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A New 3-D Memristive Time-delay Chaotic System with Multi-scroll and Hidden Attractors
5th International Conference on Advances in Control and 5th International Conference on 5th International Conference on Advances Advances in in Control Control and and Optimization of Dynamical Systems 5th International Conference on Advances in Control Optimization of Systems Optimization of Dynamical Dynamical Systems Available onlineand at www.sciencedirect.com February 18-22, 2018. Hyderabad, India 5th International Conference on Advances in Control and Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India 5th International Conference on Advances in Control and February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India
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IFAC PapersOnLine 51-1 (2018) 580–585
A A New New 3-D 3-D Memristive Memristive Time-delay Time-delay Chaotic Chaotic System System with with Multi-scroll Multi-scroll and and Hidden Hidden A New 3-D Memristive Time-delay Chaotic System with Multi-scroll and Hidden Attractors A Chaotic System with Multi-scroll and Hidden A New New 3-D 3-D Memristive Memristive Time-delay Time-delay Attractors Chaotic System with Multi-scroll and Hidden Attractors 1 1 Attractors Nalini Prasad Mohanty1*,Rajeeb Dey1**, Attractors
1*,Rajeeb Dey1**, Nalini Mohanty 1 *,Rajeeb Dey1**, Nalini Prasad Prasad Mohanty BinoyMohanty Krishna11Roy 1 1*** Nalini Prasad *,Rajeeb *** Dey1**, Binoy Krishna BinoyMohanty Krishna1Roy Roy *,Rajeeb Nalini Prasad 1*** Dey1**, *,Rajeeb Dey **, Nalini Prasad Mohanty Binoy Krishna Roy *** Binoy Krishna Roy11*** 1 National Institute of Technology Silchar, Silchar788010, Assam Binoy Krishna Roy 1 1National Institute of Technology Silchar,*** Silchar788010, Assam National Institute of Technology Silchar, Silchar788010, Assam 1 , ** [email protected] , ***[email protected]) (E-mail:*[email protected] Silchar, Silchar788010, Assam 1National Institute of Technology (E-mail:*[email protected] , ** [email protected] , *** [email protected]) , **[email protected] , ***[email protected]) (E-mail:*[email protected] Silchar, Silchar788010, Assam 1National Institute of Technology National Institute of Technology Silchar, Silchar788010, Assam , **[email protected] , *** [email protected]) (E-mail:*[email protected] (E-mail:*[email protected], **[email protected], ***[email protected]) , ** [email protected] , *** [email protected]) (E-mail:*[email protected] Abstract: In the recent era, memristor and time-delay are potential candidates for the construction of Abstract: In In the the recent recent era, era, memristor memristor and and time-delay time-delay are are potential potential candidates candidates for for the the construction construction of of Abstract: complex dynamical systems with special and features. This paper proposescandidates a novel time-delay chaotic system Abstract: In the recent era, memristor time-delay are potential for the construction of complex dynamical systems with special features. This paper proposes a novel time-delay chaotic system complex dynamical special features. This paper proposes a novel time-delay chaotic system Abstract: In theofrecent era, with memristor are potential candidates for thechaotic construction of in the presence asystems memristive device.and Thetime-delay proposed system generates multi-scroll attractors Abstract: In theofrecent era, with memristor are potential candidates for thechaotic construction of complex dynamical special features. This proposes aa novel time-delay chaotic system in the presence presence memristive device.and Thetime-delay proposed system generates multi-scroll attractors in the of aasystems memristive device. The proposed system generates multi-scroll chaotic attractors complex dynamical systems with special features. This paper paper proposes novel time-delay chaotic system without any equilibrium point. Such nature of a system is not reported in the literature yet to the best of complex dynamical systems with special features. This paper proposes a novel time-delay chaotic system in the presence of a memristive device. The proposed system generates multi-scroll chaotic attractors without any equilibrium equilibrium point. Such Such nature of proposed system is issystem not reported reported in the the literature chaotic yet to to the the best of of without any point. of aa system not in literature yet best in the presence of a The memristive device. generates multi-scroll attractors authors’ knowledge. existence ofnature richThe chaotic behaviour in the proposed system ischaotic investigated by in the presence of a The memristive device. proposed system generates multi-scroll attractors without any equilibrium point. Such of aa system is not reported in the literature to the best of authors’ knowledge. existence ofnature richThe chaotic behaviour in the proposed proposed system is isyet investigated by authors’ knowledge. The existence of rich chaotic behaviour in the system investigated by without any equilibrium point. Such nature of system is not reported in the literature yet to the best of different numerical tools whichSuch include bifurcation diagram, frequency instantaneous phase without any equilibrium point. nature of a system is not in reported in spectrum, the literature yet to the best of authors’ knowledge. The of chaotic the system is by different numerical tools which include include bifurcation diagram, frequency spectrum, instantaneous phase different numerical tools which bifurcation diagram, frequency spectrum, instantaneous phase authors’ knowledge. The existence existence of rich rich chaotic behaviour behaviour in the proposed proposed system is investigated investigated by plot and attractors plot. authors’ knowledge. The existence of rich chaotic behaviour in the proposed system is investigated by different numerical tools which include bifurcation diagram, frequency spectrum, instantaneous phase plot and and attractors attractors plot. plot different numericalplot. tools which include bifurcation diagram, frequency spectrum, instantaneous phase different numerical toolssystem, which include bifurcation diagram, frequency spectrum, instantaneous phase plot and attractors plot. Keywords: New chaotic time-delay, memristor, hidden attractors, no equilibrium © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. plot and attractors plot. Keywords: New chaotic system, time-delay, memristor, hidden attractors, no equilibrium equilibrium Keywords: New chaotic system, time-delay, memristor, hidden attractors, no plot and attractors plot. Keywords: Keywords: New New chaotic chaotic system, system, time-delay, time-delay, memristor, memristor, hidden hidden attractors, attractors, no no equilibrium equilibrium Keywords: New chaotic system, time-delay, memristor,al., hidden equilibrium 2014;attractors, Banerjee,no2013; Wang, Duan, 2012; Pham et al., al., Banerjee, 2013; 1. INTRODUCTION al., 2014; 2014; Banerjee, 2013; Wang, Wang, Duan, Duan, 2012; 2012; Pham Pham et et al., al., 2013, Pham et al., 2016). 1. INTRODUCTION 1. INTRODUCTION al., 2014; Banerjee, 2013; Wang, Duan, 2012; Pham et al., 2013, Pham et al., 2016). 2013, Pham et al., 2016). al., 2014; Banerjee, 2013; Wang, Duan, 2012; Pham et al., 1. INTRODUCTION There has been increasing interest on chaotic systems 2013, al., 2014;dynamics Banerjee, 2013; Wang, Duan, 2012; Pham et al., 1. INTRODUCTION et can be divided as self-excited attractors There has been been increasing increasing interest on on chaotic chaotic systems systems Chaotic There has interest 2013, Pham Pham et al., al., 2016). 2016). 1. INTRODUCTION dynamics can be as attractors because of their broad range of applications, such assystems secure Chaotic Chaotic dynamics can(Leonov be divided divided as self-excited self-excited attractors 2013, Pham et al., 2016). There has been increasing interest on chaotic and hidden attractors et al., 2011; Leonov et al., because of their broad range of applications, such as secure because of their range ofinterest applications, such as(Pecora, secure Chaotic There has beenbroad increasing on robotics chaotic systems dynamics can be as self-excited attractors and attractors et 2011; et communication, biology, cryptography, and hidden hidden attractors (Leonov et al., al., 2011;etLeonov Leonov et al., al., Chaotic dynamics can (Leonov be divided divided as self-excited attractors There has beenbroad increasing interest on robotics chaotic systems because of their range of applications, such as secure 2012; Leonov and Kuznetsov, 2013; Leonov al., 2014).The communication, biology, cryptography, (Pecora, communication, biology, cryptography, robotics Chaotic dynamics can (Leonov be divided as self-excited attractors because of their Boccaletti broad range ofal., applications, such as(Pecora, secure and hidden attractors et al., 2011; Leonov et al., 2012; Leonov and Kuznetsov, 2013; Leonov et al., 2014).The Carroll 1990; et 2002; Strogatz et al., 2012; Leonov and Kuznetsov, 2013; Leonov et al., 2014).The and hidden attractors (Leonov et al., 2011; Leonov et al., because of their broad range of applications, such as secure communication, biology, cryptography, (Pecora, attractor belongs to the 2013; category of systems having no Carroll Boccaletti et al., 2002; Strogatz et Carroll 1990; 1990; chaotic Boccaletti et are al.,introduced 2002; robotics Strogatz et al., al., hidden and hidden attractors (Leonov et al., 2011; Leonov et al., communication, biology, cryptography, robotics (Pecora, 2012; and Kuznetsov, Leonov et al., hidden attractor belongs to the category of having no 1994).Various systems in the literature hiddenLeonov attractor belongs tostable the 2013; category of systems systems having no 2012; Leonov and Kuznetsov, Leonov etpoints al., 2014).The 2014).The communication, biology, cryptography, robotics (Pecora, Carroll 1990; Boccaletti et al., 2002; Strogatz et al., equilibrium point, only equilibrium and an 1994).Various chaotic systems are introduced in the literature 1994).Various chaotic systems are in the literature Leonov and Kuznetsov, 2013; Leonov etpoints al., 2014).The Carroll 1990; Boccaletti et1963), al.,introduced 2002; Strogatz et al., 2012; hidden attractor belongs to the category of systems having no equilibrium point, only stable equilibrium and an like Lorenz system (Lorenz, Rössler System (Rössle, equilibrium point, only stable equilibrium points and an hidden attractor belongs to the category of systems having no Carroll 1990; Boccaletti et al., 2002; Strogatz et al., 1994).Various chaotic systems are introduced in the literature infinite number of equilibrium points(Wang and Chen, 2013; like Lorenz system (Lorenz, 1963), Rössler System (Rössle, like Lorenz system (Lorenz, 1963), Rössler System (Rössle, hidden attractor belongs tostable the category of systems having no 1994).Various chaotic (Matsumoto, systems are introduced in System the literature equilibrium point, only equilibrium points and an infinite number of equilibrium points(Wang and Chen, 2013; 1976), Chua system 1984), Lü (Lü, infinite number of equilibrium points(Wang and Chen, 2013; equilibrium point, only stable equilibrium points and an 1994).Various chaotic systems are introduced in the literature like Lorenz system (Lorenz, 1963), Rössler System (Rössle, and Sprott, 2013). A time-delay 3-Dand chaotic system 1976), Chua system (Matsumoto, Lü System (Lü, 1976), Chuasystem system (Matsumoto, 1984), Lü System (Lü, Jafari equilibrium point, only stable equilibrium points and an like Lorenz 1963), 1984), Rössler1999), System infinite number of points(Wang Chen, 2013; Jafari and Sprott, 2013). A time-delay 3-D chaotic system Chen, 2002), Chen (Lorenz, system (Chen, Ueta, etc.