Chaos, Solitons and Fractals 113 (2018) 69–78 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos A fractional order chaotic system with a 3D grid of variable attractors J.M. Munoz-Pacheco a , E. Zambrano-Serrano a,∗ , Ch. Volos b , O.I. Tacha b , I.N. Stouboulos b , V.-T. Pham c a Faculty of Electronics Sciences, Autonomous University of Puebla,Puebla Av. San Claudio y 18 Sur, Edif. FCE2, 72570, Mexico b Laboratory of Nonlinear Systems, Circuits & Complexity, Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, GR-54124, Greece c School of Electronics and Telecommunications, Hanoi University of Science and Technology, 01 Dai Co Viet, Hanoi, Vietnam a r t i c l e i n f o Article history: Received 24 January 2018 Revised 5 May 2018 Accepted 18 May 2018 Keywords: Fractional order Chaotic attractor Adjustable variables Grid-scroll Finance system a b s t r a c t A novel fractional order dynamical system with a variable double-scroll attractor on a line, lattice and 3D grid is introduced. This system belongs to a class of chaotic systems with adjustable variables but with fractional order. Chaos generation only depends on the value of fractional order. As a result, a chaotic attractor is discovered and propagated in y-line. By introducing two extra control parameters, we also observed that the chaotic attractor varies in x-line, z-line, x − y-lattice, x − z-lattice, y − z-lattice, and 3D- grid. Dynamics of the new system are discovered by using phase portraits, bifurcation diagrams, Lyapunov spectrum, Kaplan–Yorke dimension, dissipative measure. Finally, the proposed fractional order system is designed with analog electronic circuits. Circuit results are in concordance with theoretical findings. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Chaotic systems have been studied from many years ago [1,2]. Its random-like behavior is quite interesting for several chaos- based applications ranging from living systems [3,4], to nonliving systems [5–8]. Currently, continuous-time chaotic systems are clas- sified, depending on their attractors, in two main categories: self- excited [9–13], and hidden [14–19]. In both categories, the chaotic behavior have been found not only for the integer-order domain but also for fractional order. A fractional order dynamical system is considered as a gen- eralization of the integer-order versions [10,20,21]. Several works showing the fractional order form of well-known chaotic systems such as Chua system, Chen, Lorenz and so on, were reported [22–25]. Also, new chaotic systems with fractional derivatives have been proposed [10–12,26,27]. The main research interest is related to hereditary properties, i.e., memory, that the fractional order dy- namical systems can exhibit. Indeed, those special features are very useful to describe more accurately many real world phenomena in different fields such as biology [12,28], economics [29], botany [30], hidden dynamics [14], digital circuits [27], cryptography [31], ciphers [32,33], control [34], telecommunications [35–37], image processing [38], wind turbines [39], viscoelastic studies [40], ferro- electric materials [41], and so forth. Therefore, each new fractional ∗ Corresponding author. E-mail address: erneszambrano@gmail.com (E. Zambrano-Serrano). order chaotic system is a potential candidate to improve or pro- pose chaos-based applications. Likewise, the concept of imperfect uncertain system has been in- troduced [7,8,42]. Those systems are far from ideality but coex- ist with imperfections, which are needed for a proper behavior [7,8,42–46]. The nonidealities conducts to dynamics hidden or im- perfect dynamics, for instance, a chaotic behavior was observed by exploiting those imperfections in Refs. Buscarino et al. [7] and Yim et al. [43]. Therefore, the presence of imperfections, nonidealities, and uncertainties has to be explicitly taken into account in the model, so that it represents the actual behavior observed in the physical realizations. In this context, the aforementioned character- istics of the fractional order systems provides an approach to study imperfect systems, e.g. the imperfect (fractional) order can lead to find complex dynamics, which would be not possible to observe, in some cases, by considering only integer-order derivatives. Additionally, a class of chaotic systems with variable attractors have been recently proposed [47–51]. These systems are pointed out as a proper solution for chaos-based applications since it can reduce the number of electronic components required for signal conditioning due to the chaotic signals can be designed with any polarity. It means that the location of chaotic attractor is variable on phase space and it can be chosen arbitrarily according to offset control parameters. Up to now, the majority of published works have described integer-order chaotic systems with one boostable variable [47–51], and two boostable variables [52]. In this framework, a novel fractional order chaotic system with a 3D grid of variable attractors is introduced. The proposed sys- https://doi.org/10.1016/j.chaos.2018.05.015 0960-0779/© 2018 Elsevier Ltd. All rights reserved.