Fractional-Order Control for a Novel Chaotic System Without Equilibrium Shu-Yi Shao and Mou Chen Member, IEEE

Total Page:16

File Type:pdf, Size:1020Kb

Fractional-Order Control for a Novel Chaotic System Without Equilibrium Shu-Yi Shao and Mou Chen Member, IEEE IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1 Fractional-Order Control for a Novel Chaotic System without Equilibrium Shu-Yi Shao and Mou Chen Member, IEEE, Abstract—The control problem is discussed for a chaotic sys- portant results have been reported. In the early 1990s, the tem without equilibrium in this paper. On the basis of the linear synchronization of chaotic systems was achieved by Pecora mathematical model of the two-wheeled self-balancing robot, a and Carroll[14;15], which was a trailblazing result, and the novel chaotic system which has no equilibrium is proposed. The basic dynamical properties of this new system are studied via result promoted the development of chaos control and chaos [16;17] Lyapunov exponents and Poincare´ map. To further demonstrate synchronization . In recent years, different chaos control the physical realizability of the presented novel chaotic system, a and chaos synchronization strategies have been developed chaotic circuit is designed. By using fractional-order operators, for chaotic systems. The sliding mode control method was a controller is designed based on the state-feedback method. applied to chaos control[18;19] and chaos synchronization[20]. According to the Gronwall inequality, Laplace transform and Mittag-Leffler function, a new control scheme is explored for the In [21], the feedback control method and the adaptive control whole closed-loop system. Under the developed control scheme, method were used to realize chaos control for the energy the state variables of the closed-loop system are controlled to resource chaotic system. The chaos control problems were stabilize them to zero. Finally, the numerical simulation results investigated for Lorenz system, Chen system and Lu¨ system of the chaotic system with equilibrium and without equilibrium based on backstepping design method in [22]. By using illustrate the effectiveness of the proposed control scheme. adaptive control method, the problems of chaos control[23;24] Index Terms—Chaotic system, Circuit implementation, and chaotic synchronization[25] were studied for chaotic sys- Fractional-order, Stabilization. tems. The neural adaptive control method was developed for a class of chaotic systems with uncertain dynamics, input I. INTRODUCTION and output constraints in [26]. In [27], on the basis of ROM From the simplified equation of convection roll- impulsive control method, the problems of the stabilization F s in the equations of the atmosphere, the first three- and synchronization were explored for Lorenz systems. The dimensional chaotic system was derived by Lorenz in 1963[1]. synchronization problem was resolved for a class of chaotic With the development and applying of chaos theory, a number systems by using a fractional-order observer-based method and of chaotic systems, hyperchaotic systems, fractional-order the synchronization was applied to secure communication in chaotic systems and fractional-order hyperchaotic systems [28]. In [29], the synchronization was studied for fractional- have been proposed, such as Rossler¨ chaotic system[2], Liu order systems based on the output feedback sliding mode chaotic system[3], hyperchaotic Chen system[4], hyperchaotic control method. A new synchronization strategy was presented Lu¨ system[5], fractional-order financial system[6], fractional- for two fractional-order systems and the synchronization was order Lotka-Volterra equation[7], fractional-order hyperchaos applied in image encryption in [30]. The above literature works Lorenz system[8], a modified four-dimensional fractional order focused on chaos control and chaos synchronization in prac- hyperchaotic system[9] and so on. The above mentioned chaot- tical chaotic systems with equilibrium points. However, the ic systems have equilibrium. In addition, there are a number of control of chaotic systems without equilibrium has rarely been [13] chaotic systems without equilibrium which have been studied investigated . Meanwhile, for most of the above mentioned by [10-13]. As a result, chaos control became one of the works, fractional-order controllers have rarely been used to important issues for chaotic systems. Due to great potential realize the chaos control of integer-order chaotic systems, application in electrical engineering, information processing although some important results on the fractional-order con- − and secure communication, it is important to investigate new trollers have been proposed for various systems[31 33]. In [31], control methods for chaotic systems. a well-known fractional-order controller