Analysis of a Class of Complex System Without Equilibria Via

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The following article appeared in IFAC-PapersOnLine 51(13): 526-531 (2018); and may be found at: https://doi.org/10.1016/j.ifacol.2018.07.333 Proceedings, 2nd IFAC Conference on Proceedings, 2nd IFAC Conference on Proceedings,Modelling, Identification 2nd IFAC Conference and Control on of Nonlinear Systems Modelling, Identification and Control ofAvailable Nonlinear online Systems at www.sciencedirect.com Modelling,Proceedings,Guadalajara, Identification Mexico,2nd IFAC June Conference and 20-22, Control 2018 on of Nonlinear Systems 2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018 Guadalajara,Modelling, Identification Mexico, June and 20-22, Control 2018 of Nonlinear Systems Guadalajara, Mexico, June 20-22, 2018 Guadalajara, Mexico, June 20-22, 2018 ScienceDirect

IFAC PapersOnLine 51-13 (2018) 526–531 should be at least linear affine of the formx ˙ = Ax + B, 0 Analysis of a Class of Complex System however, a correct vector B should be chosen and the Analysis of a Class of Complex System absence of equilibria does not imply a bounded flow in a withoutAnalysis Equilibria of a Class via of ComplexSwitched System Control region of the state space. Thus, to get a chaotic attractor without Equilibria via Switched Control in a system without equilibria is necessary to consider a without EquilibriaLaw via Switched Control PWL vector fields. This can be achived by considering a Law Law switched control law that is applied to generate a set of R.J. Ecalante-González∗ E. Campos-Cantón∗∗ linear affine systems. R.J. Ecalante-González∗ E. Campos-Cantón∗∗ R.J. Ecalante-González∗ E. Campos-Cantón∗∗ (a) (b) ∗ ∗∗ In section 2 basic theory of PWL systems and switched R.J. Ecalante-GonzálezDivisiónde Matemáticas∗ E. Campos-Cantón Aplicadas, ∗∗ Divisiónde Matemáticas Aplicadas, control laws is given. In section 3 a possible electronic Instituto PotosinoDivisiónde de InvestigaciónCient´ıfica Matemáticas Aplicadas, y Tecnológica A.C. Instituto Potosino de InvestigaciónCient´ıfica y Tecnológica A.C. realization for a chaotic attractor in a system without CaminoInstituto a Potosino la PresaDivisiónde de San InvestigaciónCient´ıfica José2055 Matemáticas Col. Lomas Aplicadas, y 4a Tecnológica Sección,78216, A.C. Camino a la Presa San José2055 Col. Lomas 4a Sección,78216, equilibria with one scroll is introduced. In section 4 a CaminoInstituto a Potosino la PresaSan de San Luis InvestigaciónCient´ıfica José2055 Potos´ı, S.L.P., Col. Lomas México y 4a Tecnológica Sección,78216, A.C. San Luis Potos´ı, S.L.P., México new class of dynamical systems without equilibria with Camino a la∗ Presae-mail:San San Luis [email protected] José2055 Potos´ı, S.L.P., Col. Lomas México 4a Sección,78216, ∗e-mail: [email protected] multiscroll attractors is presented along with a particular ∗e-mail:∗∗e-mail:San Luis [email protected] [email protected] Potos´ı, S.L.P., México ∗∗e-mail: [email protected] case and its electronic realization. In section 5 a new class ∗e-mail:∗∗e-mail: [email protected] [email protected] ∗∗ of complex system without equilibria with a multiscroll ∗∗e-mail: [email protected] (c) (d) Abstract: attractor in each node is presented. Abstract: Abstract:In this article we introduce a new class of complex system without equilibria which exhibits a In this article we introduce a new class of complex system without equilibria which exhibits a Fig. 1. Attractor of the system (2) with the switched InAbstract:chaotic this article multiscroll we introduce attractor a in new each class node. of The complex number system of scrolls without in the equilibria attractor which is determined exhibits a chaotic multiscroll attractor in each node. The number of scrolls in the attractor is determined 2. PIECEWISE LINEAR SYSTEMS control law (3) with A, B1 and B2 given in (4) for chaoticInby this a switched article multiscroll we control introduce attractor law to a in newallow each classthe node. ofoperation The complex number of system different of scrolls without linear in the equilibria affine attractor systems. which is determined Thus,exhibits the a T by a switched control law to allow the operation of different linear affine systems. Thus, the the initial condition x0 = (0, 0, 0) in the space bychaoticsystem a switched is multiscroll composed control attractor of many law to subsystems in allow eachthe node. whichoperation The interact number of different with of scrolls each linear otherin the affine to attractor generate systems. is a determined multiscroll Thus, the system is composed of many subsystems which interact with each other to generate a multiscroll A dynamical system