The Lorenz System : Hidden Boundary of Practical Stability and the Lyapunov Dimension
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Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System
Hindawi Complexity Volume 2020, Article ID 8175639, 17 pages https://doi.org/10.1155/2020/8175639 Research Article Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System Qiang Lai ,1 Paul Didier Kamdem Kuate ,2 Huiqin Pei ,1 and Hilaire Fotsin 2 1School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China 2Laboratory of Condensed Matter, Electronics and Signal Processing Department of Physics, University of Dschang, P.O. Box 067, Dschang, Cameroon Correspondence should be addressed to Qiang Lai; [email protected] Received 17 January 2020; Revised 18 February 2020; Accepted 26 February 2020; Published 28 March 2020 Guest Editor: Chun-Lai Li Copyright © 2020 Qiang Lai et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. *is paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. *e circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method. 1. Introduction system may generate hidden attractor. Also, a lot of pre- vious studies have shown that chaotic systems with mul- Encouraging progress has been made on chaos in the past tiple unstable equilibria usually have richer dynamic few decades. -
Numerical Bifurcation Analysis N 6329
Numerical Bifurcation Analysis N 6329 71. Popov G (2004) KAM theorem for Gevrey Hamiltonians. Ergod Books and Reviews Theor Dynam Syst 24:1753–1786 Braaksma BLJ, Stolovitch L (2007) Small divisors and large multi- 72. Ramis JP (1994) Séries divergentes et théories asymptotiques. pliers (Petits diviseurs et grands multiplicateurs). Ann l’institut Panor Synth pp 0–74 Fourier 57(2):603–628 73. Ramis JP, Schäfke R (1996) Gevrey separation of slow and fast Broer HW, Levi M (1995) Geometrical aspects of stability theory for variables. Nonlinearity 9:353–384 Hill’s equations. Arch Rat Mech An 131:225–240 74. Roussarie R (1987) Weak and continuous equivalences for fam- Gaeta G (1999) Poincaré renormalized forms. Ann Inst Henri ilies of line diffeomorphisms. In: Dynamical systems and bifur- Poincaré 70(6):461–514 cation theory, Pitman research notes in math. Series Longman Martinet J, Ramis JP (1982) Problèmes des modules pour les èqua- 160:377–385 tions différentielles non linéaires du premier ordre. Publ IHES 75. Sanders JA, Verhulst F, Murdock J (1985) Averaging methods 5563–164 in nonlinear dynamical systems. Revised 2nd edn, Appl Math Martinet J Ramis JP (1983) Classification analytique des équations Sciences 59, 2007. Springer différentielles non linéaires résonnantes du premier ordre. Ann 76. Siegel CL (1942) Iteration of analytic functions. Ann Math Sci École Norm Suprieure Sér 416(4):571–621 43(2):607–612 Vanderbauwhede A (2000) Subharmonic bifurcation at multiple 77. Siegel CL, Moser JK (1971) Lectures on celestial mechanics. resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh Springer, Berlin M (eds) Proceedings of the mathematics conference, Birzeit, 78. -
Phase-Locked Loops with Active PI Filter: the Lock-In Range Computation JYVÄSKYLÄ STUDIES in COMPUTING 239
JYVÄSKYLÄ STUDIES IN COMPUTING 239 Konstantin Aleksandrov Phase-Locked Loops with Active PI Filter: the Lock-In Range Computation JYVÄSKYLÄ STUDIES IN COMPUTING 239 Konstantin Aleksandrov Phase-Locked Loops with Active PI Filter: the Lock-In Range Computation Esitetään Jyväskylän yliopiston informaatioteknologian tiedekunnan suostumuksella julkisesti tarkastettavaksi yliopiston Agora-rakennuksen Alfa-salissa kesäkuun 17. päivänä 2016 kello 10. Academic dissertation to be publicly discussed, by permission of the Faculty of Information Technology of the University of Jyväskylä, in building Agora, Alfa hall, on June 17, 2016 at 10 o’clock. UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 2016 Phase-Locked Loops with Active PI Filter: the Lock-In Range Computation JYVÄSKYLÄ STUDIES IN COMPUTING 239 Konstantin Aleksandrov Phase-Locked Loops with Active PI Filter: the Lock-In Range Computation UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 2016 Editors Timo Männikkö Department of Mathematical Information Technology, University of Jyväskylä Pekka Olsbo, Ville Korkiakangas Publishing Unit, University Library of Jyväskylä URN:ISBN:978-951-39-6688-1 ISBN 978-951-39-6688-1 (PDF) ISBN 978-951-39-6687-4 (nid.) ISSN 1456-5390 Copyright © 2016, by University of Jyväskylä Jyväskylä University Printing House, Jyväskylä 2016 ABSTRACT Aleksandrov, Konstantin Phase-locked loops with active PI filter: the lock-in range computation Jyväskylä: University of Jyväskylä, 2016, 38 p.(+included articles) (Jyväskylä Studies in Computing ISSN 1456-5390; 239) ISBN 978-951-39-6687-4 (nid.) ISBN 978-951-39-6688-1 (PDF) Finnish summary Diss. The present work is devoted to the study of the lock-in range of phase-locked loop (PLL). The PLL concept was originally described by H. -
