Hidden bifurcation in the Multispiral Chua Attractor Menacer Tidjani, René Lozi, Leon Chua To cite this version: Menacer Tidjani, René Lozi, Leon Chua. Hidden bifurcation in the Multispiral Chua Attractor. International journal of bifurcation and chaos in applied sciences and engineering , World Scientific Publishing, 2016, 26 (14), pp.1630039-1 1630023-26. 10.1142/S0218127416300391. hal-01488550 HAL Id: hal-01488550 https://hal.archives-ouvertes.fr/hal-01488550 Submitted on 13 Mar 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. January 6, 2017 11:8 WSPC/S0218-1274 1630039 Published in International Journal of Bifurcation and Chaos, Personal file Vol. 26, No. 14, (2016), 1630039-1 to 1630039-26 DOI 10.1142/S0218127416300391 Hidden Bifurcations in the Multispiral Chua Attractor Tidjani Menacer Department of Mathematics, University Mohamed Khider, Biskra, Algeria [email protected] Ren´eLozi Universit´eCˆote d’Azur, CNRS, LJAD, Parc Valrose 06108, Nice C´edex 02, France [email protected] Leon O. Chua Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA [email protected] Received August 20, 2016 We introduce a novel method revealing hidden bifurcations in the multispiral Chua attractor in the case where the parameter of bifurcation c which determines the number of spiral is discrete. This method is based on the core idea of the genuine Leonov and Kuznetsov method for searching hidden attractors (i.e. applying homotopy and numerical continuation) but used in a very different way. Such hidden bifurcations are governed by a homotopy parameter ε whereas c is maintained constant. This additional parameter which is absent from the initial problem is perfectly fitted to unfold the actual structure of the multispiral attractor. We study completely the multispiral Chua attractor, generated via sine function, and check numerically our method for odd and even values of c from 1 to 12. In addition, we compare the shape of the attractors obtained for the same value of parameter ε while varying the parameter c. Keywords: Chua circuit; multispiral attractors; hidden bifurcations. 1. Introduction differential equations which causes a sudden “qual- itative” or topological change in its behavior. The study of changes in the qualitative or topolog- However, when a system depends on a discrete ical structure of a given family of solutions of dif- parameter, no bifurcation can occur. In this article ferential equations which depends on a continuous we overcome this dilemma by, showing that in some parameter is known as bifurcation theory [Sat- cases, “hidden bifurcation” instead of conventional tinger, 1973; Lozi, 1975]. Initially the name “point bifurcation arises from a discrete parameter. of bifurcation” was defined as the value of the parameter for which the number of solutions of such The Chua attractor a family of differential equations is increased from 1.1. one to two, like the bifurcation of a branch of a Fifty years ago, the discovery of the first chaotic tree, or when a road divides suddenly in two dif- strange attractor by E. N. Lorenz [Lorenz, 1963], ferent ways. Nowadays bifurcation is understood introduced a drastic change in the mathematical as a small smooth change made to the parameter study of oscillations. Fifty years after, only few values (the bifurcation parameters) of a system of chaotic attractors involving differential equations page 1 January 6, 2017 11:8 WSPC/S0218-1274 1630039 remain actively explored. Among them Chua chaotic attractors belong either to the class of self- attractor is nowadays incredibly used, because of excited attractors or to the class of hidden attrac- both its realizations: electronic circuit or system tors [Leonov et al., 2011; Leonov & Kuznetsov, of differential equations can be combined for mul- 2011]. The basin of attraction of a self-excited tiple purposes [Duan et al., 2004]. Chua attractor attractor overlaps with the neighborhood of an equi- is the asymptotic attractor of solutions of the sys- librium point, therefore self-excited attractors are tem of differential equations modeling the dynam- very easy to be found. On the contrary a hidden ics of the Chua circuit. The Chua circuit is the attractor has a basin of attraction that does not simplest electronic circuit exhibiting chaos, and intersect with small neighborhoods of any equi- moreover possesses a very rich dynamical behav- librium points thereby making it very difficult to ior which was verified from numerous laboratory find; it is why one can call it hidden. Hidden experiments [Zhong & Ayrom, 1985], computer sim- attractors are important in engineering applications ulations [Matsumoto, 1984], and rigorous mathe- because they