(Rössle, These Jafarihidden and Sprott, 2013). A new time-delay 3-D chaotic system infinite number of equilibrium equilibrium points(Wang and Chen, 2013; like Lorenz system (Lorenz, 1963), 1984), Rössler1999), System (Rössle, 1976), Chua system (Matsumoto, Lü System (Lü, with attractors is aA research3-D topic. Memristive Chen, 2002), Chen system (Chen, Ueta, etc. These Chen, 2002), Chen system (Chen, Ueta, 1999), etc. These infinite number of equilibrium points(Wang and Chen, 2013; 1976), Chua system (Matsumoto, 1984), Lü System (Lü, Jafari and Sprott, 2013). time-delay chaotic system with attractors is aaA new research topic. Memristive classical chaotic systems have a 1984), countable number of Jafari with hidden hidden attractors is new research topic. Memristive and Sprott, 2013). time-delay 3-D chaotic system 1976), 2002), Chua system (Matsumoto, Lü System (Lü, Chen, Chen system (Chen, 1999), etc. These time-delay 3-D chaotic systems with hidden attractors (being classical chaotic aa Ueta, countable number of classical chaotic systems have countable number of Jafari chaotic system and Sprott, 2013). A new time-delay 3-D Chen, 2002), Chensystems system have (Chen, Ueta, 1999), etc.can These with hidden attractors is a research topic. Memristive time-delay 3-D chaotic systems with hidden attractors (being equilibrium points. Hence, chaos in such systems be time-delay 3-D chaotic systems with hidden attractors (being withequilibrium hidden attractors is are a new research topic. Memristive Chen, 2002), Chensystems system have (Chen, Ueta, 1999), etc.can These classical chaotic a countable number of no point) not found in the literature. equilibrium points. Hence, chaos in such systems be equilibrium points. Hence, chaos in such systems can be with hidden attractors is a new research topic. Memristive classical chaotic systems have a countable number of time-delay 3-D chaotic systems with hidden attractors (being no equilibrium point) are not found in the literature. discovered by the conventional criteria (Shil'nikov, 2001) but no equilibrium point) are not found in the literature. time-delay 3-D chaotic systems with hidden attractors (being classical chaotic systems have a countable number of equilibrium Hence, in systems can be by abovesystems fact, this paper attempts construct discovered by the criteria (Shil'nikov, 2001) discovered bypoints. the conventional conventional criteria (Shil'nikov, 2001) but time-delay 3-Dthe chaotic hidden (being equilibrium points. Hence,itchaos chaos inbesuch such can but be Fascinated notwith found inattractors theto literature. no equilibrium are by above fact, paper attempts to construct in a no-equilibrium system, cannot used.systems Fascinated by the the point) above fact, this paper attempts to literature. construct no equilibrium point) are this not found in the equilibrium points. Hence,itchaos inbesuch systems can but be aFascinated discovered by the conventional criteria (Shil'nikov, 2001) memristive-based time-delay chaotic system having multiin a no-equilibrium system, cannot used. in a no-equilibrium system, it cannot be (Shil'nikov, used. nomemristive-based equilibrium point) are not found in the literature. discovered by the conventional criteria 2001) but aaFascinated by the above fact, this paper attempts to construct chaotic system having multitime-delay memristive-based time-delay chaotic multiFascinated byattractors. the above fact, this paper system attemptshaving to construct discovered byO.the conventional criteria (Shil'nikov, 2001) but scroll in a no-equilibrium system, it cannot be used. hidden In 1971, L. Chua introduced a new two-terminal circuit byattractors. the above fact, this paper system attemptshaving to construct in a1971, no-equilibrium system, it cannot be used. aFascinated memristive-based time-delay chaotic multiscroll hidden In L. O. Chua introduced aa new two-terminal circuit scroll hidden attractors. In 1971, L. O. Chua introduced new two-terminal circuit a memristive-based time-delay chaotic system having multiin a no-equilibrium system, it cannot be used. element named as memristor, which is two-terminal characterisedcircuit by a scroll a memristive-based time-delayaschaotic having hidden attractors. In 1971, L. Chua introduced aa new rest of paper is organised follows.system In Section 2, amultibrief element as which characterised by element named as memristor, memristor, which isflux characterised by aa The scroll hidden attractors. In 1971, named L. O. O. Chua introduced newis two-terminal circuit The rest of paper is organised as follows. In Section 2, aa brief brief relationship between the charge and linkage (Chua, The rest of paper is organised as follows. In Section 2, scroll hidden attractors. In 1971, L. O. Chua introduced a new two-terminal circuit element named as memristor, which is characterised by a introduction of a memristor is given. Section 3 deals with the relationship between the charge and flux linkage (Chua, relationship between the charge andwas linkage (Chua, element named as memristor, which isflux characterised by a The rest of paper is organised as follows. In Section 2,with a brief introduction of a memristor is given. Section 3 deals with the 1971). The realisation of memristor done at Hewlettintroduction of a memristor is given. Section 3 deals the The rest of paper is organised as follows. In Section 2, a brief element named as memristor, which is characterised by a relationship between the charge and flux linkage (Chua, proposed novel memristor-based time-delay chaotic system 1971). The realisation of memristor was done at Hewlett1971). The realisation of memristor at HewlettThe rest ofnovel paper is organised follows. In Section a brief relationship between charge andwas fluxdone linkage (Chua, introduction of a memristor-based memristor is as given. Section 3chaotic deals2,with the proposed memristor-based time-delay chaotic system Packard laboratory by the aof group of researchers (Strukov et al., introduction proposed novel time-delay system of a memristor is given. Section 3 deals with the relationship between the charge and flux linkage (Chua, 1971). The realisation memristor was done at Hewlettno equilibrium point. isIngiven. Section 4, basic dynamical Packard laboratory by of researchers (Strukov et PackardIt laboratory by aaarea group of researchers (Strukov et al., al., with introduction of a memristor-based memristor Section 3chaotic deals with the 1971). The realisation ofgroup memristor was at Hewlettproposed novel time-delay system with no no equilibrium equilibrium point. In In Section Section 4, basic basic dynamical 2008). opens a new of of research fordone memristor-based with point. 4, dynamical proposed novel memristor-based time-delay chaotic system 1971). The realisation ofgroup memristor was done at HewlettPackard laboratory by a researchers (Strukov et al., analysis of the system is discussed. In Section 5, numerical 2008). It opens a new area of research for memristor-based 2008). It opens a new area of research for memristor-based proposed novel memristor-based time-delay chaotic system Packard laboratory by a group of researchers (Strukov et al., with no equilibrium point. In Section 4, basic dynamical analysis of the system is discussed. In Section 5, numerical applications like neural network, adaptive filter, high-speed analysis the systempoint. isChaotic discussed. In Section 5, dynamical numerical with no of equilibrium In Section 4, of basic Packard laboratory by aarea group researchers (Strukov et al., analyses 2008). It new of of research for filter, memristor-based are presented. behaviour the system is applications like network, adaptive high-speed applications likeaa neural neural network, adaptive filter, high-speed with no of equilibrium In Section 4, of basic 2008). It opens opens newand area memristor-based analysis the systempoint. isChaotic discussed. In Section 5, dynamical numerical analyses are presented. Chaotic behaviour the system is is low power processor soof onresearch (Yang etfor al., 2013). For the analysis analyses are presented. behaviour of the system of the system is discussed. In Section 5, numerical 2008). It opens a new area of research for memristor-based applications like neural network, adaptive filter, high-speed confirmed by bifurcation plot, attractor analysis, low power processor and so on (Yang et al., 2013). For the low power processor and so on (Yang et al., 2013). For the analysis of the system is discussed. In Section 5, numerical applications like neural network, adaptive filter, high-speed analyses are presented. Chaotic behaviour of the system is confirmed by bifurcation plot, attractor analysis, inherent nonlinear nature of memristor, it has been widely confirmedare by plot, attractor analysis, analyses presented. behaviour of the system is applications like neural network, adaptiveital., filter, high-speed low power processor and (Yang For the phasebifurcation plotChaotic and frequency spectrum analysis. inherent nonlinear nature of memristor, has been widely inherent nonlinear nature ofofon memristor, has2013). been widely analyses are by presented. behaviour of the system is low power processor and so so on (Yang et et ital., 2013). the instantaneous confirmed bifurcation plot, attractor analysis, instantaneous phasebifurcation plotChaotic and frequency frequency spectrum analysis. used for nonlinear the construction chaotic withFor many instantaneous phase plot and spectrum analysis. confirmed by plot, attractor analysis, low power processor and so on (Yang etsystems al., 2013). For the Finally, inherent nature of memristor, it has been widely conclusions of the paper are drawn in Section 6. used for the construction of chaotic systems with many used for the construction of chaotic systems with many confirmed by bifurcation plot, attractor analysis, inherent nonlinear nature of memristor, it has been widely instantaneous phase plot and frequency spectrum analysis. Finally, conclusions conclusions ofplot the and paperfrequency are drawn drawnspectrum in Section Sectionanalysis. 