is said to be a piecewise linear if it (a)x1 x2 x3 and its projections onto the planes: systembyattractor. a switched is composed This control new of class many law of to subsystems piecewise allow the linear whichoperation (PWL) interact of different system with each presents linear other affine tono generate positive systems. a real multiscroll Thus, part the in − − attractor. This new class of piecewise linear (PWL) system presents no positive real part in has an associated vector field of the form (b) x1 x2, (c) x1 x3 and (d) x2 x3. attractor.systemthe eigenvalues is composed This newof the of class many Jacobian of subsystems piecewise matrix linear whichas opposed (PWL) interact to system with the reported each presents other systems tono generate positive with a multiscrolls. real multiscroll part in − − − the eigenvalues of the Jacobian matrix as opposed to the reported systems with multiscrolls. A1x + B1, if x D1; theattractor.The eigenvalues scrolls This present newof the a class complex Jacobian of piecewise behavior matrix linear since as opposed these (PWL) don’t to system the unwrap reported presents and systems don’t no positive appear with multiscrolls. real close part to an in ∈ Consider the dynamical system introduced in Escalante- theThe eigenvalues scrolls present of the a complex Jacobian behavior matrix since as opposed these don’t to the unwrap reported and systems don’t appear with multiscrolls. close to an A2x + B2, if x D2; Thethe eigenvalues scrolls present of the a complex Jacobian behavior matrix since as opposed these don’t to the unwrap reported and systems don’t appear with multiscrolls. close to an ∈ Theunstable scrolls manifold present of a a complex saddle-focus behavior equilibrium since these point. don’t A particularunwrap and case don’t is taken appear as close case study to an x˙ = f(x)= . . . (1) González and Campos-Cantón(2017) given by (2) with unstableThe scrolls manifold present of a a complex saddle-focus behavior equilibrium since these point. don’t A particularunwrap and case don’t is taken appear as close case study to an  . . . unstableand simulation manifold plots of ofa saddle-focus the attractor equilibrium are provided. point. A particular case is taken as case study  the switched control law: unstableand simulation manifold plots of ofa saddle-focus the attractor equilibrium are provided. point. A particular case is taken as case study  A x + B , if x D ; and simulation plots of the attractor are provided. n m m B1, if x1 <σ; and© 2018, simulation IFAC (International plots of the Federation attractor of areAutomatic provided. Control) Hosting by Elsevier Ltd. All rights reserved. ∈ B(x)= (3) Keywords: Chaos theory, chaotic behaviour, piecewise linear controllers, piecewise linear where Bi could be zero, Di with i =1, 2, ..., m are B2, if x1 σ. Keywords: Chaos theory, chaotic behaviour, piecewise linear controllers, piecewise linear  m n ≥ Keywords:systems, systemsChaos without theory, chaotic equilibria behaviour, piecewise linear controllers, piecewise linear the domains of each sub-system, and i=1 Di = R y systems, systems without equilibria m and: systems,Keywords: systemsChaos without theory, chaotic equilibria behaviour, piecewise linear controllers, piecewise linear i=1 Di = . systems, systems without equilibria ∅ 01 1 0.5 1 1. INTRODUCTION PWL dynamical systems are known to be capable of n n − 1. INTRODUCTION PWL dynamical systems are known to be capable of The vector field defined by f : R R , for a PWL A = 0 0.53 ,B1 = 0 ,B2 = 10 . (4) 1. INTRODUCTION PWLproducing dynamical chaos, ansystems example are of known this is to the be well capable studied of → − producing chaos, an example of this is the well studied system without equilibria, should satisfy certain require- 0 3 0.5 0 0 1. INTRODUCTION producingPWLChua system. dynamical chaos, Furthermore ansystems example are hyperchaotic of known this is to the piecewise be well capable studied linear of − − Chua system. Furthermore hyperchaotic piecewise linear ments in order to avoid equilibria of the system.There are In this casex ˙ = Ax + B andx ˙ = Ax + B are UDSWE Chuaproducingsystems system. without chaos, Furthermore equilibria an example have hyperchaotic of been this reported, is the piecewise well as the studied linear four- 1 2 systems without equilibria have been reported, as the four- two approaches to get this. The former is by considering systems and have no equilibria in all 3. This system systemsChuadimensional system. without hyperchaotic Furthermore equilibria havesystem hyperchaotic been introduced reported, piecewise in as theLi linear et four- al. R The study of dynamical systems without equilibria has dimensional hyperchaotic