was presented. In Over the past few decades, chaos control and chaos syn- [32], the concept of a fractional-order PIλDµ controller was chronization have received much attention and many im- proposed and the fractional-order controller included fractional order integrator and fractional-order differentiator. In [33], on This article has been accepted for publication in a future issue of this the basis of the Lyapunov stability theory, a novel fractional- journal, but has not been fully edited. Content may change prior to final publication. order controller was given, and fractional-order chaotic and hy- This work is supported by National Natural Science Foundation of China perchaotic systems were controlled by the proposed fractional- (61573184), Jiangsu Natural Science Foundation of China (SBK20130033), order controller. The fractional-order controllers are effective Six Talents Peak Project of Jiangsu Province (2012-XXRJ-010) and Funda- mental Research Funds for the Central Universities (NE2016101). Recom- to control systems, which have been proved in the mentioned mended by Associate Editor Antonio Visioli. works. Therefore, it is valuable to further explore the chaos Shu-Yi Shao and Mou Chen are with the College of Automation Engineer- control of integer-order chaotic systems without equilibrium ing, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: [email protected], [email protected]). by using fractional-order controllers. Digital Object Identifier 10.1109/JAS.2016.7510124 Inspired by the above discussions, the objective of this paper 2 IEEE/CAA JOURNAL OF AUTOMATICA SINICA is to design an efficient fractional-order controller and the control input u, (1) can be described as stability is realized for the closed-loop system. A novel chaotic 2 3 2 3 2 3 x_ 0 1 0 0 x system without equilibrium is proposed based on the model of r r 6 x¨ 7 6 0 0 −23:6701 0 7 6 x_ 7 two-wheeled self-balancing robot. Meanwhile, the presented 6 r 7 = 6 7 6 r 7 4 θ_ 5 4 0 0 0 1 5 4 θ 5 new system is used to verify the effectiveness of the proposed p p θ¨ 0 0 124:5128 0 θ_ control scheme. p 2 3 p 0 The organization of the paper is as follows. Section II 6 7 details the problem formulation. A novel chaotic system is 6 4:5974 7 _ ¨ + 4 5 Φ(xr; x_ r; θp; θp) + u (2) proposed and the chaotic system circuit is designed in Section 0 III. Section IV presents the fractional-order controller based on −19:0414 the state-feedback method. The numerical simulation studies T where the control input u = [u1; u2; u3; u4] . are presented to demonstrate the effectiveness of the developed This paper aims at constructing a novel chaotic system control method in Section V, followed by some concluding without equilibrium and developing a fractional-order control remarks in Section VI. scheme, so that the stabilization of the whole closed-loop sys- tem is realized based on the designed control strategy. Under designed fractional-order controller, the state variables of the II. PROBLEM STATEMENT AND PRELIMINARIES closed-loop system will be asymptotically stable. To develop In this paper, a novel chaotic system will be proposed by the fractional-order control scheme, we firstly introduce the following definitions and lemmas: only considering the straight line position xr and the pitch Definition 1[35]. The Caputo fractional derivative operator, angle θp of the two-wheeled self-balancing robot of Googol Technology as shown in Fig. 1. A mathematical model related which is one of the most widely used fractional derivative to the two-wheeled self-balancing robot of Googol Technology operators, is defined for the function f(t) as follows : Z was established in [34]. The linear mathematical model for t (m) α 1 f (τ) x and θ of the two-wheeled self-balancing robot of Googol D f(t) = − dτ; (3) r p Γ(m − α) (t − τ)α m+1 Technology is described in the form t0 2 3 2 3 2 3 where α is the fractional order and m − 1 < α < m, m = x_ r 0 1 0 0 xr [α] + 1, [α] denotes the integer part of α, and Γ(·) is gamma R 1 6 x¨ 7 6 − 7 6 7 − m−α−1 −t 6 r 7 6 0 0 23:6701 0 7 6 x_ r 7 function, which is defined as Γ(m α) = 0 t e dt. 4 _ 5 = 4 5 4 5 θp 0 0 0 1 θp The main advantage of (3) is that Caputo derivative of a ¨ _ ≤ θp 0 0 124:5128 0 θp constant is equal to zero. Particularly, when 0 < α 1, 2 3 L f α g α − α−1 0 we have D f(t) = s F (s) s f(0): The Laplace 6 7 transform of fractional integral at t = 0 has the following 6 4:5974 7 0 + 4 5 Cθ (1) form: 0 { } −19:0414 L D−αf(t) = s−αL ff(t)g = s−αF (s); (α > 0); (4) where Cθ denotes the pitch torque. where t and s are the variables in the time domain and Laplace domain, respectively. F (s) = L(f(t)) and L(·) stands for the Laplace transform. In this paper, the fractional-order controller will be de- scribed by using Caputo definition with lower limit of integral t0 = 0 and the order 1 < α < 2.