6. features (Muthuswamy, Kokate, 2009; Muthuswamy, Finally, the paper are in 6. instantaneous phase of inherent nonlinear nature ofofmemristor, it has been 2010). widely used for the systems many features Kokate, 2009; features (Muthuswamy, Kokate, 2009; Muthuswamy, Muthuswamy, 2010). phase ofplot used for(Muthuswamy, the construction construction of chaotic chaotic systems with with2010). many instantaneous Finally, conclusions the and paperfrequency are MEMRISTOR drawnspectrum in Sectionanalysis. 6. 2. INTRODUCTION TO Finally, conclusions of the paper are drawn in Section 6. used for the construction of chaotic systems with many features (Muthuswamy, 2009; Muthuswamy, 2010). 2. INTRODUCTION INTRODUCTION TO MEMRISTOR Time-delay emerges in aKokate, dynamical due to switching 2. Finally, conclusions of the paperTO are MEMRISTOR drawn in Section 6. features (Muthuswamy, Kokate, 2009;system Muthuswamy, 2010). Time-delay emerges in dynamical due switching Time-delay emerges in aaKokate, dynamical system due to toTime-delay switching features (Muthuswamy, 2009;system Muthuswamy, 2010). 2. INTRODUCTION TO MEMRISTOR speed, processing speed, signal propagation time. After the invention by L.O.Chua in 1971(Chua, 1971), a 2. INTRODUCTION TO MEMRISTOR Time-delay emerges in a dynamical system due to switching After the the 2. invention by L.O.Chua L.O.Chua in 1971(Chua, 1971(Chua, 1971), 1971), aa speed, speed, propagation time. speed, processing processing speed, signal propagation time. Time-delay After invention in TO MEMRISTOR Time-delay emerges in active a signal dynamical due toTime-delay switching system has become an area ofsystem research and has varied memristor isINTRODUCTION knownby the fourth circuit 1971), element.a Time-delay emerges in active a signal dynamical system due toTime-delay switching After the invention byas L.O.Chua in basic 1971(Chua, speed, processing speed, propagation time. memristor is known knownby as L.O.Chua the fourth fourthin basic circuit 1971), element.a system has become an area of research and has varied system has become an active area of research and has varied memristor is as the basic circuit element. speed, processing speed, signal propagation time. Time-delay After the invention 1971(Chua, applications in the field of biology, physics and engineering Memristor deals with the relationship between charge anda After the invention by L.O.Chua in 1971(Chua, 1971), speed, processing speed, signal propagation time. Time-delay memristor is known as the fourth basic circuit element. system has become an active area of research and has varied Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering applications in the field of Yongzhen biology, and Memristor deals with thethe relationship between charge and system has become an active area ofphysics research andengineering hasSystems varied memristor is known as fourth basic circuit element. flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; et al., 2011). memristor is known as the fourth basic circuit element. system has become an active area of research and has varied Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; Yongzhen et al., 2011). Systems (Ikeda, Matsumoto, 1987; et al.,can 2011). Systems flux whichof arememristor. two fundamental circuit variables.and There are applications in athe field of Yongzhen biology, physics and engineering deals with theOne relationship between charge and two kinds is flux controlled another represented by delay differential equation exhibit chaos Memristor Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; Yongzhen et al., 2011). Systems two kinds kinds ofarememristor. memristor. One is is flux flux controlled and another represented by aa delay differential equation exhibit chaos represented by2010; delay differential equation can exhibit chaos flux two of One controlled and another (Ikeda, Matsumoto, 1987; Yongzhen et 2007). al.,can 2011). Systems which two fundamental circuit variables. There are one kinds is charge controlled. A flux physical memristor was (Kilinç al.,by Tamaševičius et al., The presence flux which two fundamental circuit variables. There are (Ikeda, et Matsumoto, 1987; Yongzhen et 2007). al.,can 2011). Systems two ofarememristor. One is controlled and another represented a delay differential equation exhibit chaos one is charge controlled. A physical memristor was (Kilinç et al., 2010; Tamaševičius et al., The presence (Kilinç et al., 2010; Tamaševičius et al., 2007). The presence one is charge controlled. A physical memristor was represented by a in delay differential equation can exhibitinfinite chaos two kinds offabricated memristor. One 2008 is flux(Strukov controlled and2008). another successfully during et al., In of time-delay a system makes the system two kinds of memristor. One is flux controlled and another represented by a delay differential equation can exhibit chaos one is charge controlled. A physical memristor was (Kilinç et al., 2010; Tamaševičius et al., 2007). The presence successfully fabricated during 2008 2008 (Strukov memristor et al., al., 2008). 2008).was In of in aaTamaševičius system makes system infinite of time-delay time-delay in system makes the system infinite successfully fabricated during (Strukov et In one work, is charge controlled. A physical (Kilinç et al.,and 2010; et nonlinearity, al., the 2007). The presence this a flux-controlled memristor which is characterised dimensional with the addition of it may lead one is charge controlled. A physical was (Kilinç et al.,and 2010; Tamaševičius et nonlinearity, al., the 2007). The presence successfully fabricated during 2008 (Strukov et al., In of time-delay system makes system infinite this work, flux-controlled memristor which memristor is characterised characterised dimensional with addition of it lead dimensional and in withaa the the addition of etc. nonlinearity, it may may lead this work, aa flux-controlled memristor which is successfully fabricated during 2008(Strukov (Strukov et al., 2008). 2008). In of time-delay in system makes the system infinite by its incremental memductance et al., 2008) In is to chaos, hyperchaos, multistability, Infinite dimensional successfully fabricated during 2008 (Strukov et al., 2008). of time-delay in a system makes the system infinite this work, a flux-controlled memristor which is characterised dimensional and with the addition of nonlinearity, it may lead by its incremental memductance (Strukov etis characterised al., 2008) 2008) is is to chaos, hyperchaos, multistability, etc. Infinite dimensional to chaos, hyperchaos, multistability, etc. Infinite dimensional by its incremental memductance (Strukov et al., this work, a flux-controlled memristor which dimensional and with the addition of nonlinearity, it may lead used. The nonlinear dynamics describing a flux-controlled dynamics of time-delay chaotic systems are reported in the this work, anonlinear flux-controlled memristor which isflux-controlled characterised dimensional and with the additionsystems of etc. nonlinearity, it mayinlead by incremental memductance (Strukov al., to chaos, multistability, Infinite dimensional used. The dynamics describing dynamics of chaotic are reported dynamics ofintime-delay time-delay chaotic systems are reported in the the used. dynamics describing by its its The incremental memductance (Strukov aaet etflux-controlled al., 2008) 2008) is is to chaos, hyperchaos, hyperchaos, multistability, etc. dimensional memristor isnonlinear described as literature (Srinivasan et al., 2011; LeInfinite et al., 2012; Valliet by its The incremental memductance (Strukov aetflux-controlled al., 2008) is to chaos, as hyperchaos, multistability, etc. Infinite dimensional used. nonlinear dynamics describing dynamics of time-delay chaotic systems are reported in the memristor is described as literature as in (Srinivasan et al., 2011; Le et al., 2012; Valliet literature asofintime-delay (Srinivasanchaotic et al., 2011; Le are et al., 2012; Valliet describeddynamics as used. The isnonlinear describing a flux-controlled dynamics systems reported in the memristor used. The isnonlinear dynamics describing a flux-controlled dynamics ofintime-delay chaotic systems areal., reported in the memristor as literature as et Le 2012; memristor is described described asrights reserved. literature © as2018, in (Srinivasan (Srinivasan et al., al., 2011; 2011; Le et et of al.,Automatic 2012; Valliet Valliet Copyright © 2018 IFAC 612Hosting 2405-8963 IFAC (International Federation Control) by Elsevier Ltd. All memristor is described as literature as in (Srinivasan et al., 2011; Le et al., 2012; Valliet Copyright © 2018 IFAC 612
Copyright 2018 responsibility IFAC 612Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 612 10.1016/j.ifacol.2018.05.097 Copyright © 2018 IFAC 612 Copyright © 2018 IFAC 612
5th International Conference on Advances in Control and Optimization of Dynamical Systems Nalini Prasad Mohanty et al. / IFAC PapersOnLine 51-1 (2018) 580–585 February 18-22, 2018. Hyderabad, India
i W ( )v
x y y x ay z by sin( z )
(1)
where i is the current through the memristor and v is the voltage across the memristor. The nonlinear term W () is the memductance defined by (Muthuswamy, 2010) as W ( )
dq( ) d
581
2 z 1 y W ( x) y
(6)
(2)
3. MEMRISTOR-BASED TIME-DELAY CHAOTIC SYSTEM WITH NO EQUILIBRIUM POINT In this section, we describe the dynamics of the novel timedelay chaotic system. 3.1 Memristor-based chaotic system without time-delay Fig. 1: Lyapunov Exponents ( ( ) ( ) ( )) ( )
To design a memristor-based chaotic system, we choose a cubic nonlinearity for the flux dependent function q( ) same as (Muthuswamy, 2010; Bao et al., 2010; Fitch et al., 2012) and given in (3).