system introduced in Li et al. that linear operators Ai are not singular, so each domain generates a chaotic attractor located around the plane The study of dynamical systems without equilibria has dimensionalsystems(2014). This without hyperchaoticsystem equilibria was obtained havesystem been fromintroduced reported, an approximation in as theLi et four- al. Theattracted study the of dynamical attention of systems the scientific without community equilibria has in (2014). This system was obtained from an approximation Di, assigned to a linear or linear affine transformation x = σ. The resulting attractor for σ = 0 is shown in attracted the attention of the scientific community in (2014).dimensionalmade to This an hyperchaoticsystem extended was system obtained system based fromintroduced onan approximation the in diffusion- Li et al. 1 attractedTherecent study years. the of One dynamical attention of the first of systems dynamical the scientific without systems community equilibria presenting has in made to an extended system based on the diffusion- Aix + Bi, must not contain an equilibrium point given Figure 1. recent years. One of the first dynamical systems presenting made(2014).less Lorenz to This an system. system extended was On system theobtained other based from hand, onan there approximation the isdiffusion- a class 1 recentattractedoscillations years. the and One attention no of the equilibria first of dynamical the was scientific described systems community by presenting Arnold in less Lorenz system. On the other hand, there is a class by x∗ = A− Bi , i.e., there is not equilibrium point due oscillations and no equilibria was described by Arnold lessmadeof systems Lorenz to an that system. extended show On complex system the other baseddynamics hand, on there different the isdiffusion- a fromclass − 1 oscillationsrecentSommerfeld years. and in One 1902 no of the (Kiselevaequilibria first dynamical et was al. described(2016)). systems However by presenting Arnold the of systems that show complex dynamics different from to x∗ / Di, instead, Ai− Bi Dj with i = j. The second Sommerfeld in 1902 (Kiseleva et al. (2016)). However the oflessthe systems Lorenzmultiscroll that system. attractors, show On complex the instead, other dynamics hand, they there display different is a multi- fromclass ∈ − ∈ 3.1 Electronic realization Sommerfeldoscillationsfirst system and in without 1902 no (Kiselevaequilibria equilibria et was with al. described(2016)). a chaotic However by flow Arnold was the the multiscroll attractors, instead, they display multi- approach is given by considering that linear operators Ai first system without equilibria with a chaotic flow was theofwing systems multiscroll attractors. that attractors, For show example, complex instead, a dynamicsfour-dimensional they display different chaotic multi- from firstSommerfeldreported system by Sprott in without 1902 in (Kiseleva 1994equilibria (Sprott et with al. (1994)) (2016)). a chaotic and However was flow called was the wing attractors. For example, a four-dimensional chaotic are singular, so the linear affine vector fieldx ˙ = Aix + Bi reported by Sprott in 1994 (Sprott (1994)) and was called wingthesystems multiscroll attractors. capable ofattractors, For producing example, instead, a multiwinga four-dimensional they butterfly display chaotic multi- n In this subsection a possible electronic realization for the reportedfirstSprott system case by A.Sprott without The in associated 1994equilibria (Sprott vector with (1994)) afield chaotic and of this was flow system called was systems capable of producing a multiwing butterfly chaotic has no equilibria in all R . In this work we consider the Sprott case A. The associated vector field of this system systemswingattractor attractors. capable was presented of For producing example, in Tahir a multiwinga four-dimensionalet al. (2015). butterfly chaotic system (2) with the switched control law (3) with A, B Sprottreportedpresents case twoby A.Sprott quadratic The in associated 1994 nonlinearities (Sprott vector (1994)) fieldand isand of a this was particular system called attractor was presented in Tahir et al. (2015). last approach. 