Control of chaos by self-adapted delayed feedback

Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control

A time-delayed feedback control (TDFC) system is by nature a rather special version of the familiar autoregressive moving-average (ARMA) control, or the canonical state-space control systems. Despite some of its inherent limitations, TDFC can be quite successful in many chaos control applications. To understand to what extent the TDFC method is useful, some analytic (sufficient) conditions for chaos control from the TDFC approach are derived in this paper, for both stabilization and tracking problems. A gradient descent based search algorithm is incorporated with the TDFC to estimate the time delay constant for tracking unstable periodic orbits. The established theoretical results and estimation method are further clarified via a case study of the typical chaotic Rossler system with computer simulations.

It was recently shown (Keating J P and Prado S D 2001 Proc. R. Soc. A 457 1855–72) that, in the semiclassical limit, the scarring of quantum eigenfunctions by classical periodic orbits in chaotic systems may be dramatically enhanced when the orbits in question undergo bifurcation. Specifically, a bifurcating orbit gives rise to a scar with an amplitude that scales as ℏα and a width that scales as ℏω, where α and ω are bifurcation-dependent scar exponents whose values are typically smaller than those (α = ω = ½) associated with isolated and unstable periodic orbits. We here analyse the influence of bifurcations on the autocorrelation function of quantum eigenstates, averaged with respect to energy. It is shown that the length-scale of the correlations around a bifurcating orbit scales semiclassically as ℏ1−α, where α is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps.

Signal correlator with programmable variable time delay based on optical coherent transients

Recently, there has been an extensive interest in research on the analysis and control of chaotic behavior in nonlinear systems. Various control problems can be defined for such systems, such as targeting the trajectories to a desired point, stabilizing unstable periodic orbits etc. In this paper we present the necessity and possibility to control chaotic dynamic systems. The first part of this paper describes the methods for the identification of Unstable Periodic Orbits (UPO) of a chaotic dynamic system. Some methods of control are applied to stabilize UPO embedded in the dynamics of the system. The chaos control was first advanced by Ott, Grebogi and York (OGY method). Finally, the resonant parametric perturbation method is applied to control chaos in buck converter. In particular, the effect of phase shift in the perturbing signal is studied.

In this work a feed-forward NN based NAR model for forecasting time series is presented. The learning rule used to adjust the NN weights is based on the Levenberg-Marquardt method. In function of the long or short term stochastic dependence of the time series, we propose an online heuristic law to set the training process and to modify the NN topology. The approach is tested over five time series obtained from samples of the Mackey-Glass delay differential equations and from monthly cumulative rainfall. Three sets of parameters for MG solution were used, whereas the monthly cumulative rainfall belongs to two different sites and times period, La Perla 1962-1971 and Santa Francisca 200-2010, both located at Cordoba, Argentina. The approach performance presented is shown by forecasting the 18 future values from each time series simulated by a Monte Carlo of 500 trials with fractional Gaussian noise to specify the variance.

Introduction. Growing learner diversity, expanding biomedical knowledge, and rapid clinical digitization have exposed the limits of on-size-fits-all health sciences training. Competency-Based Education (CBE) has sharpened the focus on outcomes, yet many programs still move learners through fixed sequences, infrequent assessments, and retrospective remediation. Precision Education (PE) offers a complementary, data-enabled approach that uses learning analytics, artificial intelligence (AI), and continuous feedback loops. Embedded within CBE, PE converts competencies from static milestones into dynamic trajectories that can be measured, visualized, and adjusted in real time across the curriculum. This abstract highlights the need for an interprofessional strategy to design, fund, and scale PE across medicine, nursing, rehabilitation, and allied health care professions. Without shared standards, coordinated governance, and sustained investment, small uncoordinated pilot programs risk widening disparities between well-resources and resource-limited institutions. With coordination, however, PE can personalize learning while upholding rigorous, profession-wide expectations and advancing equity. Discussion. PE operationalizes a cyclical process of problem analysis, planning, learning, and data-informed adjustment, applied at the micro (learner), meso (program), and macro (institution) levels to align experiences with competency outcomes. AI and learning analytics enable early risk detection, adaptive content, richer simulation, and timely coaching, while dashboards make progress visible to learners and faculty. Trustworthy use demands transparent data governance, interpretable models, where feasible, human-in-the-loop decisions for consequential actions, and routine equity audits to detect and mitigate bias. There are several illustrative initiatives that already show feasibility. For example, national priorities that elevate PE, program exemplars that analyze clinical exposure, and EHR activity to tailor coaching and rebalance workloads, in addition to competency-based programs that advance students according to mastery all demonstrate feasible initiatives for health science education. Anticipated benefits include improved competence and confidence for learners, more targeted teaching and resource allocation for faculty and institutions, and better clinical readiness for patients and systems. Major barriers remain fragmented platforms and limited infrastructure; faculty capability gaps and change fatigue; and ethical concerns about privacy, fairness, and explainability. To move from pilots to infrastructure, we recommend a coordinated, interprofessional strategy. This includes a national consortium to set vocabulary and reference architectures; shared benchmarks focused on outcomes and equity; stackable micro-credentials in learning analytics and AI-enhanced instruction; and aligned policy and funding to equip resource-limited programs. Immediate steps include forming steering committees, mapping data sources, launching low-stakes pilots with equity metrics, and sharing competency dashboards. Conclusion. PE is a timely, practical extension of CBE that personalizes learning while preserving rigorous standards. By integrating interoperable data, AI-enabled analytics, and continuous feedback loops, programs can provide precise support, improve progression equity, and strengthen clinical readiness. Establishing a consortium, codifying shared frameworks, investing in faculty capability, and tying funding to equity and readiness gains can accelerate adoption. Acting now with focused pilots, transparent dashboards, and learning communities will move the field from isolated innovations to a durable, equitable infrastructure in which every learner receives the right support at the right moment, and every patient benefits from a workforce prepared for practice.