Recommended publications
  • Role of Nonlinear Dynamics and Chaos in Applied Sciences
    v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings.
    [Show full text]
  • Instructional Experiments on Nonlinear Dynamics & Chaos (And
    Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline..
    [Show full text]
  • International Journal of Engineering and Advanced Technology (IJEAT)
    IInntteerrnnaattiioonnaall JJoouurrnnaall ooff EEnnggiinneeeerriinngg aanndd AAddvvaanncceedd TTeecchhnnoollooggyy ISSN : 2249 - 8958 Website: www.ijeat.org Volume-2 Issue-5, June 2013 Published by: Blue Eyes Intelligence Engineering and Sciences Publication Pvt. Ltd. ced Te van ch d no A l d o n g y a g n i r e e IJEat n i I E n g X N t P e L IO n O E T r R A I V n NG NO f IN a o t l i o a n n r a u l J o www.ijeat.org Exploring Innovation Editor In Chief Dr. Shiv K Sahu Ph.D. (CSE), M.Tech. (IT, Honors), B.Tech. (IT) Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Dr. Shachi Sahu Ph.D. (Chemistry), M.Sc. (Organic Chemistry) Additional Director, Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., Bhopal (M.P.), India Vice Editor In Chief Dr. Vahid Nourani Professor, Faculty of Civil Engineering, University of Tabriz, Iran Prof.(Dr.) Anuranjan Misra Professor & Head, Computer Science & Engineering and Information Technology & Engineering, Noida International University, Noida (U.P.), India Chief Advisory Board Prof. (Dr.) Hamid Saremi Vice Chancellor of Islamic Azad University of Iran, Quchan Branch, Quchan-Iran Dr. Uma Shanker Professor & Head, Department of Mathematics, CEC, Bilaspur(C.G.), India Dr. Rama Shanker Professor & Head, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea Dr. Vinita Kumari Blue Eyes Intelligence Engineering & Sciences Publication Pvt. Ltd., India Dr. Kapil Kumar Bansal Head (Research and Publication), SRM University, Gaziabad (U.P.), India Dr.
    [Show full text]
  • Control of Chaos: Methods and Applications
    Automation and Remote Control, Vol. 64, No. 5, 2003, pp. 673{713. Translated from Avtomatika i Telemekhanika, No. 5, 2003, pp. 3{45. Original Russian Text Copyright c 2003 by Andrievskii, Fradkov. REVIEWS Control of Chaos: Methods and Applications. I. Methods1 B. R. Andrievskii and A. L. Fradkov Institute of Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia Received October 15, 2002 Abstract|The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system ex- citation, the method of Poincar´e map linearization (OGY method), and the method of time- delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications. 1. INTRODUCTION The term control of chaos is used mostly to denote the area of studies lying at the interfaces between the control theory and the theory of dynamic systems studying the methods of control of deterministic systems with nonregular, chaotic behavior. In the ancient mythology and philosophy, the word \χαωσ" (chaos) meant the disordered state of unformed matter supposed to have existed before the ordered universe. The combination \control of chaos" assumes a paradoxical sense arousing additional interest in the subject.
    [Show full text]
  • Math Morphing Proximate and Evolutionary Mechanisms
    Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales.
