scholarly journals Control of Chaos in the Rössler System

1997 ◽  
Vol 50 (2) ◽  
pp. 263 ◽  
Author(s):  
Stuart Corney
Keyword(s):  
Periodic Orbits ◽  
Time Series Data ◽  
Control Method ◽  
Three Dimensional ◽  
Dynamical Variable ◽  
Series Data ◽  
External Parameter ◽  
Unstable Periodic Orbits ◽  
System A ◽  
Linear Differential

The control method of Ott, Grebogi and Yorke (1990) as applied to the Rössler system, a set of three-dimensional non-linear differential equations, is examined. Using numerical time series data for a single dynamical variable the method was successfully employed to control several of the unstable periodic orbits in a three-dimensional embedding of the data. The method also failed for a number of unstable periodic orbits due to difficulties in linearising about the orbit or the tangential coincidence of the stable manifold and the motion of the orbit with external parameter.

Extracting unstable periodic orbits from chaotic time series data

1997 ◽  
Vol 55 (5) ◽  
pp. 5398-5417 ◽  
Author(s):  
Paul So ◽  
Edward Ott ◽  
Tim Sauer ◽  
Bruce J. Gluckman ◽  
Celso Grebogi ◽  
...  
Keyword(s):  
Time Series ◽  
Periodic Orbits ◽  
Time Series Data ◽  
Chaotic Time Series ◽  
Series Data ◽  
Unstable Periodic Orbits

Container port throughput analysis and active management using control theory

2021 ◽  
pp. 147509022110208
Author(s):  
Cuong Truong Ngoc ◽  
Xiao Xu ◽  
Hwan-Seong Kim ◽  
Duy Anh Nguyen ◽  
Sam-Sang You
Keyword(s):  
Control Theory ◽  
Time Series Data ◽  
Sliding Mode ◽  
Container Terminal ◽  
Three Dimensional ◽  
Maritime Transportation ◽  
Container Terminals ◽  
Active Management ◽  
Dynamical Analysis ◽  
Series Data

This paper deals with three-dimensional (3D) model of competitive Lotka-Volterra equation to investigate nonlinear dynamics and control strategy of container terminal throughput and capacity. Dynamical behaviors are intensely explored by using eigenvalue evaluation, bifurcation analysis, and time-series data. The dynamical analysis is to show the stability with bifurcation of the competition and collaboration of multiple container terminals in the maritime transportation. Based on the chaotic analysis, the sliding mode control theory has been utilized for optimization of port operations under disruptions. Extensive numerical simulations have been conducted to validate the efficacy and reliability of the presented control algorithms. Particularly, the closed-loop system has been assessed through chaotic suppression and synchronization strategies for port management. Finally, the presented fundamental techniques can be utilized to provide managerial insights and solutions on efficient seaport operations that allow more timely and cost-effective decision making for port authorities in such a highly competitive environment.

scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

scholarly journals Detecting and stabilizing periodic orbits of chaotic Henon map through predictive control

2018 ◽  
Vol 27 (2018) ◽  
pp. 73-78
Author(s):  
Dumitru Deleanu
Keyword(s):  
Periodic Orbits ◽  
Predictive Control ◽  
Nonnegative Integer ◽  
Strange Attractor ◽  
Discrete System ◽  
Control Method ◽  
Nonlinear Mapping ◽  
Unstable Periodic Orbits ◽  
Hénon Map ◽  
Henon Map

The predictive control method is one of the proposed techniques based on the location and stabilization of the unstable periodic orbits (UPOs) embedded in the strange attractor of a nonlinear mapping. It assumes the addition of a small control term to the uncontrolled state of the discrete system. This term depends on the predictive state ps + 1 and p(s + 1) + 1 iterations forward, where s is the length of the UPO, and p is a large enough nonnegative integer. In this paper, extensive numerical simulations on the Henon map are carried out to confirm the ability of the predictive control to detect and stabilize all the UPOs up to a maximum length of the period. The role played by each involved parameter is investigated and additional results to those reported in the literature are presented.

scholarly journals Track Irregularity Time Series Analysis and Trend Forecasting

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Jia Chaolong ◽  
Xu Weixiang ◽  
Wang Futian ◽  
Wang Hanning
Keyword(s):  
Time Series ◽  
Time Series Data ◽  
Series Data ◽  
Short Term ◽  
Measuring Point ◽  
Trend Forecasting ◽  
Changing Trend ◽  
Track Irregularity ◽  
Linear Differential

The combination of linear and nonlinear methods is widely used in the prediction of time series data. This paper analyzes track irregularity time series data by using gray incidence degree models and methods of data transformation, trying to find the connotative relationship between the time series data. In this paper, GM(1,1)is based on first-order, single variable linear differential equations; after an adaptive improvement and error correction, it is used to predict the long-term changing trend of track irregularity at a fixed measuring point; the stochastic linear AR, Kalman filtering model, and artificial neural network model are applied to predict the short-term changing trend of track irregularity at unit section. Both long-term and short-term changes prove that the model is effective and can achieve the expected accuracy.

