Detection Of Unstable Periodic Orbits Research Articles

Nonlinear energy harvester with coupled Duffing oscillators

Structural vibrations are very common in aerospace and mechanical engineering systems, where dynamic analysis of modern aerospace structures and industrial machines has become an indispensable step in their design. Suppression of unwanted vibrations and their exploitation for energy harvesting at the same time would be the most desirable scenario. The dynamical system presented in this communication is based on a discrete model of energy harvesting device realized in such a manner as to achieve both vibration suppression and harvesting of vibration energy by introducing the nonlinear energy sink concept. The mechanical model is formed as a two-degree of freedom nonlinear oscillator with an oscillating magnet and harmonic base excitation. The corresponding mathematical model is based on the system of nonlinear nonhomogeneous Duffing type differential equations. To explore complex dynamical behaviour of the presented model, periodic solutions and their bifurcations are found by using the incremental harmonic balance (IHB) and continuation methods. For the detection of unstable periodic orbits, the Floquet theory is applied and an interesting harmonic response of the presented nonlinear dynamical model is detected. The main advantage of the presented approach is its ability to obtain approximated periodic responses in terms of Fourier series and estimate the voltage output of an energy harvester for a system with strong nonlinearity. The accuracy of the presented methodology is verified by comparing the results obtained in this work with those obtained by a standard numerical integration method and results from the literature. Numerical examples show the effects of different physical parameters on amplitude-frequency, response amplitude - base amplitude and time response curves, where a qualitative change is explored and studied in detail. Presented theoretical results demonstrate that the proposed system has advanced performance in both system requirements - vibration suppression, and energy harvesting.

Detection of unstable periodic orbits in mineralising geological systems.

Worldwide, mineral exploration is suffering from rising capital costs, due to the depletion of readily recoverable reserves and the need to discover and assess more inaccessible or geologically complex deposits. For gold exploration, this problem is particularly acute. We propose an innovative approach to mineral exploration and orebody characterisation, based on the analysis of geological core data as a spatial dynamical system, using the mathematical tools of dynamical system analysis. This approach is highly relevant for orogenic gold deposits, which-in contrast to systems formed at chemical equilibrium-exhibit many features of nonlinear dynamical systems, including episodic fluctuations on various length and time scales. Feedback relationships between thermo-chemical and deformation processes produce recurrent fluid temperatures and pressures and the deposition of vein-filling minerals such as pyrite and gold. We therefore relax the typical assumption of chemical equilibrium and analyse the underlying processes as aseismic, non-adiabatic, and inherent to a hydrothermal, nonlinear dynamical open-flow chemical reactor. These processes are approximated using the Gray-Scott model of reaction-diffusion as a complex toy system, which captures some of the features of the underlying mineralisation processes, including the spatiotemporal Turing patterns of unsteady chemical reactions. By use of this analysis, we demonstrate the capability of recurrence plots, recurrence power spectra, and recurrence time probabilities to detect underlying unstable periodic orbits as one sign of deterministic dynamics and their robustness for the analysis of data contaminated by noise. Recurrence plot based quantification is then applied to three mineral concentrations in the core data from the Sunrise Dam gold deposit in the Yilgarn region of Western Australia. Using a moving window, we reveal the episodic recurring low-dimensional dynamic structures and the period doubling route to instability with depth, embedded in and originating from higher-dimensional processes of the complex mineralisation system.

Detection of Unstable Periodic Orbits and Chaos Control in a Passive Biped Model

Dynamic analysis of passive biped models plays a significant role both in understanding human locomotion and in developing humanoid robots. In this investigation, two chaos control algorithms based on linearization of Poincare map (OGY method) and artificial neural networks (ANNs) are utilized to control the motion of a passive biped model. To this end, the chaotic characteristics of the system are analyzed using several nonlinear dynamics tools such as Poincare map, bifurcation diagram, and Lyapunov exponents, and then, unstable periodic orbits (UPOs) of the system are detected. Detection of these orbits helps to extract a desired walking pattern and also is utilized for chaos elimination of the system. The robustness of the proposed ANN-based control algorithm is verified by applying toe-off impulses to the biped during the gait. Furthermore, the effect of network parameters on the biped walking performance is investigated to get design guidelines for the ANN-based controller.