    [Show full text]
  • Chaos Theory: the Essential for Military Applications
    U.S. Naval War College U.S. Naval War College Digital Commons Newport Papers Special Collections 10-1996 Chaos Theory: The Essential for Military Applications James E. Glenn Follow this and additional works at: https://digital-commons.usnwc.edu/usnwc-newport-papers Recommended Citation Glenn, James E., "Chaos Theory: The Essential for Military Applications" (1996). Newport Papers. 10. https://digital-commons.usnwc.edu/usnwc-newport-papers/10 This Book is brought to you for free and open access by the Special Collections at U.S. Naval War College Digital Commons. It has been accepted for inclusion in Newport Papers by an authorized administrator of U.S. Naval War College Digital Commons. For more information, please contact [email protected]. The Newport Papers Tenth in the Series CHAOS ,J '.' 'l.I!I\'lt!' J.. ,\t, ,,1>.., Glenn E. James Major, U.S. Air Force NAVAL WAR COLLEGE Chaos Theory Naval War College Newport, Rhode Island Center for Naval Warfare Studies Newport Paper Number Ten October 1996 The Newport Papers are extended research projects that the editor, the Dean of Naval Warfare Studies, and the President of the Naval War CoJIege consider of particular in terest to policy makers, scholars, and analysts. Papers are drawn generally from manuscripts not scheduled for publication either as articles in the Naval War CollegeReview or as books from the Naval War College Press but that nonetheless merit extensive distribution. Candidates are considered by an edito­ rial board under the auspices of the Dean of Naval Warfare Studies. The views expressed in The Newport Papers are those of the authors and not necessarily those of the Naval War College or the Department of the Navy.
    [Show full text]
  • Program of CHAOS2011 Conference
    4th Chaotic Modeling and Simulation International Conference (CHAOS2011) May 31 - June 3, 2011 Agios Nikolaos Crete Greece Program Session / Date / Time Event Talk Title / Event Room Hermes 17.00-20.00 Monday May 30 Registration Hermes 8.30-10.00 Tuesday May 31 Registration Room 1 10.00-10.40 Opening Ceremony Keynote Session (Chair: D. Sotiropoulos) Room 1 10.40-11.30 Extension of Poincare's program for integrability and chaos in Hamiltonian systems Professor Ferdinand Verhulst Room 1 11.30-12.00 Coffee Break SCS1 SPECIAL AND CONTRIBUTED SESSIONS SCS1 Room 1 31.05.11: 12.00-13.40 Chair: G. I. Burde Chaos and solitons Spontaneous generation of solitons from steady state {exact solutions to the higher order KdV G.I. Burde equations on a half-line} Posadas-Castillo C., Garza-González E., Cruz- Chaotic synchronization of complex networks with Rössler oscillators in Hamiltonian form like Hernández C., Alcorta-García E., Díaz-Romero D.A. nodes Vladimir L. Kalashnikov Dissipative solitons: the structural chaos and the chaos of destruction V.Yu.Novokshenov Tronqu'ee solutions of the Painleve' II equation Stefan C. Mancas, Harihar Khanal 2D Erupting Solitons in Dissipative Media Room 2 31.05.11: 12.00-13.40 Chair: G. Feichtinger CHAOS and Applications in social and economic life Gustav Feichtinger Multiple Equilibria, Binges, and Chaos in Rational Addiction Models David Laroze, J. Bragard, H Pleiner Chaotic dynamics of a biaxial anisotropic magnetic particle Oleksander Pokutnyi Chaotic maps in cybernetics Room 3 31.05.11: 12.00-13.40 Chair: D. Sotiropoulos Chaos and time series analysis Hannah M.
