Toward the Use of Chua's Circuit in Education, Art and Interdisciplinary

Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00641 by guest on 03 October 2021 general article Toward the Use of Chua’s Circuit in Education, Art and Interdisciplinary Research: Some Implementation and a b s t r a c t Opportunities This paper considers the merging of Chaos with art, including such forms as digital images, sounds and music, based on dynamic systems derived from Francesca Bertacchini, Chua’s Circuit and using appro- priate coding methods. Design Eleonora Bilotta, Lorella elements, logos, musical instru- ments, software environments, Gabriele, Pietro Pantano multimedia theater performances and virtual museums with strange attractors have and Assunta Tavernise also been realized. In the field of education, the paper intro- duces environments that have foreseen the virtual manipulation of patterns derived from Chua’s Circuit, which has fostered a ccording to Greek mythology, Chaos (or Chua’s Circuit was the first circuit deeper understanding of the A implementation expressly designed evolution of dynamic systems Khaos) was the gaping shapeless void from which the cosmos and the gods originated. The underlying principle was that to exhibit Chaos and the first to be through computer simulation. the world had too many complex phenomena for one to grasp rigorously proven to be chaotic [4]. all of them. From a scientific point of view, Chaos has been Chua’s Circuit is also the simplest hailed as one of the major discoveries of the 20th century [1]. physical system in which Chaos can Chaos theory is pervasive in many fields of science, including be observed [5], and it is able to create a large number of experimental ones. In mathematics it is applied to nonlinear structures of many different patterns and sizes [6]. dynamic systems with deterministic behavior that evolve in an In this paper, the milestones of Chaos history, from the use apparently random sensitive dependence on initial conditions of linear models to explain natural events to the educational [2]. The word nonlinear refers to the mathematical model used and artistic use of strange attractors, are described. Imple- for describing the system, and dynamic to essential information mentations and initial findings from research on the connec- about the system (the data), as well as to the rule that describes tions among Chaos, art and education are presented as well. the system’s condition over time. Deterministic here means that In particular, the following sections present the development the behavior of the systems is determined by initial conditions, of Chaos studies in the field of mathematics and then show without any involvement of random elements. That is, these the application of the mathematical work to educational en- systems evolve in different ways according to the given set of vironments and art (i.e. music and graphics). Finally, further initial conditions and parameters; as a result, chaotic systems developments in the research, linked to the merging of science are able to create a number of unpredictable shapes, known as with art, are clarified. “strange attractors.” Geometrically, an attractor can be a point, a limit cycle, or a fractally structured shape; from an artistic point of view, it affords the opportunity to create infinite forms The MilesTones of Chaos hisTory using a digital tool [3]. and The eMerging of MusiC In the 20th century, more precisely during the 1960s, several combinations of technological achievements, discoveries and insights brought revolutionary change to science. Thanks to the pioneering work of Fermi in the 1950s, scientists began to Francesca Bertacchini (researcher), Università della Calabria, via P. Bucci, cubo 17/B, use the electronic calculator, understanding the role of non- 87036 Arcavacata di Rende—Cosenza, Italy. E-mail: <[email protected]>. linear differential equations. However, even if these mysterious Eleonora Bilotta (researcher), Università della Calabria, via P. Bucci, cubo 17/B, 87036 Arcavacata di Rende—Cosenza, Italy. E-mail: <[email protected]>. equations were descriptive structures that allowed the behavior Lorella Gabriele (researcher), Università della Calabria, via P. Bucci, cubo 17/B, 87036 of natural and biological phenomena to be understood, their Arcavacata di Rende—Cosenza, Italy. E-mail: <[email protected]>. solutions were only accessible by computer simulations. Pietro Pantano (researcher), Università della Calabria, via P. Bucci, cubo 17/B, 87036 Also in these years, Edward Lorenz used computer simula- Arcavacata di Rende—Cosenza, Italy. E-mail: <[email protected]>. tions to describe meteorological phenomena through three Assunta Tavernise (researcher), Università della Calabria, via P. Bucci, cubo 17/B, 87036 Arcavacata