1 presents two quadratic nonlinearities and is a particular attractorsystems capable was presented of producing in Tahir a multiwing et al. (2015). butterfly chaotic and B given in (4) is presented. The electronic circuit is presentsSprottcase of casethe two Nose-Hoover A. quadratic The associated nonlinearities system vector (Hoover fieldand (1995)). is of a this particular system Recently, new approaches to get chaotic attractor in sys- 2 case of the Nose-Hoover system (Hoover (1995)). Recently,attractor was new presented approaches in to Tahir get et chaotic al. (2015). attractor in sys- Without loss of generality, we will assume the same A designed to be energized by DC power sources of 12V casepresents of the two Nose-Hoover quadratic nonlinearities system (Hoover and (1995)). is a particular Recently,tems without new approaches equilibria based to get on chaotic piecewise attractor linear in sys- The attractors exhibited by these systems without equi- Recently,tems without new approaches equilibria based to get on chaotic piecewise attractor linear in sys- matrix for all domains Di, with i =1,...,n. Therefore, ± Thecase ofattractors the Nose-Hoover exhibited system by these (Hoover systems (1995)). without equi- temsRecently, without new approaches equilibria based to get on chaotic piecewise attractor linear in sys- and 15V and is shown in Figure 2. The circuit diagram Thelibria attractors fall into the exhibited category by of these hidden systems attractors without according equi- tems were without introduced equilibria in Escalante-González based on piecewise and linear Campos- sys- the system (1) can be rewritten as follows: ± libria fall into the category of hidden attractors according tems were without introduced equilibria in Escalante-González based on piecewise and linear Campos- sys- has been divided in five sub-diagrams. libriaTheto the attractors fall definition into the exhibited category given by by of Leonov these hidden systems etattractors al. without (2011), according equi-since temsCantón(2017) were introduced and Escalante-González in Escalante-González et al. (2017).and Campos- In the to the definition given by Leonov et al. (2011), since Cantón(2017)tems were introduced and Escalante-González in Escalante-González et al. (2017).and Campos- In the tolibriathe the basin fall definition into of the attraction category given by does of Leonov hidden not intersect etattractors al. (2011), with according small since Cantón(2017)same spirit that and the Escalante-González aforementioned works, et al. (2017). in this Inpaper the The sub-circuits in the sub-diagrams in Figures 2a and 2b the basin of attraction does not intersect with small sameCantón(2017) spirit that and the Escalante-González aforementioned works, et al. (2017). in this Inpaper the x˙ = Ax + B(x), (2) thetoneighborhoods the basin definition of attraction of equilibria.given by does Leonov There not areintersect et alsoal. (2011), more with recent small since samea new spirit class that of piecewise the aforementioned linear (PWL) works, system in this without paper consist of an operational amplifier (op amp) configured as neighborhoods of equilibria. There are also more recent asame new spirit class that of piecewise the aforementioned linear (PWL) works, system in this without paper where B(x) is handled by the switched control law. neighborhoodsthethree-dimensional basin of attraction of equilibria. dynamical does There systems not areintersect without also more with equilibria recent small aequilibria new class is reported. of piecewise The linear new class (PWL) is interesting system without since adder-subtractor followed by an op amp configured as an three-dimensional dynamical systems without equilibria equilibriaa new class is reported. of piecewise The linear new class (PWL) is interesting system without since three-dimensionalneighborhoodsreported with quadratic of equilibria. dynamical nonlinearities There systems are without in also its more description, equilibria recent equilibriathese systems is reported. present a The multiscroll new class attractor is interesting whose scrolls since integrator, these are the responsible circuits for the signals reported with quadratic nonlinearities in its description, theseequilibria systems is reported. present a The multiscroll new class attractor is interesting whose scrolls since 3. SYSTEM WITH ONE SCROLL reportedthree-dimensionalas the one with in Wei quadratic (2011) dynamical which nonlinearities systems is based without onin itsthe description, Sprott equilibria case thesedon’t systemsunwrap closepresent to a an multiscroll unstable manifold attractor of whose an equilib- scrolls x1 and x2, respectively. as the one in Wei (2011) which is based on the Sprott case don’tthese systemsunwrap closepresent to a an multiscroll unstable manifold attractor of whose an equilib- scrolls asreportedDsystem the one with or in the Wei quadratic seventeen (2011) which nonlinearities flows is found based by onin three itsthe description, Sprott numerical case don’trium point unwrap as theclose current to an unstable reported manifold systems. ofMoreover an equilib- the D system or the seventeen flows found by three numerical riumdon’t point unwrap as