q( ) c d 3 Thus, the nonlinearity function
W ( )
of
system
(5)
with
(3)
W ( )
is given as
dq( ) c 3d 2 d
(4)
where c and d are constants. To design the memristor-based time-delay chaotic system, we started with the 3D system given by Sajad Jafari in 2016 (Jafari et al., 2016). The memristor-based system is described as:
x y y x ayz by sin( z ) z 1 y 2 W ( x) y
10
(5)
where a and b are positive parameters, W (x) is the memductance as defined in (4). Here, the memristor-based chaotic system defined in (5) is a three-dimensional system. The Lyapunov exponents (LEs) of system (5) with , and initial conditions ( ) are ( ( ) ( ) ( )) which is given in Fig.1. So the Lyapunov ) is calculated as 3.0. The ( ) nature of dimension ( LEs conforms that the system in (5) is a chaotic system. The chaotic multi-scroll attractors of system (5) with same parameters and initial conditions are shown in Fig.2.
b
5
Z
0
-5
-1 0
3.2 Memristor-based chaotic system with time-delay
-1 5
-4
Based on the memristive chaotic system (5), a memristive time-delay chaotic system is proposed as follows: Fig. 2: Chaotic ( ( ) ( ) ( ))
-2
X
attractors ( )
0
of
2
system
4
(5)
with
are constants and is the time-delay, y denotes y(t ) . The maximum Lyapunov exponent of the where
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5th International Conference on Advances in Control and 582 Optimization of Dynamical Systems Nalini Prasad Mohanty et al. / IFAC PapersOnLine 51-1 (2018) 580–585 February 18-22, 2018. Hyderabad, India
system in (6) with a = 0.1, b = 2.9, c = 0.5, d = 0.01 and 0.2 is 0.136 which is calculated by the algorithm given in (Farmer, 1982). It can be noted that the proposed system is more chaotic than the system in (5) because it has the larger first Lyapunov exponent. The chaotic attractors of the proposed system in (6) with ( ( ) ( ) ( )) ( ), and 0.2 are shown in Fig.3. It is seen from Fig. 3 that the proposed system has multi-scroll chaotic attractors.
1 y 2 W ( x) y 0
(9)
From (7), we get y 0 which is inconsistent with (9). As a result, the proposed memristor-based time-delay system has no equilibrium point. The absence of an equilibrium point makes the proposed system significantly different from other time-delay systems reported in the literature. 5. NUMERICAL ANALYSIS OF THE MEMRISTORBASED TIME-DELAY CHAOTIC SYSTEM This section describes some numerical analysis of the proposed system in (6). 5.1 Bifurcation Analysis Bifurcation diagram is plotted by varying one of the parameters of the proposed system and keeping other parameters fixed. Figure 4 shows the bifurcation diagram for the variation of the parameter [0.15,0.3] and keeping . The bifurcation diagram by varying a [0,0.3] and keeping 0.2 is shown in Fig.5. The bifurcation diagram by varying b [0,5] and keeping
0.2 is shown in Fig.6. Figure 7 shows the bifurcation diagram for the variation of the parameter and keeping c [0,2] . The bifurcation diagram by varying 0.2 and keeping d [0,0.07] 0.2 is shown in Fig.8. It is seen from Fig. 4, Fig. 7 and Fig. 8 that the system has periodic and chaotic behaviour in the specified ranges. It is observed from Fig. 5 and Fig. 6 that the system has chaotic behaviour within the specified ranges.
Fig. 3: Chaotic ( ( ) ( ) ( ))
attractors ( )
of
system
(6)
with
4. BASIC DYNAMICAL ANALYSIS OF THE MEMRISTOR-BASED TIME-DELAY CHAOTIC SYSTEM
Fig. 4: Bifurcation diagram of system (6) with variation of parameter and ( ( ) ( ) ( )) ( )
The equilibrium points of the three-dimensional memristorbased time-delay system in (6) are calculated by solving (7) to (9).
y0
(7)
x ayz by sin( z) 0
(8) 614
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Fig. 5: Bifurcation diagram of system (6) with variation of parameter a and ( ( ) ( ) ( )) ( )
583
Fig. 8: Bifurcation diagram of system (6) with variation of parameter d and ( ( ) ( ) ( )) ( ) 5.2 Attractor Analysis It is apparent from the bifurcation diagram in Fig. 4, the system in (6) shows a periodic attractor, which is shown in Fig.9, with and . The bifurcation diagram in Fig.8 reveals that the system in (6) shows a periodic attractor with period two for the parameter , 0.2 and is shown in Fig. 10. 2 1
Fig. 6: Bifurcation diagram of system (6) with variation of parameter b and ( ( ) ( ) ( )) ( )
Y
0 -1 -2 -3 -1
0
Fig. 9: A periodic ( ( ) ( ) ( )) (
X attractor )
1
of
2
system
(6)
with
2 1
Y
0 -1 -2
Fig. 7: Bifurcation diagram of system (6) with variation of parameter c and ( ( ) ( ) ( )) ( )
-3 -4 -2
-1
0
X
1
2
Fig. 10:A periodic attractor with period two of system (6) with ( ( ) ( ) ( )) ( ) 615
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1
Spectrum of x
In this subsection, the instantaneous phase (IP) of the proposed system is calculated by using Hilbert transformation (HT) (Lauterborn, 1997). Here, we have used state of system (6) to calculate the instantaneous phase of the proposed system using MATLAB software. The IP of a chaotic system increases monotonically with respect to time, whereas for a periodic system it remains constant. The IP of the proposed system with and 0.2 for is shown in Fig. 11. It is seen from Fig. 11 the IP is increasing monotonically with respect to time. Thus, the proposed system is a chaotic system.
1
5.3 Instantaneous phase plot
a
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
Frequency (Hz)
Spectrum of x
1
Fig. 11: IP plot of state (𝑡𝑡) of system (8) using HT with ( ( ) ( ) ( ) ( )).
0.6 0.4 0.2 0
5.4 Frequency Spectrum analysis
b
0.8
0
0.5
1
Frequency (Hz)
1.5
In this subsection, we represent the randomness of a signal which is validated by its frequency spectrum. The randomness of a signal is another measure of chaoticness of a signal. Chaotic signals are wideband signals and can easily distinguish from regular/periodic signals by their frequency spectrum (Singh, Roy, 2015).
Fig. 12: (a) Frequency spectrum of state (𝑡𝑡) of system given in (Jafari et al., 2016) for and (b) Frequency spectrum of state (𝑡𝑡)of system (6) with and 0.2 , and ( ( ) ( ) ( ) ( )).
Here, we compare the frequency spectrum of the proposed system with the system given by (Jafari et al., 2016). Frequency spectrum of state x(t) of the system given in (Jafari et al., 2016) for a = 0.1, b = 2.9 is shown in Fig. 12 (a). Frequency spectrum of state (𝑡𝑡) of the system in (6) with and 0.2 , and ( ( ) ( ) ( ) ( )) given in Fig. 12 (b). By considering the spectrum of (𝑡𝑡) till 10% of its highest component, the system given in (Jafari et al., 2016) has 0.424 Hz whereas our proposed system has 0.685 Hz. Also the spectrum of proposed system is denser than the system given in (Jafari et al., 2016). Hence, the proposed system has wider spectrum compare with the system given in (Jafari et al., 2016).
6. CONCLUSIONS A memristor-based time-delay 3-D chaotic system with four nonlinear terms is reported in this paper. The proposed system has no equilibrium point which satisfies the characteristics of a hidden attractor system. The chaotic behaviour of the proposed system is analysed by various numerical tools such as bifurcation diagram, attractor analysis, instantaneous phase plot and frequency spectrum analysis. The first Lyapunov exponent of the proposed system is much more than the original memristor-based chaotic system considered in this paper. The frequency spectrum reveals the spectrum of the proposed system has wider band in comparison with the non-time-delay original system. REFERENCES Pecora, L.M. and Carroll, T.L., 1990. Synchronization in chaotic systems. Physical review letters, 64(8), p.821.