    [Show full text]
  • Fourth SIAM Conference on Applications of Dynamical Systems
    Utah State University DigitalCommons@USU All U.S. Government Documents (Utah Regional U.S. Government Documents (Utah Regional Depository) Depository) 5-1997 Fourth SIAM Conference on Applications of Dynamical Systems SIAM Activity Group on Dynamical Systems Follow this and additional works at: https://digitalcommons.usu.edu/govdocs Part of the Physical Sciences and Mathematics Commons Recommended Citation Final program and abstracts, May 18-22, 1997 This Other is brought to you for free and open access by the U.S. Government Documents (Utah Regional Depository) at DigitalCommons@USU. It has been accepted for inclusion in All U.S. Government Documents (Utah Regional Depository) by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. tI...~ Confers ~'t' '"' \ 1I~c9 ~ 1'-" ~ J' .. c "'. to APPLICAliONS cJ May 18-22, 1997 Snowbird Ski and Summer Resort • Snowbird, Utah Sponsored by SIAM Activity Group on Dynamical Systems Conference Themes The themes of the 1997 conference will include the following topics. Principal Themes: • Dynamics in undergraduate education • Experimental studies of nonlinear phenomena • Hamiltonian systems and transport • Mathematical biology • Noise in dynamical systems • Patterns and spatio-temporal chaos Applications in • Synchronization • Aerospace engineering • Biology • Condensed matter physics • Control • Fluids • Manufacturing • Me;h~~~~nograPhY 19970915 120 • Lasers and o~ • Quantum UldU) • 51a m.@ Society for Industrial and Applied Mathematics http://www.siam.org/meetingslds97/ds97home.htm 2 " DYNAMICAL SYSTEMS Conference Prl Contents A Message from the Conference Chairs ... Get-Togethers 2 Dear Colleagues: Welcoming Message 2 Welcome to Snowbird for the Fourth SIAM Conference on Applications of Dynamica Systems. Organizing Committee 2 This highly interdisciplinary meeting brings together a diverse group of mathematicians Audiovisual Notice 2 scientists, and engineers, all working on dynamical systems and their applications.
    [Show full text]
  • Frequently Asked Questions About Nonlinear Science J.D
    Frequently Asked Questions about Nonlinear Science J.D. Meiss Sept 2003 [1] About Sci.nonlinear FAQ This is version 2.0 (Sept. 2003) of the Frequently Asked Questions document for the newsgroup s ci.nonlinear. This document can also be found in Html format from: http://amath.colorado.edu/faculty/jdm/faq.html Colorado, http://www-chaos.engr.utk.edu/faq.html Tennessee, http://www.fen.bris.ac.uk/engmaths/research/nonlinear/faq.html England, http://www.sci.usq.edu.au/mirror/sci.nonlinear.faq/faq.html Australia, http://www.faqs.org/faqs/sci/nonlinear-faq/ Hypertext FAQ Archive Or in other formats: http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.pdf PDF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.rtf RTF Format, http://amath.colorado.edu/pub/dynamics/papers/sci.nonlinearFAQ.tex old version in TeX, http://www.faqs.org/ftp/faqs/sci/nonlinear-faq the FAQ's site, text version ftp://rtfm.mit.edu/pub/usenet/news.answers/sci/nonlinear-faq text format. This FAQ is maintained by Jim Meiss [email protected]. Copyright (c) 1995-2003 by James D. Meiss, all rights reserved. This FAQ may be posted to any USENET newsgroup, on-line service, or BBS as long as it is posted in its entirety and includes this copyright statement. This FAQ may not be distributed for financial gain. This FAQ may not be in cluded in commercial collections or compilations without express permission from the author. [1.1] What's New? Fixed lots of broken and outdated links. A few sites seem to be gone, and some new sites appeare d.
    [Show full text]
  • Control of Chaos in Strongly Nonlinear Dynamic Systems
    Control of Chaos in Strongly Nonlinear Dynamic Systems Lev F. Petrov Plekhanov Russian University of Economics Stremianniy per., 36, 115998, Moscow, Russia [email protected] Abstract We consider the dynamic processes models based on strongly nonlin- ear systems of ordinary differential equations and various stable and unstable solutions of such systems. The behavior and evolution of so- lutions are studied when the system parameters and external influences change. As a tool for controlling the behavior of solutions is considered a system parameters change. Particular attention is paid to the param- eter of dissipation as a tool for controlling chaos. For simple systems, a controlled transition from stable solutions to chaos and back is consid- ered. The generalization of the solutions chaotic behavior for complex systems and the criteria for optimal level of chaos in such systems are discussed. 1 Introduction For complex systems such as Economics, social relations, communication, chaos is a natural form of behavior. For these systems mathematical models allow qualitatively, but not quantitatively evaluate the system behavior. In real complex systems, both a regular ordered behavior and chaotic states of systems are observed. Complete elimination of chaos in a complex system, as a rule, leads to system degradation. For the economy this is stagnation, lack of competition, initiatives and new prospective. For social relations, this is the lack of democracy. For information transfer systems, this is regulated and metered information and misinformation, information vacuum, the lack of the possibility of obtaining information from alternative sources. The lack of complex systems regulation leads to an excessive level of chaos.