theclose current to an unstable reported manifold systems. ofMoreover an equilib- the The sub-diagram in Figure 2c consists of an op amp Dasmethods system the one proposed or in the Wei seventeen (2011) in Jafari which flows et is foundal. based (2013). by on three the The Sprott numerical previous case riumswitched point control as the law current ensures reported the absence systems. of Moreover equilibrium the Definition 1. (UDSWE). A dynamical system with an as- methods proposed in Jafari et al. (2013). The previous switchedrium point control as the law current ensures reported the absence systems. of Moreover equilibrium the configured as an adder followed by an op amp configured methodsDworks system have proposed or reported the seventeen in three-dimensional Jafari flows et foundal. (2013). by complex three The numerical systems,previous switchedpoints and control determine law ensures the number the absence of scrolls of exhibited equilibrium by sociated vector field of the form f(x)=Ax + B is works have reported three-dimensional complex systems, pointsswitched and control determine law ensures the number the absence of scrolls of exhibited equilibrium by as an integrator and its designed to produce the signal x3. worksmethodsbut higher have proposed dimentions reported in three-dimensional Jafarihave also et al.been (2013). explored, complex The for systems,previous exam- pointsthe attractor. and determine the number of scrolls exhibited by an Unbounded Dissipative System Without Equilibria but higher dimentions have also been explored, for exam- thepoints attractor. and determine the number of scrolls exhibited by 3 3 butworksple, higherfour-dimensional have dimentions reported three-dimensionalchaotic have also and been hyperchaotic explored, complex dynamical for systems, exam- the attractor. (UDSWE) if A =[a1,a2,a3] R × whose characteristic The gains of the adder in the sub-circuit of x3 were chosen ple, four-dimensional chaotic and hyperchaotic dynamical theIt is attractor. well known that no system without equilibria has polynomial P (λ) has the roots∈ λ = p iq, λ =0 ple,butsystems higherfour-dimensional without dimentions equilibria chaotic have through also and been hyperchaotic the explored, use of dynamical forquadratic exam- It is well known that no system without equilibria has A 1,2 3 according to the third row vector of the matrix A, while systems without equilibria through the use of quadratic Ita linear is well associated known that vector no system field of without the form equilibriax ˙ = Ax, has in with p<0, q = 0 and B is a linear{ combination± of the} systemsple,and/or four-dimensional cubic without nonlinearities equilibria chaotic through have and hyperchaotic also the been use of reported dynamical quadratic in aIt linear is well associated known that vector no system field of without the form equilibriax ˙ = Ax, has in the gains of the adder-subtractor of the sub-circuits of x1 and/or cubic nonlinearities have also been reported in aorder linear to haveassociated a system vector without field equilibria of the form thex ˙ vector= Ax field, in and/orsystems cubic without nonlinearities equilibria through have also the been use of reported quadratic in order to have a system without equilibria the vector field form and x2 were designed according to the first two row vectors and/orWang et cubic al. (2012), nonlinearities Pham et al. have (2016), also Tahir been et reported al. (2015). in ordera linear to haveassociated a system vector without field equilibria of the form thex ˙ vector= Ax field, in Wangand/or et cubic al. (2012), nonlinearities Pham et al. have (2016), also Tahir been et reported al. (2015). in B = k1a1 + k2a2 + k3a3 + k4v, of the matrix A and considering an additional gain of 1 Wang et al. (2012), Pham et al. (2016), Tahir et al. (2015). order to have a system without equilibria the vector field − Wang et al. (2012), Pham et al. (2016), Tahir et al. (2015). where k4 = 0 and v fulfills Av = 0 and v = 0. reserved for signals A and B, respectively. Proceedings,2405-8963 © 2018, 2nd IFACIFAC Conference (International on Federation of Automatic Control)526 Hosting by Elsevier Ltd. All rights reserved. Proceedings,Peer review under 2nd IFACresponsibility Conference of International on Federation of Automatic526 Control. Proceedings,Modelling, Identification 2nd IFAC Conference and Control on of Nonlinear 526 Modelling,10.1016/j.ifacol.2018.07.333 Identification and Control of Nonlinear Modelling,Proceedings,Systems Identification 2nd IFAC Conference and Control on of Nonlinear 526 527 Systems SystemsModelling,Guadalajara, Identification Mexico, June and 20-22, Control 2018 of Nonlinear Guadalajara, Mexico, June 20-22, 