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IFAC PapersOnLine 51-1 (2018) 580–585
A A New New 3-D 3-D Memristive Memristive Time-delay Time-delay Chaotic Chaotic System System with with Multi-scroll Multi-scroll and and Hidden Hidden A New 3-D Memristive Time-delay Chaotic System with Multi-scroll and Hidden Attractors A Chaotic System with Multi-scroll and Hidden A New New 3-D 3-D Memristive Memristive Time-delay Time-delay Attractors Chaotic System with Multi-scroll and Hidden Attractors 1 1 Attractors Nalini Prasad Mohanty1*,Rajeeb Dey1**, Attractors
1*,Rajeeb Dey1**, Nalini Mohanty 1 *,Rajeeb Dey1**, Nalini Prasad Prasad Mohanty BinoyMohanty Krishna11Roy 1 1*** Nalini Prasad *,Rajeeb *** Dey1**, Binoy Krishna BinoyMohanty Krishna1Roy Roy *,Rajeeb Nalini Prasad 1*** Dey1**, *,Rajeeb Dey **, Nalini Prasad Mohanty Binoy Krishna Roy *** Binoy Krishna Roy11*** 1 National Institute of Technology Silchar, Silchar788010, Assam Binoy Krishna Roy 1 1National Institute of Technology Silchar,*** Silchar788010, Assam National Institute of Technology Silchar, Silchar788010, Assam 1 , ** [email protected] , ***[email protected]) (E-mail:*[email protected] Silchar, Silchar788010, Assam 1National Institute of Technology (E-mail:*[email protected] , ** [email protected] , *** [email protected]) , **[email protected] , ***[email protected]) (E-mail:*[email protected] Silchar, Silchar788010, Assam 1National Institute of Technology National Institute of Technology Silchar, Silchar788010, Assam , **[email protected] , *** [email protected]) (E-mail:*[email protected] (E-mail:*[email protected], **[email protected], ***[email protected]) , ** [email protected] , *** [email protected]) (E-mail:*[email protected] Abstract: In the recent era, memristor and time-delay are potential candidates for the construction of Abstract: In In the the recent recent era, era, memristor memristor and and time-delay time-delay are are potential potential candidates candidates for for the the construction construction of of Abstract: complex dynamical systems with special and features. This paper proposescandidates a novel time-delay chaotic system Abstract: In the recent era, memristor time-delay are potential for the construction of complex dynamical systems with special features. This paper proposes a novel time-delay chaotic system complex dynamical special features. This paper proposes a novel time-delay chaotic system Abstract: In theofrecent era, with memristor are potential candidates for thechaotic construction of in the presence asystems memristive device.and Thetime-delay proposed system generates multi-scroll attractors Abstract: In theofrecent era, with memristor are potential candidates for thechaotic construction of complex dynamical special features. This proposes aa novel time-delay chaotic system in the presence presence memristive device.and Thetime-delay proposed system generates multi-scroll attractors in the of aasystems memristive device. The proposed system generates multi-scroll chaotic attractors complex dynamical systems with special features. This paper paper proposes novel time-delay chaotic system without any equilibrium point. Such nature of a system is not reported in the literature yet to the best of complex dynamical systems with special features. This paper proposes a novel time-delay chaotic system in the presence of a memristive device. The proposed system generates multi-scroll chaotic attractors without any equilibrium equilibrium point. Such Such nature of proposed system is issystem not reported reported in the the literature chaotic yet to to the the best of of without any point. of aa system not in literature yet best in the presence of a The memristive device. generates multi-scroll attractors authors’ knowledge. existence ofnature richThe chaotic behaviour in the proposed system ischaotic investigated by in the presence of a The memristive device. proposed system generates multi-scroll attractors without any equilibrium point. Such of aa system is not reported in the literature to the best of authors’ knowledge. existence ofnature richThe chaotic behaviour in the proposed proposed system is isyet investigated by authors’ knowledge. The existence of rich chaotic behaviour in the system investigated by without any equilibrium point. Such nature of system is not reported in the literature yet to the best of different numerical tools whichSuch include bifurcation diagram, frequency instantaneous phase without any equilibrium point. nature of a system is not in reported in spectrum, the literature yet to the best of authors’ knowledge. The of chaotic the system is by different numerical tools which include include bifurcation diagram, frequency spectrum, instantaneous phase different numerical tools which bifurcation diagram, frequency spectrum, instantaneous phase authors’ knowledge. The existence existence of rich rich chaotic behaviour behaviour in the proposed proposed system is investigated investigated by plot and attractors plot. authors’ knowledge. The existence of rich chaotic behaviour in the proposed system is investigated by different numerical tools which include bifurcation diagram, frequency spectrum, instantaneous phase plot and and attractors attractors plot. plot different numericalplot. tools which include bifurcation diagram, frequency spectrum, instantaneous phase different numerical toolssystem, which include bifurcation diagram, frequency spectrum, instantaneous phase plot and attractors plot. Keywords: New chaotic time-delay, memristor, hidden attractors, no equilibrium © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. plot and attractors plot. Keywords: New chaotic system, time-delay, memristor, hidden attractors, no equilibrium equilibrium Keywords: New chaotic system, time-delay, memristor, hidden attractors, no plot and attractors plot. Keywords: Keywords: New New chaotic chaotic system, system, time-delay, time-delay, memristor, memristor, hidden hidden attractors, attractors, no no equilibrium equilibrium Keywords: New chaotic system, time-delay, memristor,al., hidden equilibrium 2014;attractors, Banerjee,no2013; Wang, Duan, 2012; Pham et al., al., Banerjee, 2013; 1. INTRODUCTION al., 2014; 2014; Banerjee, 2013; Wang, Wang, Duan, Duan, 2012; 2012; Pham Pham et et al., al., 2013, Pham et al., 2016). 1. INTRODUCTION 1. INTRODUCTION al., 2014; Banerjee, 2013; Wang, Duan, 2012; Pham et al., 2013, Pham et al., 2016). 2013, Pham et al., 2016). al., 2014; Banerjee, 2013; Wang, Duan, 2012; Pham et al., 1. INTRODUCTION There has been increasing interest on chaotic systems 2013, al., 2014;dynamics Banerjee, 2013; Wang, Duan, 2012; Pham et al., 1. INTRODUCTION et can be divided as self-excited attractors There has been been increasing increasing interest on on chaotic chaotic systems systems Chaotic There has interest 2013, Pham Pham et al., al., 2016). 2016). 1. INTRODUCTION dynamics can be as attractors because of their broad range of applications, such assystems secure Chaotic Chaotic dynamics can(Leonov be divided divided as self-excited self-excited attractors 2013, Pham et al., 2016). There has been increasing interest on chaotic and hidden attractors et al., 2011; Leonov et al., because of their broad range of applications, such as secure because of their range ofinterest applications, such as(Pecora, secure Chaotic There has beenbroad increasing on robotics chaotic systems dynamics can be as self-excited attractors and attractors et 2011; et communication, biology, cryptography, and hidden hidden attractors (Leonov et al., al., 2011;etLeonov Leonov et al., al., Chaotic dynamics can (Leonov be divided divided as self-excited attractors There has beenbroad increasing interest on robotics chaotic systems because of their range of applications, such as secure 2012; Leonov and Kuznetsov, 2013; Leonov al., 2014).The communication, biology, cryptography, (Pecora, communication, biology, cryptography, robotics Chaotic dynamics can (Leonov be divided as self-excited attractors because of their Boccaletti broad range ofal., applications, such as(Pecora, secure and hidden attractors et al., 2011; Leonov et al., 2012; Leonov and Kuznetsov, 2013; Leonov et al., 2014).The Carroll 1990; et 2002; Strogatz et al., 2012; Leonov and Kuznetsov, 2013; Leonov et al., 2014).The and hidden attractors (Leonov et al., 2011; Leonov et al., because of their broad range of applications, such as secure communication, biology, cryptography, (Pecora, attractor belongs to the 2013; category of systems having no Carroll Boccaletti et al., 2002; Strogatz et Carroll 1990; 1990; chaotic Boccaletti et are al.,introduced 2002; robotics Strogatz et al., al., hidden and hidden attractors (Leonov et al., 2011; Leonov et al., communication, biology, cryptography, robotics (Pecora, 2012; and Kuznetsov, Leonov et al., hidden attractor belongs to the category of having no 1994).Various systems in the literature hiddenLeonov attractor belongs tostable the 2013; category of systems systems having no 2012; Leonov and Kuznetsov, Leonov etpoints al., 2014).The 2014).The communication, biology, cryptography, robotics (Pecora, Carroll 1990; Boccaletti et al., 2002; Strogatz et al., equilibrium point, only equilibrium and an 1994).Various chaotic systems are introduced in the literature 1994).Various chaotic systems are in the literature Leonov and Kuznetsov, 2013; Leonov etpoints al., 2014).The Carroll 1990; Boccaletti et1963), al.,introduced 2002; Strogatz et al., 2012; hidden attractor belongs to the category of systems having no equilibrium point, only stable equilibrium and an like Lorenz system (Lorenz, Rössler System (Rössle, equilibrium point, only stable equilibrium points and an hidden attractor belongs to the category of systems having no Carroll 1990; Boccaletti et al., 2002; Strogatz et al., 1994).Various chaotic systems are introduced in the literature infinite number of equilibrium points(Wang and Chen, 2013; like Lorenz system (Lorenz, 1963), Rössler System (Rössle, like Lorenz system (Lorenz, 1963), Rössler System (Rössle, hidden attractor belongs tostable the category of systems having no 1994).Various chaotic (Matsumoto, systems are introduced in System the literature equilibrium point, only equilibrium points and an infinite number of equilibrium points(Wang and Chen, 2013; 1976), Chua system 1984), Lü (Lü, infinite number of equilibrium points(Wang and Chen, 2013; equilibrium point, only stable equilibrium points and an 1994).Various chaotic systems are introduced in the literature like Lorenz system (Lorenz, 1963), Rössler System (Rössle, and Sprott, 2013). A time-delay 3-Dand chaotic system 1976), Chua system (Matsumoto, Lü System (Lü, 1976), Chuasystem system (Matsumoto, 1984), Lü System (Lü, Jafari equilibrium point, only stable equilibrium points and an like Lorenz 1963), 1984), Rössler1999), System infinite number of points(Wang Chen, 2013; Jafari and Sprott, 2013). A time-delay 3-D chaotic system Chen, 2002), Chen (Lorenz, system (Chen, Ueta, etc.