DOCSLIB.ORG

Intermittency Induced in a Bistable Multiscroll Attractor by Means of Stochastic Modulation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Intermittency Induced in a Bistable Multiscroll Attractor by Means of Stochastic Modulation CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019, 114–120 INTERMITTENCY INDUCED IN A BISTABLE MULTISCROLL ATTRACTOR BY MEANS OF STOCHASTIC MODULATION Jose´ L. Echenaus´ıa-Monroy Guillermo Huerta-Cuellar∗ Laboratory of Dynamical Systems Laboratory of Dynamical Systems CULagos, Universidad de Guadalajara CULagos, Universidad de Guadalajara Mexico´ Mexico´ [email protected] ∗Corresponding author: [email protected] Rider Jaimes-Reategui´ Juan H. Garc´ıa-Lopez´ Hector´ E. Gilardi-Velazquez´ Laboratory of Dynamical Systems Laboratory of Dynamical Systems Facultad de Ingenier´ıa CULagos, Universidad de CULagos, Universidad de Universidad Panamericana Guadalajara, Mexico´ Guadalajara, Mexico´ Mexico´ [email protected] [email protected] [email protected] Article history: Received 16.10.2019, Accepted 26.11.2019 Abstract inverse period-doubling bifurcations, respectively, and In this work, numerical results of a nonlinear switch- crisis-induced intermittency with a crisis of chaotic at- ing system that presents bistable attractors subjected to tractors [Manneville & Pomeau, 1979; Hirsh, Nauen- stochastic modulation are shown. The system exhibits a berg, & Scalpino , 1982; Hirsh, Huberman, & Scalpino; dynamical modification of the bistable attractor, giving Hu, & Rudnick]. For systems that show multistable be- rise to an intermit behavior, which depends of modula- havior, noise presence can be useful to influence inter- tion strength. The resulting attractor converge to an in- esting dynamics as hopping attractor [Kraut, & Feudel; termittent double-scroll, for low amplitude modulation, Huerta-Cuellar et. al.; Pisarchik et. al.], as physical and and a 9-scroll attractor for a higher applied noise ampli- natural phenomenons [Huerta-Cuellar et. al.; Pisarchik tude. A Detrended Fluctuation Analysis (DFA) applied et. al; Gelens et. al.]. Also, noise can induce on-off inter- to the x state variable, shows a perturbations robust- mittency in those systems that exhibit bistable behavior ness region, since the increase of noise does not present [Campos-Mejia, 2013]. A system with periodic poten- changes. Due to the applied noise, the final obtained tial in the high frequency regime could shows the occur- system has higher randomness, compared with the orig- rence of intermittency as the case of a pendulum, where inal one. The understanding of the dynamical behavior the linear-response theory yields maximum frequency- of multiscrolls systems is highly important for advanc- dependent mobility as noise strength function [Saikia et ing technology in communications, as well in memory al., 2011]. systems applications. In the case of systems that generates intermittent ac- tivity, an analysis and characterization of the equilib- Key words rium points number in a Chua multiscroll system by ap- plied noise was shown by [Arathi, Rajasekar, & Kurths]. Multiscroll, Intermittency, Dynamics, Bistability. In that sense, the generation of systems with scrolls in their phase space, such as the Lorenz and Chua systems 1 Introduction [Lorenz, 1963; Chua, 1992], have been extensively stud- In many nonlinear dynamical systems, intermittency is ied from a dynamical point of view, being the Lorenz at- a common behavior, characterized by irregular burst that tractor a particular case with intermittent behavior [Man- modify regular behavior [Manneville & Pomeau, 1979]. neville & Pomeau, 1979]. Over the past few years, Different types of intermittency can be mentioned, e.g., the design and control of systems with multiple scrolls type I and on-off intermittency are related with