2018 Guadalajara,Systems Mexico, June 20-22, 2018 Guadalajara,10.1016/j.ifacol.2018.07.333 Mexico, June 20-22, 2018 2405-8963 2018 IFAC MICNON Guadalajara, Mexico, June 20-22, 2018R.J. Ecalante-González et al. / IFAC PapersOnLine 51-13 (2018) 526–531 527 should be at least linear affine of the formx ˙ = Ax + B, 0 however, a correct vector B should be chosen and the absence of equilibria does not imply a bounded flow in a region of the state space. Thus, to get a chaotic attractor in a system without equilibria is necessary to consider a PWL vector fields. This can be achived by considering a switched control law that is applied to generate a set of linear affine systems. (a) (b) In section 2 basic theory of PWL systems and switched control laws is given. In section 3 a possible electronic realization for a chaotic attractor in a system without equilibria with one scroll is introduced. In section 4 a new class of dynamical systems without equilibria with multiscroll attractors is presented along with a particular case and its electronic realization. In section 5 a new class of complex system without equilibria with a multiscroll (c) (d) attractor in each node is presented. Fig. 1. Attractor of the system (2) with the switched 2. PIECEWISE LINEAR SYSTEMS control law (3) with A, B1 and B2 given in (4) for T the initial condition x0 = (0, 0, 0) in the space A dynamical system is said to be a piecewise linear if it (a)x x x and its projections onto the planes: 1 − 2 − 3 has an associated vector field of the form (b) x1 x2, (c) x1 x3 and (d) x2 x3. A x + B , if x D ; − − − 1 1 ∈ 1 A2x + B2, if x D2; Consider the dynamical system introduced in Escalante- x˙ = f(x)= . . ∈. (1) González and Campos-Cantón(2017) given by (2) with  . . . the switched control law:  Anx + Bm, if x Dm; B , if x <σ; ∈ B(x)= 1 1 (3) where Bi could be zero, Di with i =1, 2, ..., m are B2, if x1 σ.  m n ≥ the domains of each sub-system, and i=1 Di = R y m and: Di = . i=1 ∅ 01 1 0.5 1 n n − The vector field defined by f : R R , for a PWL A = 0 0.53 ,B1 = 0 ,B2 = 10 . (4) system without equilibria, should satisfy→ certain require- 0 − 3 0.5 0 0 ments in order to avoid equilibria of the system.There are − − In this casex ˙ = Ax + B1 andx ˙ = Ax + B2 are UDSWE two approaches to get this. The former is by considering systems and have no equilibria in all R3. This system that linear operators Ai are not singular, so each domain generates a chaotic attractor located around the plane D , assigned to a linear or linear affine transformation i x1 = σ. The resulting attractor for σ = 0 is shown in Aix + Bi, must not contain an equilibrium point given 1 Figure 1. by x∗ = A− Bi , i.e., there is not equilibrium point due − 1 to x∗ / Di, instead, A− Bi Dj with i = j. The second ∈ − i ∈ 3.1 Electronic realization approach is given by considering that linear operators Ai are singular, so the linear affine vector fieldx ˙ = Aix + Bi has no equilibria in all Rn. In this work we consider the In this subsection a possible electronic realization for the last approach. system (2) with the switched control law (3) with A, B1 and B2 given in (4) is presented. The electronic circuit is Without loss of generality, we will assume the same A designed to be energized by DC power sources of 12V matrix for all domains Di, with i =1,...,n. Therefore, and 15V and is shown in Figure 2. The circuit diagram± the system (1) can be rewritten as follows: has been± divided in five sub-diagrams. The sub-circuits in the sub-diagrams in Figures 2a and 2b x˙ = Ax + B(x), (2) consist of an operational amplifier (op amp) configured as where B(x) is handled by the switched control law. adder-subtractor followed by an op amp configured as an integrator, these are the responsible circuits for the signals 3. SYSTEM WITH ONE SCROLL x1 and x2, respectively.

Definition 1. (UDSWE). A dynamical system with an as- The sub-diagram in Figure 2c consists of an op amp sociated vector field of the form f(x)=Ax + B is configured as an adder followed by an op amp configured an Unbounded Dissipative System Without Equilibria as an integrator and its designed to produce the signal x3. 3 3 (UDSWE) if A =[a1,a2,a3] R × whose characteristic The gains of the adder in the sub-circuit of x were chosen ∈ 3 polynomial PA(λ) has the roots λ1,2 = p iq, λ3 =0 according to the third row vector of the matrix A, while { ± } with p<0, q = 0 and B is a linear combination of the the gains of the adder-subtractor of the sub-circuits of x 1 form and x2 were designed according to the first two row vectors B = k1a1 + k2a2 + k3a3 + k4v, of the matrix A and considering an additional gain of 1 where k = 0 and v fulfills Av = 0 and v = 0. reserved for signals A and B, respectively. − 4

