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 instruments, 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: . 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: . 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: . 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: . 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: . nonlinear equations [7]. In particular, he came to conclude that a small numerical error could change large-scale features See 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

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 numerically 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 conference 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)

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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 the ESG for schools [24,25]. models of expressive faces) in order to powerfully engage and motivate learners In particular, the approach was based on realize a play explaining Chaos concepts. lacking strong scientific backgrounds. In the idea that artifacts (including digital The approach was linked to narrative particular, experimentation with 30 high ones) can increase students’ motivation, thinking [32]. school students (aged 16–18) demon- stimulating the learning process and These explorations of specific emer- strated a significant relationship between learners’ creativity while also reducing mental effort [26]. The theory refer- Fig. 4. Image from the cover of the International Journal of Bifurcation and Chaos (Vol. 15, enced was constructivism, which held no. 2, 2005). (© ESG) that direct contact with objects, manipulation (including virtual manipulation) and working groups encourage learners to think, to formulate hypotheses and to test them, increasing involvement and satisfaction [27,28]. An experimental laboratory based on edutainment (“education” plus “entertainment”) [29], in which students had to physically build a Chua’s Circuit, was thus conceived. The construction involved the manipulation of circuit components, the users’ understanding of the functioning of the circuit diagram and the learning of the impor- tance of connections. This arrangement [30] foresaw the virtual manipulation of specific elements through the use of the interface of the software Chaos Explorer, allowing a deep understanding of the evolution of dynamic systems through computer simulation, creating changes in the patterns derived from Chua’s Circuit (Fig. 2). Students produced art forms such as 2D and 3D images, sound and music compositions . Further studies of educational/cre- ative laboratories provided students with

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00641 by guest on 03 October 2021 the experimental group’s learning and its motivation. Moreover, the significant relationship between learning scores and the motivation subscale “Interest/Enjoy- ment” suggested that the more a student enjoys him/herself the higher the learning scores. Also, qualitative results have shown that engaging experiences are a crucial element for the teaching/learning of science topics [33].

New Routes to Creativity

In studying visualization of the dynamics of Chua’s Circuit through computa- tional models, we have taken the basic features of Chaos, such as the sensitive dependence on initial conditions, fractal structure and the qualitative trans- Fig. 5. Navigation of an avatar through a room of the virtual museum devoted to chaos, formations of behavior, into account. 2010. (© ESG) We have used these characteristics to investigate a new class of systems (at- Fig. 6. Design products derived from Chaos, 2007. (© ESG) tractors of Chua Machines) [34], which manifest unexpected behavior linked to the emergence of surprising forms, in order to produce artistic/mathematical products (Fig. 3). Fractals, dynamic systems and Chaos structures have been applied in different sectors of the arts, from painting to music, such as Pollock’s works [35–36], music composition [37] and electronic music [38–40]. Also, the Bridges Conference , running annually since 1998, has improved the dialogue between the mathematical and artistic points of view in various fields where artistic and mathematical thinking and practice merge. Artistic images and digital paintings using Chaos in their design [41] have been realized by many artists, and some images realized by ESG have been selected as covers for the International Journal of Bifurcation and Chaos, which is devoted to this scientific topic (Fig. 4). We have also organized an exhibit as part of the international project SCIENAR (Scien- tific Scenarios and Art), promoted by the EACEA Culture program , revealing the astonishing beauty of these patterns in real and digital form and enjoying great success with the public. However, in order to reach a wider audience, we created a website collecting our artistic works [42]. Furthermore, we have created a perpetual exhibit in the form of an immersive environment in a virtual museum devoted to strange attractors, in which the user, via an avatar, can observe attractors as results of Chaos Theory (Fig. 5). According to the learning principles of constructivism, the user can manip-