saddle- have been a subject of interest for the scientific com- node bifurcations, type II and type III with Hopf and munity [Echenaus´ıa-Monroy, & Huerta-Cuellar],´ hav- CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 115 ing a great impact in their application, such as secure UDS type II is defined by equilibrium points with a real communication systems, neural modeling, generation positive eigenvalue with two complex conjugated with of pseudo-random systems, and deterministic Brownian negative real part, with Σ(λ1;2;3) < 0 [Campos-Canton´ motion [Kwon et al., 2011; Yalcin et al., 2004; Huerta- et al., 2010]. Cuellar et al., 2014]. Recently a bistable multiscroll fam- Multiscroll generation by means of series of saturated ily has been presented and characterized [Echenaus´ıa et functions results in a better way to control the generated al., 2018]. In such system it is possible to obtain different fixed points, than with Piece Wise Linear (PWL) systems behaviors by means of bifurcation parameter variation [Lu¨ et al., 2004; Lu¨ et al., 2006]. This paper is based on (ζ). By considering the results shown in [Echenaus´ıa et a multiscroll generator system aimed by the jerk equa- al., 2018], in this work a study on noise-induced inter- tion, with the implementation of a Saturated Non Linear mittency between coexisting regimes of 1-scroll behav- Function (SNLF), as switching law, resulting in the sys- ior, under the influence of external Gaussian noise, is tem described by eq. (3). carried out. The observed intermitent dynamics may be associated with on-off intermittency. The interest in in- termittent fluctuations arises from its usefulness for both x_ = y; technical applications and fundamental research. There y_ = z; (3) are some works that presents useful memory applica- z_ = −α31x − α32y − α33z + α31f(x; k; h; p); tions based on multiscrolls [Itoh & Chua, 2008; Pham et al., 2019], in this case the applied noise could be imple- been f(x; k; h; p) the SNLF, defined as: mented to dynamically change the system memory prop- erties. p X f(x; k; h; p) = fm(x; k; h); (4) 2 Theory m=−p Consider a set of deterministic nonlinear differential equations, with chaotic behavior, defined as in [Campos- for which k > 0 is the slope of the saturated function Canton´ et al., 2010]. series, h > 2 is the saturated delay time of the saturated function, defined by the op-amp switching speed, p is a χ_ = Aχ + B; (1) positive integer, m = 1; 2; : : : ; n, where n defines the number of scrolls to generate. The function segmenta- T 3 where χ = [x1; x2; x3] 2 R is the state variable, B = tion is defined as follows: T 3 [b1; b2; b3] 2 R stands for a real constant vector. The behavior of the system is defined by the eigenvalues of 8 3×3 2k if x > mh + 1; the matrix A 2 R which is given as a linear operator < as follows: fm(x; k; h) = k(x − mh) + k; if x − mh ≤ 1; : 0 if x < mh − 1; (5) 0 1 α11 α12 α13 A = @α21 α22 α23A : (2) 8 0 if x > mh ± 1; α31 α32 α33 < f−m(x; k; h) = k(x ± mh) − k; if x ± mh ≤ 1; Discarding any combination of eigenvalues that is not : −2k if x < mh − 1; characteristic of hyperbolic-saddle-node points, that is, (6) the system is consistent with the Unstable Dissipative The SNLF contemplated for this study is constructed System type I (UDS I), defined in [Campos-Canton´ et based on [Echenaus´ıa et al., 2018], studied with a bifur- el., 2012; Ontanon˜ et al., 2012]. cation parameter ζ. This ζ parameter works as an indi- A switched system is implemented, constituted by a vidual control gain for the nonlinear function. This al- set of equations in form of eq. (1), in order to change lows the possibility to analyze the bifurcation diagrams the dynamical behavior which is governed by a switch- for a defined set of parameters in the model and gener- ing law, Si with i = 1; :::; n, and n ≥ 