527 2018 IFAC MICNON 2018 IFAC MICNON Guadalajara,528 Mexico, June 20-22, 2018R.J. Ecalante-González et al. / IFAC PapersOnLine 51-13 (2018) 526–531 Guadalajara, Mexico, June 20-22, 2018

The switched control law is then given by: The maximum Lyapunov exponent was calculated to be 1, if x S ; MLE =0.233936 which is an indication of chaos. − ≤ σ1 σ(x)= 0, if Sσ1 Sσ2 ; 0 u = 0 . (10) In this subsection a possible electronic realization for the (a) σ(x)( + ) system (5) with A given in (6) with the switching law given 1 2 in (11) is presented. The electronic circuit is designed to be B σ(x)( + )a , if x u S; B(x, σ(x)) = 1 − 1 2 3 − ≤ energized by DC power sources of 12V and 15V. The B2 σ(x)(1 + 2)a3, if x u>S; circuit diagram has been divided± in eight sub-diagrams± − − (11) shown in Figures 6, 7 and 8. where b11 b21 The sub-circuit in the sub-diagram shown in Figure 6a consist of an op amp configured as an inverter followed by (b) B1 = b12 ,B2 = b22 , (12) b13 b23 an op amp configured as integrator and it is responsible 3 for the signal x1. such that B1,B2 R with b11 > 0, b21 < 0 and b ,b = 0. ∈ The sub-circuits in the sub-diagrams shown in Figures 6b Fig. 3. Attractor of the circuit diagram in Figure 2 gener- 22 23 and 6c consist of a an op amp configured as adder- ated by electronic simulation projected on the plane With an appropriate selection of parameters the system subtractor followed by an op amp configured as an inte- x1 x3. (5) exhibit a triple-scroll attractor. − grator and are responsible for the signals x2 and x3. (c) Since the null space of A is null(A) = span (1, 0, 0)T , A The sub-diagrams in Figures 7a, 7b and 7c consist of a and B(x, σ(x)) fulfill the definition 1 for all{ x. } comparator followed by an op amp configured as buffer It is worth notice that the number of scrolls can be easily and another one either as adder-subtractor or adder. These increased just by extending σ(x) from 1 and 1 to n and sub-circuits are responsible for the signals A, B, and C, m to have m + n + 1 number of scrolls.− − which together produce the change of the affine part based on the signal E coming from the adder-subtractor shown Consider now the particular case with B1 and B2 as in Figure 8b. This sub-circuit depends on the signal D follows: 0.5 3.5128 of the sub-circuit in Figure 8a which is composed of two − comparators followed by op amps configured as buffers and B1 = 0 ,B2 = 9.4093 , (13) 0 0.3543 an op amp configured as an adder. (d) (e) and the parameters given in Table 1. This electronic circuit also makes use of the TL082CP and LM339AN devices and as the first circuit realization Fig. 2. Sub-diagrams of the proposed electronic realization Table 1. Parameters for a particular case presented, the mathematical resistor values calculated of the system (3) with A, B1 and B2 given in (4) for were approximated to achievable values by one or two the sub-circuits that produce the output signals: (a) Parameter Value Fig. 4. Attractor of the circuit diagram in Figure 2 pro- a 0.5 resistors from the E12 series combined either in parallel x1 (b) x2, (c) x3, (d) A and (e) B. jected onto the plane x1 x3 obtained experimentally. b3 or series. − 2.6 The sub-circuits in the sub-diagrams of Figures 2d and 2e 1 An electronic simulation has been run and the result is 00 0 2 4.5 consist of a comparator followed by an op amp configured presented in Figure 9. as a buffer and an adder in the first case and an adder- A = 0 ab . (6) 0 −b a The resulting attractor is shown in Figure 5. subtractor in the second one. These two sub-circuits are − − responsible for the commutation of B1 and B2. where a, b R>0, i.e. the roots of PA(λ) are λ1,2 = a ib and λ = 0.∈ − ± The electronic circuit makes use of the general purpose 3 JFET-input dual Operational amplifier TL082CP and the For the switched control law, five commutation planes are quad differential comparator LM339AN. Mathematical considered: (a) (b) resistor values calculated were approximated to achievable 3 Su1 = x R γ1x1 + γ2x2 + γ3x3 = γ3(1 + 2) , values by one or two resistors from the E12 series combined { ∈ 3| } S = x R γ1x1 + γ2x2 + γ3x3 =0 , either in parallel or series. { ∈ 3| } Sd1 = x R γ1x1 + γ2x2 + γ3x3 = γ3(1 + 2) , (a) (b) { ∈ | − } An electronic simulation of the proposed circuit has been (7) run and the result is shown in Figure 3. with γ =0.9027, γ =0.2441, γ = 0.3543 and 1 2 3 − 3 (c) Furthermore, experimental results are shown in Figure 4. Sσ1 = x R x3 = 2 , { ∈ 3| − } (8) Sσ2 = x R x3 = 1 , { ∈ | } Fig. 6. Sub-diagrams of the proposed electronic realization 4. MULTISCROLL SYSTEM with 1,2 R>0. of the system (5) with A given in (6) and the switching ∈ law given in (11) for the sub-circuits that produce the To write the switching control, law the notation x>S The term multiscroll attractor is used to refer to three or (c) (d) output signals: (a) x1, (b) x2 and (c) x3. more scrolls in an attractor Campos-Cantónet al. (2010). will be used if x is in the set pointed by the vector (γ ,γ,γ)T , x S if x is in the opposite set or on In this section a system with a triple scroll is proposed. 