Subject of this article is visualization ofchaotic attractor in 3D space. 3D view of chaosenables create a demonstrative projection aboutrelations of chaos generated by physical circuit -Chua’s circuit. The program created in C++program language enables view on chaotic attractorboth in 2D and 3D space together with plane objectsvisualization – elements of singularities. The Chua’sattractor visualisation represents visualization oflarge graphical data set

Branislav Sobota

Milan Guzan

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 Visualization of chaos  Milan Guzan* and Branislav Sobota **  *Department of Theoretical Electrotechnics and Electrical Measurement, Technical University of Kosice, Faculty of Electrical Engineering and Informatics,  Park Komenskeho 3, 040 20 Kosice, Slovak Republic, E-Mail: milan.guzan@tuke.sk  * * Department of Computers and Informatics, Technical University of Kosice, Faculty of Electrical Engineering and Informatics,  Letna 9, 040 20 Kosice, Slovak Republic, E-Mail: branislav.sobota@tuke.sk   Abstract – Subject of this article is visualization of chaotic attractor in 3D space. 3D view of chaos enables create a demonstrative projection about relations of chaos generated by physical circuit - Chua’s circuit. The program created in C++ program language enables view on chaotic attractor both in 2D and 3D space together with plane objects visualization – elements of singularities. The Chua’s attractor visualisation represents visualization of large graphical data set.  Keywords:  chaos, Chua’s circuit, singularity, 3D.  I. INTRODUCTION  The first work aimed on Chua’s circuit [1] draws a publication explosion dealing with Chua’s circuit, chaos and its utilization. The work [2] states nearly 800 scientific articles dealing with this subject by the year 2004. Based on works [3], [4] and [5] the work [6] was published about influence of elements of singularities on chaos relations. This work motivated an idea of singularity element 3D visualization together with chaotic attractor presented in this paper.   II. CHUA’S CIRCUIT   Chua’s circuit is shown on Fig.1a, I-V characteristic of non-linear resistor (NR) is shown on Fig.1b as voltage dependence u1 – g(u1). The circuit is described by equation system (1).   ( )( )( )1 1 2 1 1 12 2 1 2 22 3( ) ( )( )C du dt G u u g u I QC du dt G u u i QL di dt u i Qρ= − − − ≡= − + ≡= − − ≡       (1)  where   ( )( )( )( )1 2 1 1 0 1 12 1 1 11( ) 212P PO Og u m u m m u B u Bm m u B u B= + − − − + ++ − − − +  (2)  Next we consider control pulse I=0, the resistance of the inductance ρ=0 and parameters (3) C1=0,1, C2=2, L=0,142857, G=0,7,                           m0=−4, m1=−0,1, m2=5, Bp=1, B0=14        (3)    III. LINEARIZATION OF THE SYSTEM AND SINGULARITIES The linearization of (1) can be written in the form                         $ [ ] $d x dt x= A                      (4) where 1 2[ , , ] tx u u i=$  and [A] is the Jacobi matrix with eigenvectors λ1, λ2, λ3, belonging to corresponding singularity P+, P-, 0 in Fig.1b. Since we are in the space R3, g(u1) in Fig. 2a) is a surface coplanar with u2. However, since the singularities are solutions to the system                  1 2 30, 0 a 0Q Q Q= = =                   (5) it is obvious that they lie in the plane u2=0 which is the plane of Monge projection in Fig. 2a). In it lies also the cross-section of the load plane Q2=0, whose a)GLC1iΙu2C2 u1(u )1g     b)       -15 -10 -5 0 5 15-10-5010Gu1iP +P -m1m2m1m2BP-B0B0-BPm0[A][V] Fig.1a) Chua’s circuit b) I - V characteristics of conductance G and NR NR g(u1) 48  cross-section at Q3=0 is a straight line. Hence its common points define three singularities P+, P-, 0 in all three projection planes in Fig. 2a) b) c). Chua’s circuit is the first physical circuit with all three singularities unstable at (2) and (3). Co-ordinates of the singularities are listed in Table 1.  TABLE 1. Co-ordinates of the singularities of Chua’s circuit  Singularity i [A] u 1 [V]  u2 [V] P− 4,55 -6,5 0 0 0 0 0 P+ -4,55 6,5 0  It is necessary to point, that existence of three unstable singularities is necessary, but not sufficient condition for presence of chaos in Chua’s circuit. Presence of 3 unstable singularities and consequently generation of chaos with help of structure similar as Chua’s circuit is presented at work [7].  