3. Each system ates more than one single attractor. The modified system 3 Si has a domain Di ⊂ R , containing the equilibrium is described by: ∗ −1 χi = −Ai Bi. Then, the switching law governs the ∗ ∗ SW dynamics by changing the equilibria from χi to χj , t 3 x_ = y; i 6= j, when the flow Φ : Di −! R crosses from the i − th to the j − th domain. y_ = z; (7) In order to generate multiscrolls dynamics, eq. (1) z_ = −α[x + y + z − ζf(x; k; h; p)]; must to accomplishes with the UDS I definition. A UDS 0 < α < 1 type I corresponds to equilibrium points with a real neg- where is a system parameter that modifies the equilibria stability, and ζ is the bifurcation param- ative eigenvalue (λ1) with two complex conjugated with positive real part (λ2, and λ3), with Σ(λ1;2;3) < 0, and a eter. This modification allows to analyze the behavior CYBERNETICS AND PHYSICS, VOL. 8, NO. 3, 2019 116 3 Methodology and Results From the dynamical system in eq. (7), with a switching law that creates a 9-scroll attractor, a bifurcation diagram constructed, generated by a gradual change of the bifur- cation parameter ζ, and a fixed parameter α = 0:45, is shown in Figure 1(a). A value of α parameter higher than 0:45 causes a more restrictive dynamical behavior of the system [Echenaus´ıa et al., 2018]. Figure 1(b), shows a zoom of the region to analyze with ζ = 0:0585, which generates a bistable behavior. Noise addition to the eq. (7) can stimulate jumps be- tween each of the bistable states. This bistable region forms a double potential well, for which the system re- sponse only have one state each time. With the noise amplitude increasing is possible to change the average residence times of the system until get an equilibrium of jumps between the two states. Without loss of generality, the equation system with the added noise is as follows: x_ = y; y_ = z; (9) z_ = −α[x + y + z − ζf(x; k; h; p)] + Nη; where N is the noise amplitude, and η represents the added noise, generated by the Box-Muller¨ method. The noise addition is made for each itteration of the numeri- cal system, in order to have a perturbation for every sim- ulation time.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Kaotik Sistemlerin Klasik Ve Zeki Yaklaşimlar Ile Kontrolü
    T.C. SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ KAOTİK SİSTEMLERİN KLASİK VE ZEKİ YAKLAŞIMLAR İLE KONTROLÜ DOKTORA TEZİ Uğur Erkin KOCAMAZ Enstitü Anabilim Dalı : ELEKTRİK-ELEKTRONİK MÜHENDİSLİĞİ Enstitü Bilim Dalı : ELEKTRONİK Tez Danışmanı : Doç. Dr. Yılmaz UYAROĞLU Kasım 2018 T.C. SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ KAOTİK SİSTEMLERİN KLASİK VE ZEKİ YAKLAŞIMLAR İLE KONTROLÜ DOKTORA TEZİ Uğur Erkin KOCAMAZ Enstitü Anabilim Dalı : ELEKTRİK-ELEKTRONİK MÜHENDİSLİĞİ Enstitü Bilim Dalı : ELEKTRONİK BEYAN Tez içindeki tüm verilerin akademik kurallar çerçevesinde tarafımdan elde edildiğini, görsel ve yazılı tüm bilgi ve sonuçların akademik ve etik kurallara uygun şekilde sunulduğunu, kullanılan verilerde herhangi bir tahrifat yapılmadığını, başkalarının eserlerinden yararlanılması durumunda bilimsel normlara uygun olarak atıfta bulunulduğunu, tezde yer alan verilerin bu üniversite veya başka bir üniversitede herhangi bir tez çalışmasında kullanılmadığını beyan ederim. Uğur Erkin KOCAMAZ 07.11.2018 TEŞEKKÜR Bu çalışmanın ortaya çıkmasında bana yardımcı olan, ilgi ve desteğini hiç eksiltmeyen, yol gösterici olan, fikirlerime önem veren, engin bilgi ve tecrübesiyle beni yönlendiren değerli danışmanım Doç. Dr. Yılmaz UYAROĞLU’na teşekkür ederim. Benim bu aşamaya gelmemde en çok emeği geçen, her zaman maddi ve manevi desteklerini arkamda hissettiğim annem babam Cemile ve Numan KOCAMAZ’a ve kardeşim Çiğdem ALTUN’a en içten saygı, sevgi ve şükranlarımı sunarım. i İÇİNDEKİLER TEŞEKKÜR ..……………………………………………………………………… i İÇİNDEKİLER …………………………………………………………………....