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Downloaded from http://www.mitpressjournals.org/doi/pdf/10.1162/LEON_a_00641 by guest on 03 October 2021 [47]; each actor is a virtual model of a fa- mous personality in the field of science, reconstructed on the basis of a set of pic- tures. In music, we have created software environments for sound and music pro- duction, as well as a CD-ROM contain- ing musical compositions realized using Chua’s Attractors [48]. The software ChaosExplorer, used in school laboratories as referred to above, is an effective tool to create music de- Fig. 7. Adolfo Adamo, theatrical performance, The Strange AttrACTOR, 2005. (© ESG, 2005). rived from Chaos. In fact, a mathematical model (a system of three differential equations in three unknowns) simulates ulate virtual objects and watch videos that on Chaos and strange attractors [46]. In Chua’s Circuit. The data derived from enrich the museum experience. Users the field of theater, Adolfo Adamo of the the simulation of the model are encoded have found the experience informative ESG has acted in and directed the ongo- and converted into images, sounds and and engaging at the same time [43–44]. ing performance “The Strange AttrAC- music through a series of computer al- They have the opportunity to expand TOR” (Fig. 7). Its main aim has been gorithms. Therefore, when one of the their own knowledge, taking charge of the use of the stage as another means five parameters changes, the attractor the complete learning experience in to research, experiment and think about shape varies significantly. Since an allow- ways that best suit their individual needs humanity, identity, infinity and art. The able range of values has been selected and interests. Users can explore the vir- intrinsic metaphor has been that psycho- for each parameter, a collection of basic tual museum as they would a museum logical dynamics are chaotic: Sometimes configurations has been provided. How- in real life (a map on the right side of they exist in a steady state and sometimes ever, the manipulation of the parameters the desktop helps users who choose they are disordered and/or disorga- is achieved through a very simple and this modality of navigation). Another nized, influencing behavioral expres- intuitive interface. In fact, if the vertex approach to the environment is based sions. Music (composed using dynamic of a star configuration is manipulated, on gestures received by the Wii wireless systems—Cellular Automata and Chua’s the attractor shape changes [49] (Fig. controller, offering an experience that is System) and videos/images created us- 8). Thus, very small modifications of pa- especially engaging for young users. In ing strange attractors (emerging from rameters radically change the structure addition to this modality, very similar to Chua’s Oscillator) have also been used of the obtained attractor. The attractor videogames, the user can use the mouse/ in the performance. shapes can be saved as JPEG images or keyboard/joystick in conventional In the area of virtual environments, as 3D models in VRML format; ChaosEx- ways to move and to explore the envi- we have realized a video on Chaos and a plorer has also been developed to work ronment. 3D animated movie faithfully re-creating on the iPad and iPhone. Other research products, such as de- the story of the genesis of Chaos theory Another tool conceived by ESG us- sign elements (Fig. 6), logos and other (Article Frontispiece). The 3D movie ing mathematical models to gener- visual communication tools [45], proto- uses virtual characters and environments ate, analyze and synthesize sounds is types of new musical instruments and based on real events and was realized in ImaginationTools music software [50]. It uses a new interdisciplinary research. sor Chua in order to celebrate his work philosophy of emergent musical software Moreover, we have also used represen- and in particular to tell the story of the together with an accessible 3D graphical tations of attractors in interactive con- birth of Chua’s Circuit and the scientific user interface; users can join a series of texts to create virtual environments and knowledge derived from it. Fifteen char- tools, graphically represented as spheres, multimedia theater performances based acters have been created for the video according to simple rules in order to

Fig. 8. The star configuration in Chaos Explorer for Windows (left) and for iPad and iPhone (right), 2012. (© ESG)