Detecting unstable periodic orbits in high-dimensional chaotic systems from time series: Reconstruction meeting with adaptation

Detecting unstable periodic orbits (UPOs) in chaotic systems based solely on time series is a fundamental but extremely challenging problem in nonlinear dynamics. Previous approaches were applicable but mostly for low-dimensional chaotic systems. We develop a framework, integrating approximation theory of neural networks and adaptive synchronization, to address the problem of time-series-based detection of UPOs in high-dimensional chaotic systems. An example of finding UPOs from the classic Mackey-Glass equation is presented.

Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

Unstable periodic orbit detection for ODEs with periodic forcing

The Davidchack–Lai iterative scheme for the complete detection of unstable periodic orbits (UPOs) in maps is applied to a second order, nonlinear ODE with periodic forcing. The modifications to the scheme required to apply it to ODEs are detailed before the results for a particular example, the varactor equation, are given.

Analyses of transient chaotic time series.

We address the calculation of correlation dimension, the estimation of Lyapunov exponents, and the detection of unstable periodic orbits, from transient chaotic time series. Theoretical arguments and numerical experiments show that the Grassberger-Procaccia algorithm can be used to estimate the dimension of an underlying chaotic saddle from an ensemble of chaotic transients. We also demonstrate that Lyapunov exponents can be estimated by computing the rates of separation of neighboring phase-space states constructed from each transient time series in an ensemble. Numerical experiments utilizing the statistics of recurrence times demonstrate that unstable periodic orbits of low periods can be extracted even when noise is present. In addition, we test the scaling law for the probability of finding periodic orbits. The scaling law implies that unstable periodic orbits of high period are unlikely to be detected from transient chaotic time series.

Towards complete detection of unstable periodic orbits in chaotic systems

We present a rigorous analysis and numerical evidence indicating that a recently developed methodology for detecting unstable periodic orbits is capable of yielding all orbits up to periods limited only by the computer precision. In particular, we argue that an efficient convergence to every periodic orbit can be achieved and the basin of attraction can be made finite and accessible for typical or particularly chosen initial conditions.

Different classifications of UPOs in the parametrically different chaotic ISI series of a neural pacemaker.

Various interspike intervals (ISIs) in a series of spontaneous discharges of experimental neural pacemakers were identified as chaotic and as lying between stable period 2 and stable period 3. The method of So et al. for detection of unstable periodic orbits (UPOs) was applied to analyze two chaotic ISI series generated under the action of [Ca2+]o in different concentrations. In the chaotic ISI series near the stable period 2, an unstable period 2 orbit was detected. In the chaotic ISI series near the stable period 3, both an unstable period 2 orbit and an unstable period 3 orbit were detected. The location of the unstable period 2 orbit was close to that of the stable period 2 in the return map, while the location of the unstable period 3 orbit was close to that of the stable period 3. The results not only revealed the structures of various chaotic ISI series, but also indicated that the classification of UPOs could reflect the experimental control parameters at which the chaotic ISI series were generated. The previously discovered period adding bifurcation in the ISI series generated by experimental neural pacemakers [12] was further described in terms of the evolution of the period orbits.

Theory and applications of the systematic detection of unstable periodic orbits in dynamical systems

A topological approach and understanding to the detection of unstable periodic orbits based on a recently proposed method [Phys. Rev. Lett. 78, 4733 (1997)] is developed. This approach provides a classification of the set of transformations necessary for finding the orbits. Applications to the Ikeda and Henon map are performed, allowing a study of the distributions of Lyapunov exponents for high periods. In particular, the properties of the least unstable orbits up to period 36 are investigated and discussed.

Detecting unstable periodic orbits from transient chaotic time series

We address the detection of unstable periodic orbits from experimentally measured transient chaotic time series. In particular, we examine recurrence times of trajectories in the vector space reconstructed from an ensemble of such time series. Numerical experiments demonstrate that this strategy can yield periodic orbits of low periods even when noise is present. We analyze the probability of finding periodic orbits from transient chaotic time series and derive a scaling law for this probability. The scaling law implies that unstable periodic orbits of high periods are practically undetectable from transient chaos.