Resonant chaos control by light in a chemiluminescent reaction

Time delayed-feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. The method is based on applying feedback proportional to the deviation of the current state of the system from its state one period in the past so that the control signal vanishes when the stabilization of the desired orbit is attained. A brief review of experimental implementations, applications for theoretical models, and most important modifications of the method is presented. Some recent results concerning the theory of the delayed feedback control as well as an idea of using unstable degrees of freedom in a feedback loop to avoid a well known topological limitation of the method are described in details.

In this paper we present a new method of controlling periodic orbits in chaotic systems. This method can be applied in situations when the chaotic system depends on one system parameter, which can be changed over a continuous interval or over a discrete, two-element set. We compare the new method to other ones, discuss its properties, and illustrate our approach with a numerical example.

WAMS (Wide-Area Measurement Systems) involve all kinds of communicating channels. Different channels have different time delays and randomicity of time-delay. So closed-loop power systems ought to be modeled as nonlinear dynamic systems with multiple time delays and varying time delays. A method is proposed in this paper to design a controller that can adapt to feedback signals with multiple time delays and varying time delays. The method is based on the topological characteristics of stable time-delay regions, the LMIs (Linear Matrix Inequalities) method, GA (Genetic Algorithm) and the balanced truncation model-reduced algorithm. Numerical results form a tow-area four-machine power system demonstrate that the controller derived from the above method can damp low-frequency oscillations in the system for the feedback signals with multiple time delays and varying time delays.

We present an efficient control scheme for stabilizing unstable periodic orbits of chaotic systems. The resulting orbits are called cupolets and have been proven to be useful in the representation of oscillatory or quasi periodic signals such as appear in music and image compression (Short et al., AES 118th Convention preprint 6446, May 2005; Short et al., AES 119th Convention preprint 6588, October 2005). In this paper we show that these cupolets can be used effectively to produce an adaptive basis for the space of real-valued functions of a discrete variable. From this basis, we construct a multiresolution analysis which allows for the approximation of signals at different resolution levels and apply it to image compression. This adaptive multiresolution analysis provides an interesting continuum between Fourier analysis and wavelet analysis.

Time-delayed feedback can be used to stabilize unstable periodic orbits (UPOs) in dynamical systems. The technique involves feedback of signals delayed by the orbit's period so that the feedback signals vanish on the target orbit and hence the UPO becomes a solution of the original system. An obvious advantage of this method is that it requires only the knowledge of the period of the target UPO. In this brief, a pulsewidth-modulated current-mode H-bridge inverter serves as a subject of investigation. Under certain parameter ranges, it exhibits higher periodic solutions and chaos. A simple time-delay feedback control method is applied to stabilize the UPOs. By using the time-delayed feedback control method, the stability limit and gain range of the system are increased.

A Nd:YAG laser with an intracavity frequency doubling crystal displays chaotic dynamics for certain ranges of operating parameters. It has been experimentally shown that dynamical control of such a laser system is possible and that a variety of periodic waveforms can be stably generated through an appropriate feedback technique. Computations performed on a numerical model establish that unstable periodic orbits can be stabilized by this procedure. The nature of the stabilization process is studied, and transients from chaotic dynamics to periodic waveforms after application of the control signal are examined experimentally and numerically. The distribution of power among the modes of the laser and the synchronization of this process by the control signal are studied. These investigations demonstrate the dynamical control of a chaotic optical system.

It is shown that we can control spatiotemporal chaos in the Frenkel–Kontorova (FK) model by a model-free control method based on reinforcement learning. The method uses Q-learning to find optimal control strategies based on the reward feedback from the environment that maximizes its performance. The optimal control strategies are recorded in a Q-table and then employed to implement controllers. The advantage of the method is that it does not require an explicit knowledge of the system, target states, and unstable periodic orbits. All that we need is the parameters that we are trying to control and an unknown simulation model that represents the interactive environment. To control the FK model, we employ the perturbation policy on two different kinds of parameters, i.e., the pendulum lengths and the phase angles. We show that both of the two perturbation techniques, i.e., changing the lengths and changing their phase angles, can suppress chaos in the system and make it create the periodic patterns. The form of patterns depends on the initial values of the angular displacements and velocities. In particular, we show that the pinning control strategy, which only changes a small number of lengths or phase angles, can be put into effect.