    [Show full text]
  • Itinerary Synchronization Between PWL Systems Coupled with Unidirectional Links
    Itinerary synchronization between PWL systems coupled with unidirectional links A. Anzo-Hern´andeza, E. Campos-Cant´onb∗ and Matthew Nicolc aC´atedrasCONACYT - Benem´eritaUniversidad Aut´onoma de Puebla, Facultad de Ciencias F´ısico-Matematicas,´ Benemerita´ Universidad Autonoma´ de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, 72570. Puebla, Puebla, Mexico.´ [email protected] bDivision´ de Matematicas´ Aplicadas, Instituto Potosino de Investigaci´onCient´ıficay Tecnol´ogicaA.C. Camino a la Presa San Jose´ 2055 col. Lomas 4a Seccion,´ 78216, San Luis Potos´ı, SLP, Mexico.´ [email protected],∗Corresponding author. cMathematics Department, University of Houston, Houston, Texas, 77204-3008, USA. [email protected]. 1 Abstract 2 In this paper the collective dynamics of N-coupled piecewise linear 3 (PWL) systems with different number of scrolls is studied. The cou- 4 pling is in a master-slave sequence configuration, with this type of cou- 5 pling we investigate the synchrony behavior of a ring-connected network 6 and a chain-connected network both with unidirectional links. Itinerary 7 synchronization is used to detect synchrony behavior. Itinerary synchro- 8 nization is defined in terms of the symbolic dynamics arising by assigning 9 different numbers to the regions where the scrolls are generated. A weaker 10 variant of this notion, -itinerary synchronization is also introduced and 11 numerically investigated. It is shown that in certain parameter regimes 12 if the inner connection between nodes takes account of all the state vari- 13 ables of the system (by which we mean that the inner coupling matrix 14 is the identity matrix), then itinerary synchronization occurs and the co- 15 ordinate motion is determined by the node with the smallest number of 16 scrolls.
    [Show full text]
  • Coexistence of Attractors in Integer- and Fractional-Order
    Pramana – J. Phys. (2019) 93:12 © Indian Academy of Sciences https://doi.org/10.1007/s12043-019-1786-3 Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation SIFEU TAKOUGANG KINGNI1, JUSTIN ROGER MBOUPDA PONE2,∗, GAETAN FAUTSO KUIATE3 and VIET-THANH PHAM4,5 1Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon 2Department of Electrical Engineering, Fotso Victor University Institute of Technology (IUT-FV), University of Dschang, P.O. Box 134, Bandjoun, Cameroon 3Department of Physics, Higher Teacher Training College, University of Bamenda, P.O. Box 39, Bamenda, Cameroon 4Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Yen Nghia, Ha Dong District, Hanoi 100000, Vietnam 5Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, 167 Hoang Ngan, Hanoi 100000, Vietnam ∗Corresponding author. E-mail: [email protected] MS received 23 August 2018; revised 7 December 2018; accepted 13 December 2018; published online 8 May 2019 Abstract. This paper reports the results of the analytical, numerical and analogical analyses of integer- and fractional-order chaotic systems with hyperbolic sine nonlinearity (HSN). By varying a parameter, the integer order of the system displays transcritical bifurcation and new complex shapes of bistable double-scroll chaotic attractors and four-scroll chaotic attractors. The coexistence among four-scroll chaotic attractors, a pair of double-scroll chaotic attractors and a pair of point attractors is also reported for specific parameter values. Numerical results indicate that commensurate and incommensurate fractional orders of the systems display bistable double-scroll chaotic attractors, four-scroll chaotic attractors and coexisting attractors between a pair of double-scroll chaotic attractors and a pair of point attractors.
    [Show full text]
  • Editorial Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability
    Hindawi Complexity Volume 2020, Article ID 8510930, 7 pages https://doi.org/10.1155/2020/8510930 Editorial Complexity, Dynamics, Control, and Applications of Nonlinear Systems with Multistability Viet-Thanh Pham ,1,2,3 Sundarapandian Vaidyanathan ,4 and Tomasz Kapitaniak3 1Faculty of Electrical and Electronic Engineering, Phenikaa Institute for Advanced Study (PIAS), Phenikaa University, Yen Nghia, Ha Dong District, Hanoi 100000, Vietnam 2Phenikaa Research and Technology Institute (PRATI), A&A Green Phoenix Group, 167 Hoang Ngan, Hanoi 100000, Vietnam 3Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, Lodz 90-924, Poland 4Research and Development Centre, Vel Tech University, No. 42, Avadi-Vel Tech Road, Avadi, Chennai, Tamil Nadu 600062, India Correspondence should be addressed to Viet-anh Pham; [email protected] Received 4 June 2020; Accepted 5 June 2020; Published 4 September 2020 Copyright © 2020 Viet-anh Pham 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. Multistability is a critical property of nonlinear dynamical complex and challenging task. Furthermore, circuit reali- systems, where a variety of phenomena such as coexisting zations (simulations and hardware design) of multistable attractors can appear for the same parameters but with systems are useful for various practical applications in different initial conditions. e flexibility in the system’s engineering. performance can be achieved without changing parameters. is special issue aims to introduce and discuss novel Complex dynamics have been observed in multistable sys- results, control techniques, and circuit simulations for tems, and we have witnessed systems with multistability in complex nonlinear systems with multistability.