allow unexpected and potentially dis- matical analysis [Chua et al., 1986; Lozi & Ushiki, astrous responses to perturbations in a structure 1993]. L. Chua invented it in the fall of 1983 like an aircraft’s control systems (windup and anti- [Chua, 1992] when he was visiting T. Matsumoto windup), drilling systems and electrical machines at Waseda University (Tokyo, Japan). This elec- and in secure (chaotic) communications. In [Leonov, tronic circuit was mimicking directly on an oscillo- 2009b; Leonov et al., 2010], N. V. Kuznetsov and scope screen a chaotic signal. Its invention was due G. A. Leonov have proposed an effective method in response to the search of two unfulfilled quests for the numerical localization of hidden attractors in among many researchers on chaos concerning two multidimensional dynamical systems. This method wanting aspects of the Lorenz equations. The first is based on homotopy and numerical continuation. quest was to demonstrate chaos as a robust physi- They construct a sequence of similar systems such cal phenomenon, and not merely an artifact of com- that for the first (starting) system the initial data puter round-off errors. The second one was to prove for numerical computation of oscillating solution that a strange attractor, which was obtained by (starting oscillation) can be obtained analytically. computer simulation, is indeed chaotic in a rigorous Then the transformation of this starting solution mathematical sense. This circuit was the first real is tracked numerically in passing from one system (concrete) example of a physical device exhibiting to another. The first example of a hidden chaotic nonperiodic oscillations. strange attractor was found in the Chua attractor Since that discovery the Chua circuit has been [Leonov et al., 2012]. generalized to several versions. One generalization substitutes the continuous piecewise-linear function 1.3. Hidden bifurcations by a smooth function, such as a cubic polynomial As mentioned before, because the parameter c gov- [Khibnik et al., 1993; Shilnikov, 1994; Huang et al., erning the number of spirals in the modified Chua 1996; Hirsch et al., 2003], etc. circuit with sine function [Tang et al., 2001]isan Complex attractors with n-double spirals were integer, it is not possible to vary it continuously reported in [Suykens & Vandewalle, 1991]byintro- and therefore it is not possible to observe bifurca- ducing additional break points in the nonlinear ele- tion of attractors from n to n + 2 spirals when the ment in the Chua circuit, or using cellular neural parameter c changes. Moreover it is not possible to networks with a piecewise-linear output function use noninteger real values for c.Toovercomethis [Arena et al., 1996]. It is also demonstrated in obstacle, we use the core idea of the Leonov and [Suykens et al., 1997; Yal¸cin et al., 1999], that odd Kuznetsov method for searching hidden attractors number scrolls can be observed by a similar modi- (i.e. homotopy and numerical continuation) in a dif- fication. In [Tang et al., 2001], it is shown that the ferent way, in order to uncover hidden bifurcations, n-spirals attractors can be obtained with a simple governed by a homotopy parameter ε while keeping sine or a cosine function. c constant. Hidden attractors This paper is organized as follows: In Sec. 2,we 1.2. recall the model of Chua’s system with multiple- Recently a new concept concerning the classifica- spiral attractors generated via sine function as pro- tion of attractors has been introduced: periodic or posed in [Tang et al., 2001]. In Sec. 3,wepresent page 2 January 6, 2017 11:8 WSPC/S0218-1274 1630039 the analytical-numerical method for hidden attrac- where c and n ∈ N and tor localization proposed by Leonov [2009a, 2010]. π, if n is odd The third section is devoted to the localization of d = (4) hidden bifurcation in Chua’s system with multiple- 0, if n is even. spiral attractors. In addition, the results of numer- ical bifurcation of the analysis are given in Sec. 4. Itcanbeeasilyverifiedthatc governs the num- Finally, the conclusion with some discussion is given ber of periods existing in the function, and hence- in Sec. 5. forth the number of equilibrium points of (1). The equilibrium points are (xeq, 0, −xeq)withxeq =2ak k , ± ,...,±c 2. Chua’s System with and =0 1 [Tang et al., 2001]. When α =10.814,β =14.0,a =1.3,b =0.11; Multiple-Spiral Attractors 2-spirals, 4-spirals, 6-spirals, 8-spirals, 10-spirals Generated via the Sine Function and 12-spirals attractors are generated by Eqs. (1) In recent years, various approaches have been pro- and (2) for odd values of parameter c: c =1,c =3, posed, for generating n-scroll in Chua’s circuit.