    [Show full text]
  • Control of Chaotic Systems - A.L
    CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION - Vol. XIII- Control of Chaotic Systems - A.L. Fradkov CONTROL OF CHAOTIC SYSTEMS A.L. Fradkov Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, RUSSIA Keywords: Chaos, chaotization, nonlinear systems, nonlinear control, partial stability, recurrence, stabilization, feedforward control, periodic excitation, Melnikov function, controlled Poincaré map, Ott-Grebogi-Yorke (OGY) method, Pyragas method, delayed feedback, Lurie system, Lyapunov function, synchronization Contents 1. Introduction 2. Notion of chaos 3. Models of controlled systems and control goals 4. Methods of controlling chaos: continuous-time systems 4.1. Feedforward Control by Periodic Signal 4.2. Linearization of Poincaré Map (OGY Method) 4.3. Delayed Feedback 4.4. Linear and Nonlinear Control 4.5. Adaptive Control 5. Discrete-time Control 6. Neural networks 7. Fuzzy systems 8. Control of chaos in distributed systems 9. Chaotic mixing 10. Generation of chaos (chaotization) 11. Other problems 12. Conclusions Acknowledgements Glossary Bibliography Biographical Sketch Summary UNESCO – EOLSS The field related to control of chaotic systems was rapidly developing during the 1990s. Its state-of-the-artSAMPLE in the beginning of the 21stCHAPTERS century is presented in this article. Necessary preliminary material is given related to notion and properties of chaotic systems, models of the controlled plants and control goals. Several major branches of research are discussed in detail: feedforward or “nonfeedback” control (based on periodic excitation of the system); OGY method (based on linearization of Poincaré map); Pyragas method (based on time-delay feedback); traditional control engineering methods of linear, nonlinear and adaptive control; neural networks; fuzzy control. Some unsolved problems concerning the justification of chaos control methods are presented.
    [Show full text]
  • La Naturaleza, Las Matemáticas, La Ingeniería Y
    Ingeniare. Revista chilena de ingeniería, vol. 18 Nº 1, 2010, pp. 2-4 EDITORIAL NATURE, MATHEMATICS, ENGINEERING AND CHAOS The beauty of the unexpected Is Nature predictable? Can we know with certainty how physical and engineering systems evolve? Can unexpected complexity come from simplicity? These are some of the questions which arise when we discover that chaos is ubiquitous in our day-to-day life. The laws and phenomena in Nature are described by means of differential or difference equations, which are approximations of the real world that allow us to describe the behaviour of the past and future states of a system by using the current state. In chaotic systems, the solution of a system of differential or difference equations may exhibit a random pattern. That is, the measurement of the current system state leads to unpredictability of the future system states, in addition to uncertainty with respect to the past states. Consequently, chaos reveals the unexpected evolution of a system. Generally speaking, a chaotic system can be defined as a deterministic system, described by nonlinear differential or difference equations which exhibit an apparently random dynamic behaviour. Models describing chaotic dynamics do not necessarily need to be complex. Chaotic behaviour may appear in deterministic nonlinear equations which are extraordinarily simple. For this reason, it is easy to find chaotic dynamics in physical systems around us, or in systems we have built. This encourages us to think that chaos is the norm, and not the exception, in the behaviour of nonlinear systems. We are within a solar system whose planetary motion may be unpredictable.
    [Show full text]