Hidden Bifurcation in the Multispiral Chua Attractor Menacer Tidjani, René Lozi, Leon Chua
Hidden bifurcation in the Multispiral Chua Attractor Menacer Tidjani, René Lozi, Leon Chua To cite this version: Menacer Tidjani, René Lozi, Leon Chua. Hidden bifurcation in the Multispiral Chua Attractor. International journal of bifurcation and chaos in applied sciences and engineering , World Scientific Publishing, 2016, 26 (14), pp.1630039-1 1630023-26. 10.1142/S0218127416300391. hal-01488550 HAL Id: hal-01488550 https://hal.archives-ouvertes.fr/hal-01488550 Submitted on 13 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. January 6, 2017 11:8 WSPC/S0218-1274 1630039 Published in International Journal of Bifurcation and Chaos, Personal file Vol. 26, No. 14, (2016), 1630039-1 to 1630039-26 DOI 10.1142/S0218127416300391 Hidden Bifurcations in the Multispiral Chua Attractor Tidjani Menacer Department of Mathematics, University Mohamed Khider, Biskra, Algeria [email protected] Ren´eLozi Universit´eCˆote d’Azur, CNRS, LJAD, Parc Valrose 06108, Nice C´edex 02, France [email protected] Leon O. Chua Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA [email protected] Received August 20, 2016 We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation c which determines the number of spiral is discrete. -
This Is an Electronic Reprint of the Original Article. This Reprint May Differ from the Original in Pagination and Typographic Detail
This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Kuznetsov, Nikolay; Leonov, G. A.; Yuldashev, M. V.; Yuldashev, R. V. Title: Hidden attractors in dynamical models of phase-locked loop circuits : limitations of simulation in MATLAB and SPICE Year: 2017 Version: Please cite the original version: Kuznetsov, N., Leonov, G. A., Yuldashev, M. V., & Yuldashev, R. V. (2017). Hidden attractors in dynamical models of phase-locked loop circuits : limitations of simulation in MATLAB and SPICE. Communications in Nonlinear Science and Numerical Simulation, 51, 39-49. doi:10.1016/j.cnsns.2017.03.010 All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Accepted Manuscript Hidden attractors in dynamical models of phase-locked loop circuits: limitations of simulation in MATLAB and SPICE Kuznetsov N. V., Leonov G. A., Yuldashev M. V., Yuldashev R. V. PII: S1007-5704(17)30090-4 DOI: 10.1016/j.cnsns.2017.03.010 Reference: CNSNS 4137 To appear in: Communications in Nonlinear Science and Numerical Simulation Received date: 30 August 2016 Revised date: 4 November 2016 Accepted date: 13 March 2017 Please cite this article as: Kuznetsov N. -
A New 4-D Chaotic System with Hidden Attractor and Its Circuit Implementation
International Journal of Engineering & Technology, 7 (3) (2018) 1245-1250 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET doi: 10.14419/ijet.v7i3.9846 Research paper A New 4-D Chaotic System with Hidden Attractor and its Circuit Implementation Aceng Sambas1*, Mustafa Mamat2, Sundarapandian Vaidyanathan3, Mohamad Afendee Mohamed2 and W. S. Mada Sanjaya4 1Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Indonesia 2Faculty of Informatics and Computing, Universiti Sultan Zainal Abidin, Malaysia 3Research and Development Centre, Vel Tech University, Avadi, Chennai, India 4Department of Physics, Universitas Islam Negeri Sunan Gunung Djati Bandung, Indonesia *[email protected] Abstract In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations. Keywords: Chaos, chaotic systems,circuit simulation, hidden attractors, Lyapunov exponents 1. Introduction chaos exponents of a chaotic system are determined using Wolf’s algorithm [53]. In the literature, there is also good interest shown Chaos theory deals with nonlinear dynamical systems that are highly in building electronic circuit designs of chaotic systems and imple- sensitive to initial conditions. -
Looking More Closely to the Rabinovich-Fabrikant System