(Rössle, These Jafarihidden and Sprott, 2013). A new time-delay 3-D chaotic system infinite number of equilibrium equilibrium points(Wang and Chen, 2013; like Lorenz system (Lorenz, 1963), 1984), Rössler1999), System (Rössle, 1976), Chua system (Matsumoto, Lü System (Lü, with attractors is aA research3-D topic. Memristive Chen, 2002), Chen system (Chen, Ueta, etc. These Chen, 2002), Chen system (Chen, Ueta, 1999), etc. These infinite number of equilibrium points(Wang and Chen, 2013; 1976), Chua system (Matsumoto, 1984), Lü System (Lü, Jafari and Sprott, 2013). time-delay chaotic system with attractors is aaA new research topic. Memristive classical chaotic systems have a 1984), countable number of Jafari with hidden hidden attractors is new research topic. Memristive and Sprott, 2013). time-delay 3-D chaotic system 1976), 2002), Chua system (Matsumoto, Lü System (Lü, Chen, Chen system (Chen, 1999), etc. These time-delay 3-D chaotic systems with hidden attractors (being classical chaotic aa Ueta, countable number of classical chaotic systems have countable number of Jafari chaotic system and Sprott, 2013). A new time-delay 3-D Chen, 2002), Chensystems system have (Chen, Ueta, 1999), etc.can These with hidden attractors is a research topic. Memristive time-delay 3-D chaotic systems with hidden attractors (being equilibrium points. Hence, chaos in such systems be time-delay 3-D chaotic systems with hidden attractors (being withequilibrium hidden attractors is are a new research topic. Memristive Chen, 2002), Chensystems system have (Chen, Ueta, 1999), etc.can These classical chaotic a countable number of no point) not found in the literature. equilibrium points. Hence, chaos in such systems be equilibrium points. Hence, chaos in such systems can be with hidden attractors is a new research topic. Memristive classical chaotic systems have a countable number of time-delay 3-D chaotic systems with hidden attractors (being no equilibrium point) are not found in the literature. discovered by the conventional criteria (Shil'nikov, 2001) but no equilibrium point) are not found in the literature. time-delay 3-D chaotic systems with hidden attractors (being classical chaotic systems have a countable number of equilibrium Hence, in systems can be by abovesystems fact, this paper attempts construct discovered by the criteria (Shil'nikov, 2001) discovered bypoints. the conventional conventional criteria (Shil'nikov, 2001) but time-delay 3-Dthe chaotic hidden (being equilibrium points. Hence,itchaos chaos inbesuch such can but be Fascinated notwith found inattractors theto literature. no equilibrium are by above fact, paper attempts to construct in a no-equilibrium system, cannot used.systems Fascinated by the the point) above fact, this paper attempts to literature. construct no equilibrium point) are this not found in the equilibrium points. Hence,itchaos inbesuch systems can but be aFascinated discovered by the conventional criteria (Shil'nikov, 2001) memristive-based time-delay chaotic system having multiin a no-equilibrium system, cannot used. in a no-equilibrium system, it cannot be (Shil'nikov, used. nomemristive-based equilibrium point) are not found in the literature. discovered by the conventional criteria 2001) but aaFascinated by the above fact, this paper attempts to construct chaotic system having multitime-delay memristive-based time-delay chaotic multiFascinated byattractors. the above fact, this paper system attemptshaving to construct discovered byO.the conventional criteria (Shil'nikov, 2001) but scroll in a no-equilibrium system, it cannot be used. hidden In 1971, L. Chua introduced a new two-terminal circuit byattractors. the above fact, this paper system attemptshaving to construct in a1971, no-equilibrium system, it cannot be used. aFascinated memristive-based time-delay chaotic multiscroll hidden In L. O. Chua introduced aa new two-terminal circuit scroll hidden attractors. In 1971, L. O. Chua introduced new two-terminal circuit a memristive-based time-delay chaotic system having multiin a no-equilibrium system, it cannot be used. element named as memristor, which is two-terminal characterisedcircuit by a scroll a memristive-based time-delayaschaotic having hidden attractors. In 1971, L. Chua introduced aa new rest of paper is organised follows.system In Section 2, amultibrief element as which characterised by element named as memristor, memristor, which isflux characterised by aa The scroll hidden attractors. In 1971, named L. O. O. Chua introduced newis two-terminal circuit The rest of paper is organised as follows. In Section 2, aa brief brief relationship between the charge and linkage (Chua, The rest of paper is organised as follows. In Section 2, scroll hidden attractors. In 1971, L. O. Chua introduced a new two-terminal circuit element named as memristor, which is characterised by a introduction of a memristor is given. Section 3 deals with the relationship between the charge and flux linkage (Chua, relationship between the charge andwas linkage (Chua, element named as memristor, which isflux characterised by a The rest of paper is organised as follows. In Section 2,with a brief introduction of a memristor is given. Section 3 deals with the 1971). The realisation of memristor done at Hewlettintroduction of a memristor is given. Section 3 deals the The rest of paper is organised as follows. In Section 2, a brief element named as memristor, which is characterised by a relationship between the charge and flux linkage (Chua, proposed novel memristor-based time-delay chaotic system 1971). The realisation of memristor was done at Hewlett1971). The realisation of memristor at HewlettThe rest ofnovel paper is organised follows. In Section a brief relationship between charge andwas fluxdone linkage (Chua, introduction of a memristor-based memristor is as given. Section 3chaotic deals2,with the proposed memristor-based time-delay chaotic system Packard laboratory by the aof group of researchers (Strukov et al., introduction proposed novel time-delay system of a memristor is given. Section 3 deals with the relationship between the charge and flux linkage (Chua, 1971). The realisation memristor was done at Hewlettno equilibrium point. isIngiven. Section 4, basic dynamical Packard laboratory by of researchers (Strukov et PackardIt laboratory by aaarea group of researchers (Strukov et al., al., with introduction of a memristor-based memristor Section 3chaotic deals with the 1971). The realisation ofgroup memristor was at Hewlettproposed novel time-delay system with no no equilibrium equilibrium point. In In Section Section 4, basic basic dynamical 2008). opens a new of of research fordone memristor-based with point. 4, dynamical proposed novel memristor-based time-delay chaotic system 1971). The realisation ofgroup memristor was done at HewlettPackard laboratory by a researchers (Strukov et al., analysis of the system is discussed. In Section 5, numerical 2008). It opens a new area of research for memristor-based 2008). It opens a new area of research for memristor-based proposed novel memristor-based time-delay chaotic system Packard laboratory by a group of researchers (Strukov et al., with no equilibrium point. In Section 4, basic dynamical analysis of the system is discussed. In Section 5, numerical applications like neural network, adaptive filter, high-speed analysis the systempoint. isChaotic discussed. In Section 5, dynamical numerical with no of equilibrium In Section 4, of basic Packard laboratory by aarea group researchers (Strukov et al., analyses 2008). It new of of research for filter, memristor-based are presented. behaviour the system is applications like network, adaptive high-speed applications likeaa neural neural network, adaptive filter, high-speed with no of equilibrium In Section 4, of basic 2008). It opens opens newand area memristor-based analysis the systempoint. isChaotic discussed. In Section 5, dynamical numerical analyses are presented. Chaotic behaviour the system is is low power processor soof onresearch (Yang etfor al., 2013). For the analysis analyses are presented. behaviour of the system of the system is discussed. In Section 5, numerical 2008). It opens a new area of research for memristor-based applications like neural network, adaptive filter, high-speed confirmed by bifurcation plot, attractor analysis, low power processor and so on (Yang et al., 2013). For the low power processor and so on (Yang et al., 2013). For the analysis of the system is discussed. In Section 5, numerical applications like neural network, adaptive filter, high-speed analyses are presented. Chaotic behaviour of the system is confirmed by bifurcation plot, attractor analysis, inherent nonlinear nature of memristor, it has been widely confirmedare by plot, attractor analysis, analyses presented. behaviour of the system is applications like neural network, adaptiveital., filter, high-speed low power processor and (Yang For the phasebifurcation plotChaotic and frequency spectrum analysis. inherent nonlinear nature of memristor, has been widely inherent nonlinear nature ofofon memristor, has2013). been widely analyses are by presented. behaviour of the system is low power processor and so so on (Yang et et ital., 2013). the instantaneous confirmed bifurcation plot, attractor analysis, instantaneous phasebifurcation plotChaotic and frequency frequency spectrum analysis. used for nonlinear the construction chaotic withFor many instantaneous phase plot and spectrum analysis. confirmed by plot, attractor analysis, low power processor and so on (Yang etsystems al., 2013). For the Finally, inherent nature of memristor, it has been widely conclusions of the paper are drawn in Section 6. used for the construction of chaotic systems with many used for the construction of chaotic systems with many confirmed by bifurcation plot, attractor analysis, inherent nonlinear nature of memristor, it has been widely instantaneous phase plot and frequency spectrum analysis. Finally, conclusions conclusions ofplot the and paperfrequency are drawn drawnspectrum in Section Sectionanalysis. 