    [Show full text]
  • A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers
    June 20, 2017 15:16 WSPC/S0218-1274 1750091 International Journal of Bifurcation and Chaos, Vol. 27, No. 6 (2017) 1750091 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127417500912 A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers Chunhua Wang, Hu Xia∗ and Ling Zhou College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, P. R. China ∗[email protected] Received November 24, 2016; Revised March 26, 2017 A memristor is the fourth circuit element, which has wide applications in chaos generation. In this paper, a four-dimensional hyperchaotic jerk system based on a memristor is proposed, where the scroll number of the memristive jerk system is controllable. The new system is constructed by introducing one extra flux-controlled memristor into three-dimensional multiscroll jerk sys- tem. We can get different scroll attractors by varying the strength of memristor in this system without changing the circuit structure. Such a method for controlling the scroll number without changing the circuit structure is very important in designing the modern circuits and systems. The new memristive jerk system can exhibit a hyperchaotic attractor, which has more com- plex dynamic behavior. Furthermore, coexisting attractors are observed in the system. Phase portraits, dissipativity, equilibria, Lyapunov exponents and bifurcation diagrams are analyzed. Finally, the circuit implementation is carried out to verify the new system. Keywords: Hyperchaos; multiscroll attractor; memristor; circuit implementation. 1. Introduction they belong to common chaotic systems. On the In recent years, the generation of chaotic attrac- other hand, switch is used to control nonlinear func- tors has become a hot topic in the investigation of tion, which decides the number of saddle focal bal- ance index of 2, so as to adjust the scroll number Int.
    [Show full text]
  • Generation of Multi-Scroll Attractors Without Equilibria Via Piecewise Linear Systems
    Generation of Multi-Scroll Attractors Without Equilibria Via Piecewise Linear Systems R.J. Escalante-Gonz´alez,1, a) E. Campos-Cant´on,1, b) and Matthew Nicol2, c) 1)Divisi´onde Matem´aticas Aplicadas, Instituto Potosino de Investigaci´on Cient´ıfica y Tecnol´ogica A. C., Camino a la Presa San Jos´e2055, Col. Lomas 4 Secci´on,C.P. 78216, San Luis Potos´ı, S.L.P., M´exico. 2)Mathematics Department, University of Houston, Houston, Texas 77204-3008, USA. (Dated: 14 March 2017) In this paper we present a new class of dynamical system without equilibria which possesses a multi scroll attractor. It is a piecewise-linear (PWL) system which is simple, stable, displays chaotic behavior and serves as a model for analogous non- linear systems. We test for chaos using the 0-1 Test for Chaos of Ref.12. a)Electronic mail: [email protected] b)Electronic mail: [email protected] c)Electronic mail: [email protected] 1 Piecewise-linear (PWL) systems are switching systems composed of linear affine subsystems along with a rule which determines the acting subsystem. These systems are known to be capable of producing chaotic attractors, as in the well studied Chua's circuit. This paper discusses the generation of multiscroll attractors without equilibria based on PWL systems. I. INTRODUCTION In recent years the study of dynamical systems with complicated dynamics but with- out equilibria has attracted attention. Since the first dynamical system of this kind with a chaotic attractor was introduced in Ref.1 (Sprott case A), several works have investigated this topic.
