on the basis of well-known dynamical systems placed inside a spherical model of the head. For each of the simulations, the correlation dimension estimates obtained by both methods were calculated and compared, when possible, with the known (or estimated) dimension of the underlying dynamical system. We show that the single-channel method fails to reliably quantify spatially extended dynamics, while the multichannel method performs better. It follows that the latter is preferable, given the known spatially distributed nature of brain processes. Hum. Brain Mapping 5:26-47, 1997. (c) 1997 Wiley-Liss, Inc.

Qualitative understanding of the Lorenz equations through a well-known second order dynamical system

In the present paper we show how it is possible to reduce the Lorenz system, modeling the onset of turbulence in the Bénard problem, to a well-known dynamical system on a two-dimensional manifold. The reduction is made by applying in a skillful way the center manifold theory. In such a way one obtains a syntetic qualitative description of some relevant aspects of the phase portraits and of the bifurcations in the original system.

On the Prevalence of Horseshoes

In this paper the symbolic dynamics of several differentiable systems are investigated. It is shown that many well-known dynamical systems, including Axiom ${\text {A}}$ systems, piecewise monotonic maps of the interval, the Lorenz attractor and Abraham-Smale examples, have inside them subsystems conjugate to subshifts of finite type. These subsystems have hyperbolic structures and hence are stable. They can also be chosen to have entropy arbitrarily close to that of the ambient system.

On the prevalence of horseshoes

In this paper the symbolic dynamics of several differentiable systems are investigated. It is shown that many well-known dynamical systems, including Axiom A {\text {A}} systems, piecewise monotonic maps of the interval, the Lorenz attractor and Abraham-Smale examples, have inside them subsystems conjugate to subshifts of finite type. These subsystems have hyperbolic structures and hence are stable. They can also be chosen to have entropy arbitrarily close to that of the ambient system.

Geometry from a Time Series

It is shown how the existence of low-dimensional chaotic dynamical systems describing turbulent fluid flow might be determined experimentally. Techniques are outlined for reconstructing phase-space pictures from the observation of a single coordinate of any dissipative dynamical system, and for determining the dimensionality of the system's attractor. These techniques are applied to a well-known simple three-dimensional chaotic dynamical system.