6. features (Muthuswamy, Kokate, 2009; Muthuswamy, Finally, the paper are in 6. instantaneous phase of inherent nonlinear nature ofofmemristor, it has been 2010). widely used for the systems many features Kokate, 2009; features (Muthuswamy, Kokate, 2009; Muthuswamy, Muthuswamy, 2010). phase ofplot used for(Muthuswamy, the construction construction of chaotic chaotic systems with with2010). many instantaneous Finally, conclusions the and paperfrequency are MEMRISTOR drawnspectrum in Sectionanalysis. 6. 2. INTRODUCTION TO Finally, conclusions of the paper are drawn in Section 6. used for the construction of chaotic systems with many features (Muthuswamy, 2009; Muthuswamy, 2010). 2. INTRODUCTION INTRODUCTION TO MEMRISTOR Time-delay emerges in aKokate, dynamical due to switching 2. Finally, conclusions of the paperTO are MEMRISTOR drawn in Section 6. features (Muthuswamy, Kokate, 2009;system Muthuswamy, 2010). Time-delay emerges in dynamical due switching Time-delay emerges in aaKokate, dynamical system due to toTime-delay switching features (Muthuswamy, 2009;system Muthuswamy, 2010). 2. INTRODUCTION TO MEMRISTOR speed, processing speed, signal propagation time. After the invention by L.O.Chua in 1971(Chua, 1971), a 2. INTRODUCTION TO MEMRISTOR Time-delay emerges in a dynamical system due to switching After the the 2. invention by L.O.Chua L.O.Chua in 1971(Chua, 1971(Chua, 1971), 1971), aa speed, speed, propagation time. speed, processing processing speed, signal propagation time. Time-delay After invention in TO MEMRISTOR Time-delay emerges in active a signal dynamical due toTime-delay switching system has become an area ofsystem research and has varied memristor isINTRODUCTION knownby the fourth circuit 1971), element.a Time-delay emerges in active a signal dynamical system due toTime-delay switching After the invention byas L.O.Chua in basic 1971(Chua, speed, processing speed, propagation time. memristor is known knownby as L.O.Chua the fourth fourthin basic circuit 1971), element.a system has become an area of research and has varied system has become an active area of research and has varied memristor is as the basic circuit element. speed, processing speed, signal propagation time. Time-delay After the invention 1971(Chua, applications in the field of biology, physics and engineering Memristor deals with the relationship between charge anda After the invention by L.O.Chua in 1971(Chua, 1971), speed, processing speed, signal propagation time. Time-delay memristor is known as the fourth basic circuit element. system has become an active area of research and has varied Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering applications in the field of Yongzhen biology, and Memristor deals with thethe relationship between charge and system has become an active area ofphysics research andengineering hasSystems varied memristor is known as fourth basic circuit element. flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; et al., 2011). memristor is known as the fourth basic circuit element. system has become an active area of research and has varied Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; Yongzhen et al., 2011). Systems (Ikeda, Matsumoto, 1987; et al.,can 2011). Systems flux whichof arememristor. two fundamental circuit variables.and There are applications in athe field of Yongzhen biology, physics and engineering deals with theOne relationship between charge and two kinds is flux controlled another represented by delay differential equation exhibit chaos Memristor Memristor deals with the relationship between charge and applications in the field of biology, physics and engineering flux which are two fundamental circuit variables. There are (Ikeda, Matsumoto, 1987; Yongzhen et al., 2011). Systems two kinds kinds ofarememristor. memristor. One is is flux flux controlled and another represented by aa delay differential equation exhibit chaos represented by2010; delay differential equation can exhibit chaos flux two of One controlled and another (Ikeda, Matsumoto, 1987; Yongzhen et 2007). al.,can 2011). Systems which two fundamental circuit variables. There are one kinds is charge controlled. A flux physical memristor was (Kilinç al.,by Tamaševičius et al., The presence flux which two fundamental circuit variables. There are (Ikeda, et Matsumoto, 1987; Yongzhen et 2007). al.,can 2011). Systems two ofarememristor. One is controlled and another represented a delay differential equation exhibit chaos one is charge controlled. A physical memristor was (Kilinç et al., 2010; Tamaševičius et al., The presence (Kilinç et al., 2010; Tamaševičius et al., 2007). The presence one is charge controlled. A physical memristor was represented by a in delay differential equation can exhibitinfinite chaos two kinds offabricated memristor. One 2008 is flux(Strukov controlled and2008). another successfully during et al., In of time-delay a system makes the system two kinds of memristor. One is flux controlled and another represented by a delay differential equation can exhibit chaos one is charge controlled. A physical memristor was (Kilinç et al., 2010; Tamaševičius et al., 2007). The presence successfully fabricated during 2008 2008 (Strukov memristor et al., al., 2008). 2008).was In of in aaTamaševičius system makes system infinite of time-delay time-delay in system makes the system infinite successfully fabricated during (Strukov et In one work, is charge controlled. A physical (Kilinç et al.,and 2010; et nonlinearity, al., the 2007). The presence this a flux-controlled memristor which is characterised dimensional with the addition of it may lead one is charge controlled. A physical was (Kilinç et al.,and 2010; Tamaševičius et nonlinearity, al., the 2007). The presence successfully fabricated during 2008 (Strukov et al., In of time-delay system makes system infinite this work, flux-controlled memristor which memristor is characterised characterised dimensional with addition of it lead dimensional and in withaa the the addition of etc. nonlinearity, it may may lead this work, aa flux-controlled memristor which is successfully fabricated during 2008(Strukov (Strukov et al., 2008). 2008). In of time-delay in system makes the system infinite by its incremental memductance et al., 2008) In is to chaos, hyperchaos, multistability, Infinite dimensional successfully fabricated during 2008 (Strukov et al., 2008). of time-delay in a system makes the system infinite this work, a flux-controlled memristor which is characterised dimensional and with the addition of nonlinearity, it may lead by its incremental memductance (Strukov etis characterised al., 2008) 2008) is is to chaos, hyperchaos, multistability, etc. Infinite dimensional to chaos, hyperchaos, multistability, etc. Infinite dimensional by its incremental memductance (Strukov et al., this work, a flux-controlled memristor which dimensional and with the addition of nonlinearity, it may lead used. The nonlinear dynamics describing a flux-controlled dynamics of time-delay chaotic systems are reported in the this work, anonlinear flux-controlled memristor which isflux-controlled characterised dimensional and with the additionsystems of etc. nonlinearity, it mayinlead by incremental memductance (Strukov al., to chaos, multistability, Infinite dimensional used. The dynamics describing dynamics of chaotic are reported dynamics ofintime-delay time-delay chaotic systems are reported in the the used. dynamics describing by its its The incremental memductance (Strukov aaet etflux-controlled al., 2008) 2008) is is to chaos, hyperchaos, hyperchaos, multistability, etc. dimensional memristor isnonlinear described as literature (Srinivasan et al., 2011; LeInfinite et al., 2012; Valliet by its The incremental memductance (Strukov aetflux-controlled al., 2008) is to chaos, as hyperchaos, multistability, etc. Infinite dimensional used. nonlinear dynamics describing dynamics of time-delay chaotic systems are reported in the memristor is described as literature as in (Srinivasan et al., 2011; Le et al., 2012; Valliet literature asofintime-delay (Srinivasanchaotic et al., 2011; Le are et al., 2012; Valliet describeddynamics as used. The isnonlinear describing a flux-controlled dynamics systems reported in the memristor used. The isnonlinear dynamics describing a flux-controlled dynamics ofintime-delay chaotic systems areal., reported in the memristor as literature as et Le 2012; memristor is described described asrights reserved. literature © as2018, in (Srinivasan (Srinivasan et al., al., 2011; 2011; Le et et of al.,Automatic 2012; Valliet Valliet Copyright © 2018 IFAC 612Hosting 2405-8963 IFAC (International Federation Control) by Elsevier Ltd. All memristor is described as literature as in (Srinivasan et al., 2011; Le et al., 2012; Valliet Copyright © 2018 IFAC 612
Copyright 2018 responsibility IFAC 612Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 612 10.1016/j.ifacol.2018.05.097 Copyright © 2018 IFAC 612 Copyright © 2018 IFAC 612
5th International Conference on Advances in Control and Optimization of Dynamical Systems Nalini Prasad Mohanty et al. / IFAC PapersOnLine 51-1 (2018) 580–585 February 18-22, 2018. Hyderabad, India
i W ( )v
x y y x ay z by sin( z )
(1)
where i is the current through the memristor and v is the voltage across the memristor. The nonlinear term W () is the memductance defined by (Muthuswamy, 2010) as W ( )
dq( ) d
581
2 z 1 y W ( x) y
(6)
(2)
3. MEMRISTOR-BASED TIME-DELAY CHAOTIC SYSTEM WITH NO EQUILIBRIUM POINT In this section, we describe the dynamics of the novel timedelay chaotic system. 3.1 Memristor-based chaotic system without time-delay Fig. 1: Lyapunov Exponents ( ( ) ( ) ( )) ( )
To design a memristor-based chaotic system, we choose a cubic nonlinearity for the flux dependent function q( ) same as (Muthuswamy, 2010; Bao et al., 2010; Fitch et al., 2012) and given in (3).