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Assunta Tavernise holds a Ph.D. in Psychol- mesis, and Analogy in Experimental Music, Chaotic ogy of Programming and Artificial Intelligence Modeling and Simulation (CMSIM) 1: 127–137. 54. F. Bertacchini, E. Bilotta, P. Pantano, Il Caos é semplice e tutti possono capirlo (Bologna: Muzzio and collaborates with the Laboratory of Psy- 41. E. Bilotta, P. Pantano, F. Stranges, “A Gallery editore, 2009). chology at University of Calabria. Her research of Chua Attractors.” Part II, International Journal interests concern various scientific topics from of Bifurcation and Chaos Vol. 17, No. 2, 293–380 55. F. Bertacchini, E. Bilotta, G. Laria, P. Pantano, (2007); E. Bilotta, F. Stranges, P. Pantano, “A Gallery “The genesis of Chua’s circuit. Connecting Science, an interdisciplinary point of view and com- of Chua Attractors.” Part III, International Journal Art and Creativity”. In A. Adamatzy & W. Cheng prise the following areas: educational technol- of Bifurcation and Chaos Vol. 17, No. 3, 657–734 (Eds.), Chaos, CNN, Memristor and beyond. Singa- ogy, virtual worlds/games, human computer (2007); E. Bilotta, G. Di Blasi, F. Stranges, P. Pantano, pore: World scientific, 108–123 (2013). interaction, edutainment and virtual agents. “A Gallery of Chua Attractors.” Part V, International Journal of Bifurcation and Chaos Vol. 17, No. 5, 56. Bilotta et al. [24]. She has worked in national and international 1383–1511 (2007). projects, including “Virtual Museum Net of Magna Graecia” (ROP 2000—003 ) and “Connecting European for Learning Chaos,” Proceedings of INTED 2010 Francesca Bertacchini is a research fellow at Culture through New Technology—NETCon- Conference, 8–10 March 2010, Valencia, Spain, nect” (Culture 2000 European Programme, 5845–5853. the University of Calabria. She collaborates with the ESG Group since 2005 and her ). 43. F. Bertacchini, E. Bilotta, P. Pantano, A. Taver- research interests are Edutainment, Virtual nise [30]. Storytelling and Communication of Science. 44. F. Bertacchini, E. Bilotta, E. Bossio, S. Vena, P. At the moment she is working on the Art & Glossary Pantano, “Learning Chaos in an Interactive Virtual Science ESG manifesto, dealing with scientific Museum,” Proceedings of VIWO 2009 (ICWL 2009), attractor parameter space—the sequence of values 19–21 August 2009, Aachen, Germany, VIII, 1–16. visualization and educational-entertaining for each parameter of the chaotic system. Over this environments for science communication. space, the qualitative and quantitative behavior of 45. F. Bertacchini, E. Bilotta, P. Pantano, “Educa- the attractor changes. tional Virtual Scenario for Learning Chaos and Complex Theories,” The International Journal of Eleonora Bilotta is a professor of cognitive psy- Chaos theory—the description of the behavior Virtual Reality Vol. 7, No. 2, 19–26 (2008). chology and coordinator of the Ph.D. course in of dynamic systems having the following features: Complex Systems at the University of Calabria. sensitive dependence on initial conditions, strange 46. A. Adamo, A. Tavernise, “Generation of Ego dy- attractors, many routes to chaos and fractal basin namics,” Proceedings of the VIII International Con- She is one of the directors of the Evolutionary boundaries. ference on Generative Art, 15–17 December 2005, Systems Group and is a member of several national and international scientific communities. third order; represents the simplest tool capable of dell’atto unico L’attrATTORE strano,” Proceedings showing chaos and the only system by which the pres- of the Convegno Matematica, Arte e Industria Cul- ence of chaotic behavior has been experimentally, turale, 19–21 May 2005, Cetraro—Italy. Lorella Gabriele is a postdoc researcher. Her mathematically and numerically proven, exhibiting 47. E. Bilotta, F. Bertacchini, G. Laria, P. Pantano, A. a rich variety of behaviors such as chaos, stochastic research interests concern psychology of pro- resonance and intermittence. Tavernise, “Virtual Humans in Education: Some im- gramming, educational technologies and their plementations from research studies,” Proceedings of EduLearn2011 Conference, Convegno IATED, use in didactic context and Human Computer fractals—mathematical objects in which the parts 4–6 July 2011, Barcelona, Spain, 6456--6464. Interaction. Since 2009 she has been a referee refer to the whole. This property is known as self- similarity. for different journals and conferences, includ- 48. E. Bilotta, P. Pantano, E. Cupellini, C. Rizzuti, “Evolutionary Methods for Melodic Sequences ing Computers & Education and Scientific strange attractors—complex mathematical patterns Generation from Non-linear Dynamic Systems,” Research and Essays. She has worked in dif- produced by a chaotic nonlinear dynamic system.

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