Looking more closely to the Rabinovich-Fabrikant system MARIUS-F. DANCA Dept. of Mathematics and Computer Science, Avram Iancu University of Cluj-Napoca, Romania and Romanian Institute of Science and Technology, Cluj-Napoca, Romania [email protected] MICHAL FECKAN˘ Department of Mathematical Analysis and Numerical Mathematics Comenius University in Bratislava, Slovakia and Mathematical Institute Slovak Academy of Sciences Bratislava, Slovakia [email protected] NIKOLAY KUZNETSOV Department of Applied Cybernetics Saint-Petersburg State University, Russia and University of Jyv¨askyl¨a,Finland [email protected] GUANRONG CHEN Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China [email protected] October 2, 2015 Abstract Recently, we look more closely into the Rabinovich-Fabrikant system, after a decade of the study in [Danca & Chen(2004)], discovering some new characteristics such as cycling chaos, transient chaos, chaotic hidden attractors and a new kind of saddles-like attractor. In addition to extensive and accurate numerical analysis, on the assumptive existence of heteroclinic orbits, we provide a few of their approximations. arXiv:1509.09206v2 [nlin.CD] 1 Oct 2015 Rabinovich-Fabrikant system; cycling chaos; transient chaos; heteroclinic orbit; LIL numerical method 1 Introduction Rabinovich & Fabrikant [1979] introduced and analyzed from physical point of view a model describing the stochasticity arising from the modulation instability in a non-equilibrium dissipative medium. This is a simplification of a complex nonlinear parabolic equation modelling different physical systems, such as the Tollmien-Schlichting waves in hydrodynamic flows, wind waves on water, concentration waves during chemical reactions in a medium where diffusion occur, Langmuir waves in plasma, etc. -
Numerical Bifurcation Theory for High-Dimensional Neural Models
Journal of Mathematical Neuroscience (2014) 4:13 DOI 10.1186/2190-8567-4-13 R E V I E W Open Access Numerical Bifurcation Theory for High-Dimensional Neural Models Carlo R. Laing Received: 10 April 2014 / Accepted: 13 June 2014 / Published online: 25 July 2014 © 2014 C.R. Laing; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determin- ing whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of in- terest satisfies a parametrised set of algebraic equations, and branches of solutions are followed as the parameter is varied. An effective way to do this is with pseudo- arclength continuation. We give an introduction to pseudo-arclength continuation and then demonstrate its use in investigating the behaviour of a number of models from the field of computational neuroscience. The models we consider are high dimen- sional, as they result from the discretisation of neural field models—nonlocal dif- ferential equations used to model macroscopic pattern formation in the cortex. We consider both stationary and moving patterns in one spatial dimension, and then trans- lating patterns in two spatial dimensions. A variety of results from the literature are discussed, and a number of extensions of the technique are given. -
Hidden Attractors on One Path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich Systems
Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems G. Chen,1 N.V. Kuznetsov,2, 3 G.A. Leonov,2, 4 and T.N. Mokaev2 1City University of Hong Kong, Hong Kong SAR, China 2Faculty of Mathematics and Mechanics, St. Petersburg State University, Peterhof, St. Petersburg, Russia 3Department of Mathematical Information Technology, University of Jyv¨askyl¨a,Jyv¨askyl¨a,Finland 4Institute of Problems of Mechanical Engineering RAS, Russia (Dated: October 2, 2018) In this report, by the numerical continuation method we visualize and connect hidden chaotic sets in the Glukhovsky-Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system. I. INTRODUCTION The Glukhovsky-Dolzhansky system describes the con- vective fluid motion inside a rotating ellipsoidal cavity. In 1963, meteorologist Edward Lorenz suggested an ap- If we set proximate mathematical model (the Lorenz system) for a < 0; σ = ar; (7) the Rayleigh-B´enardconvection and discovered numeri- − cally a chaotic attractor in this model [1]. This discovery then after the linear transformation (see, e.g., [8]): stimulated rapid development of the chaos theory, nu- 1 1 1 1 1 merical methods for attractor investigation, and till now (x; y; z) ν1− y; ν1− ν2− h x; ν1− ν2− h z) ; t ν1 t has received a great deal of attention from different fields ! ! [2{7]. The Lorenz system gave rise to various generaliza- with positive ν1; ν2; h, we obtain the Rabinovich system tions, e.g. Lorenz-like systems, some of which are also [10, 11], describing the interaction of three resonantly simplified mathematical models of physical phenomena. -
Krauskopf, B., Osinga, HM., Doedel, EJ., Henderson, ME., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., & Junge, O