q( ) c d 3 Thus, the nonlinearity function
W ( )
of
system
(5)
with
(3)
W ( )
is given as
dq( ) c 3d 2 d
(4)
where c and d are constants. To design the memristor-based time-delay chaotic system, we started with the 3D system given by Sajad Jafari in 2016 (Jafari et al., 2016). The memristor-based system is described as:
x y y x ayz by sin( z ) z 1 y 2 W ( x) y
10
(5)
where a and b are positive parameters, W (x) is the memductance as defined in (4). Here, the memristor-based chaotic system defined in (5) is a three-dimensional system. The Lyapunov exponents (LEs) of system (5) with , and initial conditions ( ) are ( ( ) ( ) ( )) which is given in Fig.1. So the Lyapunov ) is calculated as 3.0. The ( ) nature of dimension ( LEs conforms that the system in (5) is a chaotic system. The chaotic multi-scroll attractors of system (5) with same parameters and initial conditions are shown in Fig.2.
b
5
Z
0
-5
-1 0
3.2 Memristor-based chaotic system with time-delay
-1 5
-4
Based on the memristive chaotic system (5), a memristive time-delay chaotic system is proposed as follows: Fig. 2: Chaotic ( ( ) ( ) ( ))
-2
X
attractors ( )
0
of
2
system
4
(5)
with
are constants and is the time-delay, y denotes y(t ) . The maximum Lyapunov exponent of the where
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system in (6) with a = 0.1, b = 2.9, c = 0.5, d = 0.01 and 0.2 is 0.136 which is calculated by the algorithm given in (Farmer, 1982). It can be noted that the proposed system is more chaotic than the system in (5) because it has the larger first Lyapunov exponent. The chaotic attractors of the proposed system in (6) with ( ( ) ( ) ( )) ( ), and 0.2 are shown in Fig.3. It is seen from Fig. 3 that the proposed system has multi-scroll chaotic attractors.
1 y 2 W ( x) y 0
(9)
From (7), we get y 0 which is inconsistent with (9). As a result, the proposed memristor-based time-delay system has no equilibrium point. The absence of an equilibrium point makes the proposed system significantly different from other time-delay systems reported in the literature. 5. NUMERICAL ANALYSIS OF THE MEMRISTORBASED TIME-DELAY CHAOTIC SYSTEM This section describes some numerical analysis of the proposed system in (6). 5.1 Bifurcation Analysis Bifurcation diagram is plotted by varying one of the parameters of the proposed system and keeping other parameters fixed. Figure 4 shows the bifurcation diagram for the variation of the parameter [0.15,0.3] and keeping . The bifurcation diagram by varying a [0,0.3] and keeping 0.2 is shown in Fig.5. The bifurcation diagram by varying b [0,5] and keeping
0.2 is shown in Fig.6. Figure 7 shows the bifurcation diagram for the variation of the parameter and keeping c [0,2] . The bifurcation diagram by varying 0.2 and keeping d [0,0.07] 0.2 is shown in Fig.8. It is seen from Fig. 4, Fig. 7 and Fig. 8 that the system has periodic and chaotic behaviour in the specified ranges. It is observed from Fig. 5 and Fig. 6 that the system has chaotic behaviour within the specified ranges.
Fig. 3: Chaotic ( ( ) ( ) ( ))
attractors ( )
of
system
(6)
with
4. BASIC DYNAMICAL ANALYSIS OF THE MEMRISTOR-BASED TIME-DELAY CHAOTIC SYSTEM
Fig. 4: Bifurcation diagram of system (6) with variation of parameter and ( ( ) ( ) ( )) ( )
The equilibrium points of the three-dimensional memristorbased time-delay system in (6) are calculated by solving (7) to (9).
y0
(7)
x ayz by sin( z) 0
(8) 614
5th International Conference on Advances in Control and Nalini Prasad Mohanty et al. / IFAC PapersOnLine 51-1 (2018) 580–585 Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India
Fig. 5: Bifurcation diagram of system (6) with variation of parameter a and ( ( ) ( ) ( )) ( )
583
Fig. 8: Bifurcation diagram of system (6) with variation of parameter d and ( ( ) ( ) ( )) ( ) 5.2 Attractor Analysis It is apparent from the bifurcation diagram in Fig. 4, the system in (6) shows a periodic attractor, which is shown in Fig.9, with and . The bifurcation diagram in Fig.8 reveals that the system in (6) shows a periodic attractor with period two for the parameter , 0.2 and is shown in Fig. 10. 2 1
Fig. 6: Bifurcation diagram of system (6) with variation of parameter b and ( ( ) ( ) ( )) ( )
Y
0 -1 -2 -3 -1
0
Fig. 9: A periodic ( ( ) ( ) ( )) (
X attractor )
1
of
2
system
(6)
with
2 1
Y
0 -1 -2
Fig. 7: Bifurcation diagram of system (6) with variation of parameter c and ( ( ) ( ) ( )) ( )
-3 -4 -2
-1
0
X
1
2
Fig. 10:A periodic attractor with period two of system (6) with ( ( ) ( ) ( )) ( ) 615
5th International Conference on Advances in Control and 584 Optimization of Dynamical Systems Nalini Prasad Mohanty et al. / IFAC PapersOnLine 51-1 (2018) 580–585 February 18-22, 2018. Hyderabad, India
1
Spectrum of x
In this subsection, the instantaneous phase (IP) of the proposed system is calculated by using Hilbert transformation (HT) (Lauterborn, 1997). Here, we have used state of system (6) to calculate the instantaneous phase of the proposed system using MATLAB software. The IP of a chaotic system increases monotonically with respect to time, whereas for a periodic system it remains constant. The IP of the proposed system with and 0.2 for is shown in Fig. 11. It is seen from Fig. 11 the IP is increasing monotonically with respect to time. Thus, the proposed system is a chaotic system.
1
5.3 Instantaneous phase plot
a
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
Frequency (Hz)
Spectrum of x
1
Fig. 11: IP plot of state (𝑡𝑡) of system (8) using HT with ( ( ) ( ) ( ) ( )).
0.6 0.4 0.2 0
5.4 Frequency Spectrum analysis
b
0.8
0
0.5
1
Frequency (Hz)
1.5
In this subsection, we represent the randomness of a signal which is validated by its frequency spectrum. The randomness of a signal is another measure of chaoticness of a signal. Chaotic signals are wideband signals and can easily distinguish from regular/periodic signals by their frequency spectrum (Singh, Roy, 2015).
Fig. 12: (a) Frequency spectrum of state (𝑡𝑡) of system given in (Jafari et al., 2016) for and (b) Frequency spectrum of state (𝑡𝑡)of system (6) with and 0.2 , and ( ( ) ( ) ( ) ( )).
Here, we compare the frequency spectrum of the proposed system with the system given by (Jafari et al., 2016). Frequency spectrum of state x(t) of the system given in (Jafari et al., 2016) for a = 0.1, b = 2.9 is shown in Fig. 12 (a). Frequency spectrum of state (𝑡𝑡) of the system in (6) with and 0.2 , and ( ( ) ( ) ( ) ( )) given in Fig. 12 (b). By considering the spectrum of (𝑡𝑡) till 10% of its highest component, the system given in (Jafari et al., 2016) has 0.424 Hz whereas our proposed system has 0.685 Hz. Also the spectrum of proposed system is denser than the system given in (Jafari et al., 2016). Hence, the proposed system has wider spectrum compare with the system given in (Jafari et al., 2016).
6. CONCLUSIONS A memristor-based time-delay 3-D chaotic system with four nonlinear terms is reported in this paper. The proposed system has no equilibrium point which satisfies the characteristics of a hidden attractor system. The chaotic behaviour of the proposed system is analysed by various numerical tools such as bifurcation diagram, attractor analysis, instantaneous phase plot and frequency spectrum analysis. The first Lyapunov exponent of the proposed system is much more than the original memristor-based chaotic system considered in this paper. The frequency spectrum reveals the spectrum of the proposed system has wider band in comparison with the non-time-delay original system. REFERENCES Pecora, L.M. and Carroll, T.L., 1990. Synchronization in chaotic systems. Physical review letters, 64(8), p.821.
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