Calculation and Control of Unstable Periodic Orbits in Piecewise Smooth Dynamical Systems

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.

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Feedback control modulation for controlling chaotic maps

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2021.26.23052 ◽

2021 ◽

Vol 26 (3) ◽

pp. 419-439

Author(s):

Roberta Hansen ◽

Graciela A. González

Keyword(s):

Dynamical Systems ◽

Feedback Control ◽

Periodic Orbits ◽

Waiting Time ◽

Control Parameter ◽

Chaotic Maps ◽

Noise Sensitivity ◽

Control Methods ◽

Unstable Periodic Orbits ◽

Discrete Maps

Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.

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Intermittent chaos driven by nonlinear Alfvén waves

Nonlinear Processes in Geophysics ◽

10.5194/npg-11-691-2004 ◽

2004 ◽

Vol 11 (5/6) ◽

pp. 691-700 ◽

Cited By ~ 12

Author(s):

E. L. Rempel ◽

A. C.-L. Chian ◽

A. J. Preto ◽

S. Stephany

Keyword(s):

Dynamical Systems ◽

Periodic Orbits ◽

Systems Approach ◽

Alfven Waves ◽

Space Plasmas ◽

Alfvén Waves ◽

Unstable Periodic Orbits ◽

Phase Dynamics ◽

Intermittent Chaos

Abstract. We investigate the relevance of chaotic saddles and unstable periodic orbits at the onset of intermittent chaos in the phase dynamics of nonlinear Alfvén waves by using the Kuramoto-Sivashinsky (KS) equation as a model for phase dynamics. We focus on the role of nonattracting chaotic solutions of the KS equation, known as chaotic saddles, in the transition from weak chaos to strong chaos via an interior crisis and show how two of these unstable chaotic saddles can interact to produce the plasma intermittency observed in the strongly chaotic regimes. The dynamical systems approach discussed in this work can lead to a better understanding of the mechanisms responsible for the phenomena of intermittency in space plasmas.

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Computing and Controlling Basins of Attraction in Multistability Scenarios

Mathematical Problems in Engineering ◽

10.1155/2015/313154 ◽

2015 ◽

Vol 2015 ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

John Alexander Taborda ◽

Fabiola Angulo

Keyword(s):

Fixed Point ◽

Periodic Orbit ◽

Periodic Orbits ◽

Basin Of Attraction ◽

Basins Of Attraction ◽

Control Technique ◽

Unstable Periodic Orbits ◽

Control Effort ◽

Coexistence Of Attractors ◽

And Control

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

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Pattern recognition using chaotic neural networks

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022698000211 ◽

1998 ◽

Vol 2 (4) ◽

pp. 243-247 ◽

Cited By ~ 5

Author(s):

Z. Tan ◽

B. S. Hepburn ◽

C. Tucker ◽

M. K. Ali

Keyword(s):

Neural Network ◽

Neural Networks ◽

Pattern Recognition ◽

Dynamical Systems ◽

Periodic Orbits ◽

Basins Of Attraction ◽

Limited Information ◽

Unstable Periodic Orbits ◽

Chaotic Neural Networks

Pattern recognition by chaotic neural networks is studied using a hyperchaotic neural network as model. Virtual basins of attraction are introduced around unstable periodic orbits which are then used as patterns. Search for periodic orbits in dynamical systems is treated as a process of pattern recognition. The role of synapses on patterns in chaotic networks is discussed. It is shown that distorted states having only limited information of the patterns are successfully recognized.

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