EXPERIMENTAL VERIFICATION OF PYRAGAS’S CHAOS CONTROL METHOD APPLIED TO CHUA’S CIRCUIT

1994 ◽  
Vol 04 (06) ◽  
pp. 1703-1706 ◽  
Author(s):  
P. CELKA
Keyword(s):  
Periodic Orbits ◽  
Experimental Verification ◽  
Chaos Control ◽  
Control Method ◽  
Experimental Results ◽  
Experimental Setup ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit

We have built an experimental setup to apply Pyragas’s [1992, 1993] control method in order to stabilize unstable periodic orbits (UPO) in Chua’s circuit. We have been able to control low period UPO embedded in the double scroll attractor. However, experimental results show that the control method is useful under some restrictions we will discuss.

CONTROL OF n-SCROLL CHUA'S CIRCUIT

2009 ◽  
Vol 19 (11) ◽  
pp. 3813-3822 ◽  
Author(s):  
ABDELKRIM BOUKABOU ◽  
BILEL SAYOUD ◽  
HAMZA BOUMAIZA ◽  
NOURA MANSOURI
Keyword(s):  
Numerical Simulations ◽  
Fixed Points ◽  
Periodic Orbits ◽  
Predictive Control ◽  
Control Method ◽  
Sufficient Conditions ◽  
Chua’S Circuit ◽  
Unstable Periodic Orbits ◽  
Chua's Circuit ◽  
Necessary And Sufficient

This paper addresses the control of unstable fixed points and unstable periodic orbits of the n-scroll Chua's circuit. In a first step, we give necessary and sufficient conditions for exponential stabilization of unstable fixed points by the proposed predictive control method. In addition, we show how a chaotic system with multiple unstable periodic orbits can be stabilized by taking the system dynamics from one UPO to another. Control performances of these approaches are demonstrated by numerical simulations.

Modeling Interactive Behaviors While Learning With Digitized Objects in Virtual Reality Environments

2020 ◽  
pp. 215-234
Author(s):  
Eric Poitras ◽  
Kirsten R. Butcher ◽  
Matthew P. Orr
Keyword(s):  
Learning Environments ◽  
Time Series Data ◽  
Three Dimensional ◽  
Synthetic Data ◽  
Virtual Learning ◽  
Virtual Learning Environments ◽  
Series Data ◽  
Student Behaviors ◽  
Virtual Models ◽  
Interaction Behaviors

This chapter outlines a framework for automated detection of student behaviors in the context of virtual learning environments. The components of the framework establish several parameters for data acquisition, preprocessing, and processing as a means to classify different types of behaviors. The authors illustrate these steps in training and evaluating a detector that differentiates between students' observations and functional behaviors while students interact with three-dimensional (3D) virtual models of dinosaur fossils. Synthetic data were generated in controlled conditions to obtain time series data from different channels (i.e., orientation from the virtual model and remote controllers) and modalities (i.e., orientation in the form of Euler angles and quaternions). Results suggest that accurate detection of interaction behaviors with 3D virtual models requires smaller moving windows to segment the log trace data as well as features that characterize orientation of virtual models in the form of quaternions. They discuss the implications for personalized instruction in virtual learning environments.

Topological characterization of deterministic chaos: enforcing orientation preservation

2007 ◽  
Vol 366 (1865) ◽  
pp. 559-567 ◽  
Author(s):  
Marc Lefranc ◽  
Pierre-Emmanuel Morant ◽  
Michel Nizette
Keyword(s):  
Periodic Orbits ◽  
Topological Analysis ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Three Dimensions ◽  
Unstable Periodic Orbits ◽  
Topological Characterization ◽  
Triangulated Surfaces ◽  
Theoretic Characterization

The determinism principle, which states that dynamical state completely determines future time evolution, is a keystone of nonlinear dynamics and chaos theory. Since it precludes that two state space trajectories intersect, it is a core ingredient of a topological analysis of chaos based on a knot-theoretic characterization of unstable periodic orbits embedded in a strange attractor. However, knot theory can be applied only to three-dimensional systems. Still, determinism applies in any dimension. We propose an alternative framework in which this principle is enforced by constructing an orientation-preserving dynamics on triangulated surfaces and find that in three dimensions our approach numerically predicts the correct topological entropies for periodic orbits of the horseshoe map.

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