Krauskopf, B., Osinga, HM., Doedel, EJ., Henderson, ME., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., & Junge, O. (2004). A survey of methods for computing (un)stable manifolds of vector fields. http://hdl.handle.net/1983/82 Early version, also known as pre-print Link to publication record in Explore Bristol Research PDF-document University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ A survey of methods for computing (un)stable manifolds of vector fields B. Krauskopf & H.M. Osinga Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, UK E.J. Doedel Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve O., Montr´eal Qu´ebec, H3G 1M8 Canada M.E. Henderson IBM Research, P.O. Box 218, Yorktown Heights, NY 10598, USA J. Guckenheimer & A. Vladimirsky Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853{4201, USA M. Dellnitz & O. Junge Institute for Mathematics, University of Paderborn, D-33095 Paderborn, Germany Preprint of May 2004 Keywords: stable and unstable manifolds, numerical methods, Lorenz equation. Abstract The computation of global invariant manifolds has seen renewed interest in recent years. We survey different approaches for computing a global stable or unstable mani- fold of a vector field, where we concentrate on the case of a two-dimensional manifold. All methods are illustrated with the same example | the two-dimensional stable man- ifold of the origin in the Lorenz system. -
Unusual Dynamics and Hidden Attractors of the Rabinovich-Fabrikant System
Nonlinear Dynamics manuscript No. (will be inserted by the editor) Unusual dynamics and hidden attractors of the Rabinovich-Fabrikant system Marius-F. Danca · Nikolay Kuznetsov · Guanrong Chen the date of receipt and acceptance should be inserted later Abstract This paper presents some numerical evidence of unusual dynamics and hidden chaotic attractors of the Rabinovich-Fabrikant system, with some insightful descriptions and discussions. From a generalized Hamiltonian energy perspective, the attractors could be analyzed in more details. Keywords Rabinovich-Fabrikant system · Hidden chaotic attractor · Self-excited attractor · Hamiltonian energy 1 Introduction The Rabinovich-Fabrikant (RF) system, introduced by Rabinovich and Fabrikant [1] and numerically investigated in [2], was initially established as a physical model describing the stochasticity arising from the modulation instability in a non- equilibrium dissipative medium. However, recently in [2] we revealed that besides the physical properties described in [1], the system presents extremely rich com- plex dynamics. More significantly, unlike other familiar chaotic systems containing only second-order nonlinearities (such as the Lorenz system), the RF system with third-order nonlinearities presents some unusual dynamics, namely, some complex dynamics that were rarely or never seen from other systems in the past, such as \virtual" saddles in addition to several chaotic attractors with different shapes as well as hidden chaotic attractors and hidden transient chaotic attractors [3]. In fact, common interest in this system is continuously increasing (see e.g. [4{11]). Marius-F. Danca Dept. of Mathematics and Computer Science, Avram Iancu University of Cluj-Napoca, Ro- mania and Romanian Institute of Science and Technology, Cluj-Napoca, Romania Nikolay Kuznetsov Dept. -
Hidden Chaotic Attractors in a Class of Two-Dimensional Maps
ORE Open Research Exeter TITLE Hidden chaotic attractors in a class of two-dimensional maps AUTHORS Jiang, H; Liu, Y; Wei, Z; et al. JOURNAL Nonlinear Dynamics DEPOSITED IN ORE 11 October 2016 This version available at http://hdl.handle.net/10871/23849 COPYRIGHT AND REUSE Open Research Exeter makes this work available in accordance with publisher policies. A NOTE ON VERSIONS The version presented here may differ from the published version. If citing, you are advised to consult the published version for pagination, volume/issue and date of publication Hidden chaotic attractors in a class of two-dimensional maps∗ Haibo Jiangay, Yang Liub, Zhouchao Weic, Liping Zhanga a School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China bSchool of Engineering, Robert Gordon University, Garthdee Road, Aberdeen AB10 7GJ, UK cSchool of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China Abstract This paper studies the hidden dynamics of a class of two-dimensional maps inspired by the H´enonmap. A special consideration is made to the existence of fixed points and their stabilities in these maps. Our concern focuses on three typical scenarios which may generate hidden dynamics, i.e. no fixed point, single fixed point, and two fixed points. A computer search program is employed to explore the strange hidden attractors in the map. Our findings show that the basins of some hidden attractors are tiny, so the standard computational procedure for localization is unavailable. The schematic exploring method proposed in this paper could be generalized for investigating hidden dynamics of high-dimensional maps. Keywords: hidden dynamics; two-dimensional maps; fixed point; stability; coexistence.