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Analysis of a Class of Complex System Without Equilibria Via Copyright © 2018 IFAC This article may be downloaded for personal use only. Any other use requires prior permission of the author or publisher. 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-Gonz´alez∗ E. Campos-Cant´on∗∗ linear affine systems. R.J. Ecalante-Gonz´alez∗ E. Campos-Cant´on∗∗ R.J. Ecalante-Gonz´alez∗ E. Campos-Cant´on∗∗ (a) (b) ∗ ∗∗ In section 2 basic theory of PWL systems and switched R.J. Ecalante-Gonz´alezDivisi´onde Matem´aticas∗ E. Campos-Cant´on Aplicadas, ∗∗ Divisi´onde Matem´aticas Aplicadas, control laws is given. In section 3 a possible electronic Instituto PotosinoDivisi´onde de Investigaci´onCient´ıfica Matem´aticas Aplicadas, y Tecnol´ogica A.C. Instituto Potosino de Investigaci´onCient´ıfica y Tecnol´ogica A.C. realization for a chaotic attractor in a system without CaminoInstituto a Potosino la PresaDivisi´onde de San Investigaci´onCient´ıfica Jos´e2055 Matem´aticas Col. Lomas Aplicadas, y 4a Tecnol´ogica Secci´on,78216, A.C. Camino a la Presa San Jos´e2055 Col. Lomas 4a Secci´on,78216, equilibria with one scroll is introduced. In section 4 a CaminoInstituto a Potosino la PresaSan de San Luis Investigaci´onCient´ıfica Jos´e2055 Potos´ı, S.L.P., Col. Lomas M´exico y 4a Tecnol´ogica Secci´on,78216, A.C. San Luis Potos´ı, S.L.P., M´exico new class of dynamical systems without equilibria with Camino a la∗ Presae-mail:San San Luis [email protected] Jos´e2055 Potos´ı, S.L.P., Col. Lomas M´exico 4a Secci´on,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., M´exico ∗∗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 InchaoticAbstract: 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 chaoticbyIn 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 bysystemchaotic 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: systemattractor.by 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.thesystem 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; Thetheattractor. 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 ∈ 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 x˙ = f(x)= . (1) Gonz´alez and Campos-Cant´on(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 . andunstable 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 producingChuaPWL 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 Chuasystemsproducing 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 systemsdimensionalChua 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 dimensional(2014).systems This without hyperchaoticsystem equilibria was obtained havesystem been fromintroduced reported, an approximation in as theLi et four- al.
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