IV. GEOMETRIZATION OF THE SYSTEM (1) WITH PARAMETERS (3) With the above assumptions the system exhibits two attractors. One of them is a limit cycle CS and the other is a chaotic attractor CCH in Fig. 2. The projections CS are stunning in that despite the Monge projection CS in Fig.2 a) and b) the redundant view c) is perfectly elliptic. [6] The shape of this view is explained by Fig.1a since i and u2 are quantities corresponding to the ideal LC2 filter. From a technical viewpoint the task of twisting a wire so that two projections have the shape a) b) and the other projection the shape c) would be a difficult one if only one of the projections would be available. Let us remark that the chaotic attractor CCH in Fig. 2 is purposely "sparse" for clarity. The arrows on LC indicate direction of representative point movement in state space along trajectory. The symbol “+” indicates time track t=0,18s on LC in the various projections.  V. ELEMENTS OF THE SINGULARITIES There was, in work [6], defined singularity element by equation:                    1 11 1 12 2 13y u u iα α α= ∆ + ∆ + ∆           (6)  regardless of whether λ1>0 or λ1<0. When λ1>0, the element would be a REPULSOR. At λ1<0 the element would be an attractor. Eigenvalues and eigenvectors of singularities are listed in Table 2 Since the singularities and the characteristics g(u1) are symmetric, eigenvalues and eigenvectors (α11, α12, α13) of P+ are identical to those of P-.   Fig.2 Projection of attractors of the system (1) Chaos attractor Limit cycle 49  TABLE 2. The eigenvalues and eigenvectores of the singularities P (P+, P-) and 0   α11 α12 α13 λ1 -6,372793 P λ2,3 0,0113967 ±1,815248i 11,966291 -12,74558 1 λ1 33,073071 0 λ2,3 -0,211535 ±1,856749i 316,8296 66,1461426 1  In Fig. 3a are denoted the elements containing all singularities. They are uniquely defined by main lines (ML) of elements in such a way that ML passing for example through the singularity P+ is denoted in Fig.3a by the symbol MLP+. The parameters ∆i=10A and ∆i=–10A suggest that the ML are parallel with the projection plane i=0A so that they are lower than the main line MLP+, or higher by the value ∆i=10A. The interpretation of symbols with the other singularities is analogous. More intricate attractor according [1] for parameters (7) is shown on Fig.3b. Denotes and descriptions of main lines are consistent with Fig.3a. This attractor will be subject of visualisation in next part of the contribution.  3D visualisation of this attractor including elements of  singularities (E0 – is element of singularity 0, EP+ and EP- - are elements of singularities P+ and P-) is shown in Fig.4, Fig.5.  C1=1/9, C2=1, L=0,142857, G=0,7,                     m0=−0,8, m1=−0,5, m2=5, Bp=1, B0=14           (7)  Coordinates of singularities, its eigenvalues and eigenvectors are listed in Table 4 and Table 5.   TABLE 4. Co-ordinates of the singularities of Chua’s circuit for parameters (7)  Singularity i [A] u 1 [V]  u2 [V] P− 1,05 -1,5 0 0 0 0 0 P+ -1,05 1,5 0  TABLE 5. The eigenvalues of the singularities for parameters (7)   α11 α12 α13 λ1 -2,75949 P λ2,3 0,129745 ±2,13289i 2,0131973 -2,7594902 1 λ1 1,552176 0 λ2,3 -0,6760882 ±1,8978201i 1,6659961 1,55217653 1   V. VISUALISATION  During the past several years, high-performance and feature-rich PC graphics interfaces have become      Fig.3 Projection of chaotic attractors and elements of singularities in plane u2,u1 of the system (1) for parameters: a) (3), b) (7)   available at low cost. High-performance graphics devices shorten computational visualising time. High-resolution display solutions increase quality and ability of immersion into 3D explored space (e.g. in virtual labs [8]). Graphical visualising systems are often based on serial using of information. On the input side is description of visualised space (e.g. using appropriate formulas) and on output side are correct coloured pixels on the screen. Between these two stages there are some other stages partially changed the input information. A modern visualisation brings new dimensions to the understanding of actions and events still hard understandable. Chua’s circuit comes under this area. Therefore visualising system was developed [9]. It visualizes Chua’s attractor consist of appropriate curves or planes. The curves/planes can be single coloured with different colours. The visualisation of chaotic attractor in time and 3D space is graphical output of application. This visualisation depicts representative point movement and it creates chaotic attractor using two basic modes (continuous and sequential). The animation of representative point 50  movement in state space in comet form or in continuous attractor form is final result of this application.  