    [Show full text]
  • Správa O Činnosti Organizácie SAV Za Rok 2019
    Matematický ústav SAV Správa o činnosti organizácie SAV za rok 2019 Bratislava január 2020 Obsah 1. Základné údaje o organizácii 2. Vedecká činnosť 3. Doktorandské štúdium, iná pedagogická činnosť a budovanie ľudských zdrojov pre vedu a techniku 4. Medzinárodná vedecká spolupráca 5. Koncepcia dlhodobého rozvoja organizácie 6. Spolupráca s VŠ a inými subjektmi v oblasti vedy a techniky 7. Aplikácia výsledkov výskumu v spoločenskej a hospodárskej praxi 8. Aktivity pre Národnú radu SR, vládu SR, ústredné orgány štátnej správy SR a iné organizácie 9. Vedecko-organizačné a popularizačné aktivity 10. Činnosť knižnično-informačného pracoviska 11. Aktivity v orgánoch SAV 12. Hospodárenie organizácie 13. Nadácie a fondy pri organizácii SAV 14. Iné významné činnosti organizácie SAV 15. Vyznamenania, ocenenia a ceny udelené organizácii a pracovníkom organizácie SAV 16. Poskytovanie informácií v súlade so zákonom o slobodnom prístupe k informáciám 17. Problémy a podnety pre činnosť SAV PRÍLOHY A Zoznam zamestnancov a doktorandov organizácie k 31.12.2019 B Projekty riešené v organizácii C Publikačná činnosť organizácie D Údaje o pedagogickej činnosti organizácie E Medzinárodná mobilita organizácie F Vedecko-popularizačná činnosť pracovníkov organizácie SAV Správa o činnosti organizácie SAV 1. Základné údaje o organizácii 1.1. Kontaktné údaje Názov: Matematický ústav SAV Riaditeľ: doc. RNDr. Karol Nemoga, CSc. Zástupca riaditeľa: prof. RNDr. Anatolij Dvurečenskij, DrSc. Vedecký tajomník: Mgr. Marek Hyčko, PhD. Predseda vedeckej rady: Mgr. Anna Jenčová,
    [Show full text]
  • A Novel Multiscroll Chaotic System and Its Realization
    A Novel Multiscroll Chaotic System and Its Realization Simin Yu Jinhu L¨u Guanrong Chen College of Automation Institute of Systems Science Department of Electronic Engineering Guangdong University of Technology Academy of Mathematics and Systems Science City University of Hong Kong Guangzhou 510006, P. R. China Chinese Academy of Sciences Hong Kong, P. R. China Beijing 100080, P. R. China Email: [email protected] Email: [email protected] Abstract— This paper proposes a novel multiscroll chaotic sys- The paper is organized as follows. Section II introduces the tem, which is different from Chua’s circuit and all its variants in new multiscroll chaotic system, with its dynamical behaviors most aspects of the algebraic form, circuit design, and geometrical analyzed in Section III. Section IV designs a module-based structure of the attractor. In particular, the multiscroll attractor of this new system is more complex than that of the generalized circuit diagram for implementing the multiscroll chaotic at- Chua’s circuit when they both have the same number of scrolls. tractors. Section V concludes the paper. The dynamical behaviors of the new system are then analyzed, including the bifurcation diagram and the Lyapunov exponent II. A NOVEL MULTISCROLL CHAOTIC SYSTEM spectra. Moreover, a module-based circuit diagram is designed The proposed multiscroll chaotic system is described by for realizing various multiscroll attractors. Finally, experimental ⎧ dx circuits are implemented with physical observations reported. ⎨ dτ = βy− x − αf(x) dy = βx− γz (1) I. INTRODUCTION ⎩ dτ dz = ξy− z, Chua’s circuit is an important paradigm in nonlinear circuit dτ theory [1].
    [Show full text]
  • Infinite-Scroll Attractor Generated by the Complex Pendulum Model
    Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 368150, 3 pages http://dx.doi.org/10.1155/2013/368150 Research Article Infinite-Scroll Attractor Generated by the Complex Pendulum Model Sachin Bhalekar Department of Mathematics, Shivaji University, Vidyanagar, Kolhapur 416004, India Correspondence should be addressed to Sachin Bhalekar; [email protected] Received 20 November 2012; Revised 4 February 2013; Accepted 12 February 2013 Academic Editor: Rodica Costin Copyright © 2013 Sachin Bhalekar. This 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. We report the finding of the simple nonlinear autonomous system exhibiting infinite-scroll attractor. The system is generated from the pendulum equation with complex-valued function. The proposed system is having infinitely many saddle points of index two which are responsible for the infinite-scroll attractor. 1. Introduction where >0is constant and is a real valued function. We propose a complex version of (1)givenby A variety of natural systems show a chaotic (aperiodic) behaviour. Such systems depend sensitively on initial data, =−̈ sin () , (2) and one cannot predict the future of the solutions. There are various chaotic systems such as the Lorenz system [1], the where () = () + () is a complex-valued function. The Rossler system [2], the Chen system [3], and the Lusystem[¨ 4] system (2) gives rise to a coupled nonlinear system where the dependent variables are the real-valued functions. Though the chaos has been intensively studied over the past =−̈ sin () cosh () , several decades, very few articles are devoted to study the (3) complex dynamical systems.