Noisy precursors of bifurcations in a neurodynamical model for disease states of mood disorders

We study the effect of noise on a nonlinear neurodynamical model for disease states of recurrent affective disorders. Specifically, we consider how noise affects a model state with periodic event generation close to period-doubling bifurcations. Our simulations demonstrate noisy precursors of the bifurcations. The precursors can be seen in the return maps of the time series and can be quantified with a recently developed algorithm for the detection of unstable periodic orbits. Implications for the neurobiology and course of mood disorders are discussed.

Efficient algorithm for detecting unstable periodic orbits in chaotic systems

We present an efficient method for fast, complete, and accurate detection of unstable periodic orbits in chaotic systems. Our method consists of an iterative scheme and an effective technique for selecting initial points. The iterative scheme is based on the semi-implicit Euler method, which has both fast and global convergence, and only a small number of initial points is sufficient to detect all unstable periodic orbits of a given period. The power of our method is illustrated by numerical examples of both two- and four-dimensional maps.

Finding unstable periodic orbits in electroreceptors, cold receptors and hypothalamic neurons

Recently, searches for low-dimensional dynamical activity in various biological preparations have become fashionable. In particular, demonstrations of the existence of unstable periodic orbits (UPOs) and bifurcations between unstable and stable states are of interest. Here, we report the detection of UPOs in three diverse preparations: electroreceptor, cold receptor and hypothalamic neurons. Inherent, noise mediated oscillators are common to these temperature sensitive neurons, and the UPOs arise in them in the absence of external periodic stimulation. The external temperature is the bifurcation parameter. Behaviors in response to temperature transients are also shown.

Unstable periodic orbits and attractor of the barotropic ocean model

Abstract. A numerical method for detection of unstable periodic orbits on attractors of nonlinear models is proposed. The method requires similar techniques to data assimilation. This fact facilitates its implementation for geophysical models. This method was used to find numerically several low-period orbits for the barotropic ocean model in a square. Some numerical particularities of application of this method are discussed. Knowledge of periodic orbits of the model helps to explain some of these features like bimodality of probability density functions (PDF) of principal parameters. These PDFs have been reconstructed as weighted averages of periodic orbits with weights proportional to the period of the orbit and inversely proportional to the sum of positive Lyapunov exponents. The fraction of time spent in the vicinity of each periodic orbit has been compared with its instability characteristics. The relationship between these values shows the 93% correlation. The attractor dimension of the model has also been approximated as a weighted average of local attractor dimensions in vicinities of periodic orbits.

Counting unstable periodic orbits in noisy chaotic systems: A scaling relation connecting experiment with theory.

The experimental detection of unstable periodic orbits in dynamical systems, especially those which yield short, noisy or nonstationary data sets, is a current topic of interest in many research areas. Unfortunately, for such data sets, only a few of the lowest order periods can be detected with quantifiable statistical accuracy. The primary observable is the number of encounters the general trajectory has with a particular orbit. Here we show that, in the limit of large period, this quantity scales exponentially with the period, and that this scaling is robust to dynamical noise. (c) 1998 American Institute of Physics.

Characterization of low-dimensional dynamics in the crayfish caudal photoreceptor.

Attempts to detect and characterize chaos in biological systems are of considerable interest, especially in medical science, where successful demonstrations may lead to new diagnostic tools and therapies. Unfortunately, conventional methods for identifying chaos often yield equivocal results when applied to biological data, which are usually heavily contaminated with noise. For such applications, a new technique based on the detection of unstable periodic orbits holds promise. Infinite sets of unstable periodic orbits underlie chaos in dissipative systems; accordingly, the new method searches a time series only for rare events characteristic of these unstable orbits, rather than analysing the structure of the series as a whole. Here we demonstrate the efficacy of the method when applied to the dynamics of the crayfish caudal photoreceptor (subject to stimuli representative of the animal's natural habitat). Our findings confirm the existence of low-dimensional dynamics in the system, and strongly suggest the existence of deterministic chaos. More importantly, these results demonstrate the power of methods based on the detection of unstable periodic orbits for identifying low-dimensional dynamics--and, in particular, chaos--in biological systems.