Chaos visualising system provides three basic visualising modes: continuous mode (see Fig. 4a), sequential mode (see Fig. 4b) and I-V characteristics visualisation mode. System allows using four projection types (3D u1 – i - u2 projection (see Fig. 5), 2D i – u2 projection, 2D i – u1 projection and 2D u2 – u1 projection) for better examined circuit understanding.    a                                         b  Fig.4 Examples of attractor displaying in continuous mode (a) and in sequential mode (b)    Fig.5 An elements of the singularities displaying as transparent planes in 3D space  Settable basic visualising parameters for chaotic attractor visualisation are: drawing speed, points omission, chaotic attractor point size, comet length and attractor colour. The combination of these parameters defines final visualisation form of chaotic attractor. In case of 3D graphic accelerator supported acceleration of graphic interface OpenGL is drawing of 3D primitives accelerated by graphic card. It enables increasing performance and using some graphical improvements can not be used without acceleration in real time. Chaos visualising system is designed for visualization Chua’s attractor in 3D space in real time and it is based on visualising kernel developed on DCI FEEI TU Košice [10]. An application is implemented in C++ language using OpenGL graphics library in Borland Developer Studio development environment. It enables visualisation for user defined Chua‘s attractor where user has in standard visualising mode (3D projection) 6 degrees of freedom for motion in explored attractor‘s space, which mean translation in 3 axis and rotating around them. It means that system is capable to visualise the Chua’s attractor from any point and from any angle. The application works under MS Windows XP.  VI. CONCLUSIONS  Visualization of chaotic attractor provides better understanding of representative point movement in state space, what was still possible only with help of representation in projection planes.  ACKNOWLEDGMENTS  This work is supported by following grant projects: VEGA No. 1/0646/09: Tasks solution for large graphical data processing in the environment of parallel, distributed and network computer systems. KEGA No. 3/5227/07: Virtual laboratory of telemetric systems.  REFERENCES  [1] T. Matsumoto, "A Chaotic Attractor from Chua's Circuit," IEEE Transactions on Circuits & Systems, vol.CAS-31, no.12, pp.1055-1058, 1984. [2] Chua's Circuit and Chua's Equation, URL: http://www.eecs.berkeley.edu/~chua/circuitrefs.html.  [3] Špány, L. Pivka, "Invariant Manifolds and Generation of Chaos", Elektrotechnický časopis, vol. 39, pp. 417-431, 1988, (in slovak). [4] V. Špány, L. Pivka, "2-Segment Bistability and Basin Structure in 3-Segment PWL Circuits", IEE Proceedings-G, 140(1), pp. 61-67, 1993.  [5] L. Pivka, V. Špány, "Boundary Surfaces and Basin Bifurcations in Chua s Circuit" Journal of Circuits, Systems and Computers, 3(2), pp. 441-470, 1993. [6] M. Guzan, P. Galajda, L. Pivka, V. Špány, "Element of singularity is a key to laws of chaos", 15th International Czech-Slovak Scientific Conference RADIOELEKT -RONIKA 2005, pp.33-36, May 2005. [7] J. Petržela, Z. Kolka, S. Hanus, “Simple Chaotic Oscillator: from Mathematical Model to Practical Experiment“, in Radioengineering, vol. 15, No. 1,  pp.6-11, 2006. [8] I. Zolotová, J. Liguš, A. Jadlovská, "Remote and Virtual Lab – CyberVirtLab", In: Proceedings of the 17th EAEEIE Annual Conference on Innovation in Education for Electrical and Information Engineering, Craiova, Romania, pp. 339-342, ISBN 973-742-350-X, 978-973-742-350-4, June 2006.  [9] R. Mocnár, "Chaotic attractor visualisation in state space; Diploma thesis, DCI FEEI TU Košice, 2006 (in slovak). [10] B. Sobota, "Parallel hierarchical model of visualisation computing", Journal of Information, Control and Management Systems, Volume 5, pp. 345-350, ISSN 1336-1716, 2007. E0 E0 EP- E0 E0 EP+ EP- EP+ 51

Visualization of chaos

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Visualization of chaos

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