di Rende—Cosenza, Italy. E-mail: <[email protected]>. nonlinear equations [7]. In particular, he came to conclude that a small numerical error could change large-scale features See <www.mitpressjournals.org/toc/leon/46/5> for supplemental files associated with this issue. of the weather (the “Butterfly Effect”). In fact, his computer- simulated weather model was highly sensitive to initial condi- article Frontispiece. screenshots from the video “the genesis of tions, but he took more than 30 years to prove definitively chua’s circuit” (© esg, 2012). that his observations were chaotic [8]. However, Lorenz’s © 2013 ISAST doi:10.1162/LEON_a_00641 LEONARDO, Vol. 46, No. 5, pp. 456–463, 2013 457 Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00641 by guest on 03 October 2021 Lorenz’s system and two in Rössler’s), so Chua determined to build a tool with a new element, today known as the “Chua’s diode” [11]. Hence, thinking on the essence of Chaos, Chua de- vised the first circuit in which Chaos was mathematically, physically and numeri- cally proven. Chua’s Circuit gave rise to an important trend in contemporary scientific research and to the publication of thousands of scientific papers on the subject. While only two or three patterns were derived from Lorenz’s system, 40 different attractors originating from Chua’s Circuit were found [12] (Fig. 1). In fact, unlike Lorenz’s attractor, which generates just a few kinds of dynamic patterns, Chua’s Circuit and systems derived from the circuit can produce a large number of chaotic structures [13,14]. Therefore, in 1993, the circuit became recognized as the experimental paradigm of Chaos [15]. Fig. 1. Patterns derived from chua’s and lorenz’s systems, 2007. (© esg) In 2003, during an international con- ference organized at the University of Calabria, Leon Chua visited the labora- butterfly-like strange attractor, with its In the early 1980s, Leon Chua, then tory of our Evolutionary Systems Group fractal structure, similar to Mandelbrot’s a professor of electrical engineering at (ESG) at the university; at that time we set, became the prevailing icon of Chaos University of California, Berkeley, was were working on the translation of dy- and, in general, of contemporary science working at the intersection between namic systems, including Chua’s Circuit, [9]. Moreover, even if Lorenz’s discovery science and technology. Meanwhile, into music [16]. A collaboration ensued, was extraordinary and changed scientific a group of Japanese scientists, led by and applications of Chaos theory turned thinking, unfortunately it remained a Professor Matsumoto at the Univer- from subjects such as physics, engineer- mathematical abstraction, not embodied sity of Waseda (Tokyo), built a circuit ing, economics and biology to learning in engineering systems. implementing Lorenz’s equations, car- and art. As a consequence, a topic usu- In the early 1970s, Chaos became a rying out the first experiment meant to ally studied only by specialists and taught crucial topic, but both the lack of tools demonstrate the physical existence of at the university level entered secondary for experimental demonstrations and Chaos. Leon Chua attended the event. school laboratories and the art world. the impossibility of reproducing it made The experiment failed. That same night, the phenomenon of little practical inter- however, Chua realized how to repro- est. Engineers did not consider the sub- duce Chaos using an electronic circuit. learning Chaos and The ject noteworthy for their studies, but the He noticed that Lorenz’s and Rössler’s Merging wiTh arT phenomenon attracted the attention of systems [10] had at least two unstable After some specific studies [17–22] on many mathematicians and physicists. equilibrium points (three points in the relationship between learning and Fig. 2. students using the chaos explorer software to create artistic images and sounds, 2009. (© esg) 458 Bertacchini et al., Toward the Use of Chua’s Circuit in Education, Art and Research Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00641 by guest on 03 October 2021 Fig. 3. strange attractors as artistic shapes. (© esg) Chaos and important research carried a software application called Face3D, de- gence-related concepts in a hands-on out by Gandhi, Cserey, Zbrozek and voted to the creation of virtual theatri- environment, which were developed Roska [23], a teaching/learning model cal performances [31]. In this interactive out of regular school programs, dem- for the study of Chaos through the cre- learning environment, students could onstrated that specific edutainment lab ation of sounds and digital images was manipulate 3D content (parameterized work on difficult scientific topics could created by

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