    [Show full text]
  • Multiscroll in Coupled Double Scroll Type Oscillators
    MULTISCROLL IN COUPLED DOUBLE SCROLL TYPE OSCILLATORS SYAMAL KUMAR DANA Central Instrumentation, Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India [email protected] BRAJENDRA K. SINGH Department of Infectious Disease Epidemiology, Imperial College London, St Mary's Campus, London W2 1PG, UK [email protected] SATYABRATA.CHAKRABORTY1, RAM CHANDRA YADAV2 Central Instrumentation, Indian Institute of Chemical Biology, Jadavpur, Kolkata 700032, India [email protected], [email protected] JÜRGEN KURTHS Institute of Physics, University of Potsdam, Am Neuen Palais, D-14415 Potsdam, Germany [email protected] GREGORY V. OSIPOV Faculty of Computational Mathematics and Cybernetics, University of Nizhny Novgorod, 603950 Nizhny Novogorod, Russia [email protected] PRODYOT KUMAR ROY Department of Physics, Presidency College, Kolkata 700073, India [email protected] CHIN-KUN HU Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan and Center for Nonlinear and Complex Systems and Department of Physics, Chung-Yuan Christian University, Chungli 32023, Taiwan [email protected] Abstract: A unidirectional coupling scheme is investigated in double scroll type chaotic oscillators that reveal interesting multiscroll dynamics. Instead of using self-oscillatory systems, in this scheme, double scroll chaos from one oscillator is forced into another similar oscillator in a resting state. This coupling scheme is explored in the Chua oscillator, a modified Chua oscillator and the Lorenz oscillator. We have modified the Chua oscillator by simply changing its piecewise linear function a little bit and thereby derived a new 3-scroll attractor. We have observed 4-scroll, 6-scroll attractors in the driven Chua oscillator and the modified Chua oscillator respectively in an intermittency regime of weaker coupling.
    [Show full text]
  • Implementation of a New Memristor-Based Multiscroll Hyperchaotic System
    Pramana – J. Phys. (2017) 88: 34 c Indian Academy of Sciences DOI 10.1007/s12043-016-1342-3 Implementation of a new memristor-based multiscroll hyperchaotic system CHUNHUA WANG, HU XIA∗ and LING ZHOU College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China ∗Corresponding author. E-mail: [email protected] MS received 14 December 2015; revised 23 June 2016; accepted 29 July 2016; published online 17 January 2017 Abstract. In this paper, a new type of flux-controlled memristor model with fifth-order flux polynomials is presented. An equivalent circuit which realizes the action of higher-order flux-controlled memristor is also pro- posed. We use the memristor model to establish a memristor-based four-dimensional (4D) chaotic system, which can generate three-scroll chaotic attractor. By adjusting the system parameters, the proposed chaotic system performs hyperchaos. Phase portraits, Lyapunov exponents, bifurcation diagram, equilibrium points and stabil- ity analysis have been used to research the basic dynamics of this chaotic system. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design. Keywords. Memristor; hyperchaos; three-scroll chaotic attractor; circuit implementation. PACS Nos 05.40.Jc; 05.45.Pq 1. Introduction So far, memristor-based chaotic systems are lim- ited to single-scroll and double-scroll, and so chaotic Leon Chua proposed the concept of memristor as dynamics was not complicated. Memristor-based mul- the fourth circuit element in 1971 [1]. It represents a tiscroll chaotic systems have been rarely found. In relationship between the flux and the charge. Memristor 2014, Li [19] presented a memristor-based three-scroll is a two-terminal element with variable resistance called chaotic system.
    [Show full text]