1 2 3 ≤ Fig. 5. Attractor of the system (5) with A given in (6) and the plane S. the switching law given in (11) for the initial condition 5. COMPLEX SYSTEM WITHOUT EQUILIBRIA Consider a system T The notation x>Sσi will be used if x is in the set pointed x0 = (0, 0, 0) in the space (a)x1 x2 x3 and its x˙ = Ax + B(x, σ(x)), (5) T − − by the vector (0, 0, 1) , x Sσi if x is in the opposite projections onto the planes: (b) x1 x2, (c) x1 x3 In this section a new complex system without equilibria ≤ and (d) x x . − − with A given as follows: set or on the plane Sσi. 2 − 3 which exhibit a chaotic multiscroll attractor in each node

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(a) (b)

(c) Fig. 6. Sub-diagrams of the proposed electronic realization of the system (5) with A given in (6) and the switching law given in (11) for the sub-circuits that produce the (c) (d) output signals: (a) x1, (b) x2 and (c) x3. Fig. 5. Attractor of the system (5) with A given in (6) and the switching law given in (11) for the initial condition 5. COMPLEX SYSTEM WITHOUT EQUILIBRIA x = (0, 0, 0)T in the space (a)x x x and its 0 1 − 2 − 3 projections onto the planes: (b) x1 x2, (c) x1 x3 In this section a new complex system without equilibria and (d) x x . − − 2 − 3 which exhibit a chaotic multiscroll attractor in each node

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(a)

(b)

Fig. 9. Attractor of the circuit diagrams in Figures 6 and 7 generated by electronic simulation projected on the plane x x . 1 − 3 (c) Consider the complex system given by a network where Fig. 7. Sub-diagrams of the proposed electronic realization three nodes whose dynamicx ˙ = F (x),y ˙ = F (y) and of the system (5) with A given in (6) and the switching z˙ = F (z) are given by (5) with A given in (6) and the law given in (11) for the sub-circuits that produce the switching law given in (11) are connected with each node output signals: (a) A, (b) B and (c) C. (see Figure 10). Each node exhibit a chaotic multiscroll attractor and the coupling used to synchronize the attractors is given as follows: x˙ = F (x)+α E (y x)+α E (z x); 1 1 − 3 3 − y˙ = F (y)+α1E1(x y)+α2E2(z y); (16) z˙ = F (z)+α E (y − z)+α E (x − z); 2 2 − 3 3 − with α1 =0.5 and α2 = α3 = 3 and where: 100 000 100 E1 = 000 ,E2 = 010 ,E3 = 000 . (17) 000 000 001 Since the A matrix of the system has no eigenvalues with (a) (b) positive real part the coupling strengths could be restricted to α1,α2,α3 R>0. ∈ Fig. 8. Sub-diagrams of the proposed electronic realization In the Figure 11 comparisons in time of the corresponding of the system (5) with A given in (6) and the switching states of each node are shown. The initial conditions were law given in (11) for the sub-circuits that produce the chosen so each node starts in a different scroll. As it can output signals: (a) D and (b) E. be seen from the plots, the synchronization take place at is presented. The attractors synchronize due to a diffusive the beginning of the simulation starting at t = 0. coupling between the identical systems (see Pecora et al. It is worth notice that the absence of equilibria guarantees (1997)). the oscillation of the system in all the nodes for any initial The description of any two identical nodes with dynamics condition in the basin of attraction. u˙ = F (u) andv ˙ = F (v) mutually coupled can be written as u˙ = F (u)+αE(v u); (14) v˙ = F (v)+αE(u − v). − where E determines the linear combination of state compo- nents used in the difference and α is the coupling strength. In order to select appropriate values for the matrix E and α, the stability of the origin in the difference vector field w˙ =˙u v˙ is studied. The systemw ˙ can be expressed in Taylor− expansion and approximated by neglecting higher order terms. In the case of systems of the form (2) the Fig. 10. Complex system with three identical nodes whose Jacobian matrix is equal to the matrix of the system. Thus, dynamical systems present a chaotic multiscroll at- locally the stability could be studied in the system tractor and no equilibria. w˙ Aw 2Eαw. (15) ≈ −

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ACKNOWLEDGEMENTS

R.J. Escalante-Gonz´alez Ph.D. student of control and dynamical systems at IPICYT thanks CONACYT for the scolarship granted (Register number 337188).

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