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High Dimensional Chaos Tutorial Session IASTED International Workshop on Modern Nonlinear Theory (Bifurcation and Chaos) ~Montreal 2007~ Zdzislaw Musielak, Ph.D. and Dora Musielak, Ph.D. University of Texas at Arlington (UTA) Arlington, Texas (USA)

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Lecture 3 4D Rössler system Other HD Lorenz models Another HD Duffing system Double pendula Other interesting HD systems Routes to chaos Summary Objective: Review other systems that show high-dimensional chaos (HDC) and determine basic routes to HDC (HDC) and determine basic routes to HDC

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4D Rössler System First HD system with two positive Lyapunov First HD system with two positive Lyapunov exponents was introduced by Rössler (1979) exponents was introduced by Rössler (1979)

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Strange Attractor I Plane projection of the strange attractor First-return map to a Poincaré section Strange attractor is characterized by two positive, one negative and one zero Lyapunov exponents.

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Strange Attractor II AIHARE, Electrical Eng. Co., Japan

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HD Lorenz Models I Li, Tang and Chen (2005) generalized the 3D Lorenz model by adding a new variable that couples to the second equation of Lorenz’s equations and derived a 4D Lorenz model. They designed a circuit that approximates the 4D system.

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Theory vs Experiment Li, Tang and Chen (2005)

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HD Lorenz Models II 9D Lorenz model (Reiterer et al. 1998) Model describes a 3D Rayleigh-Benard convection Model does not conserve energy in dissipationless limit (Roy & Musielak 2006) Hyperchaos at R = 43.3 Period-doubling cascade

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Another HD Duffing System Savi & Pacheco (2002)

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Phase Portraits Savi & Pacheco (2002)

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Double Pendula I Ross Bannister: Initial speeds, left: Initial speeds, right: main arm = degrees/sec main arm = degrees/sec secondary arm = 0.0 degrees/sec

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Double Pendula II Bannister (2005)

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Coupled Logistic Maps General route to HDC - Harrison & Lai (1999, 2000) Pazo et al (2001)

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Coupled Rössler Systems Harrison & Lai (2000)

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Modified Chua’s Circuit Thamilmaran et al (2004) Original Chua’s circuit Modified Chua’s circuit

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Experimental Results Thamilmaran et al (2004) Phase portraits Poincaré sections Power spectra

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Theoretical Results Thamilmaran et al. (2004)

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Other Systems with HDC Coupled Ikeda maps Chaotically driven Zaslavsky map Delayed Henon maps Coupled three or more Lorenz systems Coupled two or more lasers Phonic integrated circuits Miniature eye movements Excitable physiological systems Spreading of rumor

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Types and Properties of HD Systems 1. Strange attractors with dimensions d > 3 but only one positive Lyapunov exponent - no hyperchaos. 1. Strange attractors with dimensions d cor > 3 but only one positive Lyapunov exponent - no hyperchaos. 2. Strange attractors with dimensions d > 3 and two or more positive Lyapunov exponents - systems with hyperchaos. 2. Strange attractors with dimensions d cor > 3 and two or more positive Lyapunov exponents - systems with hyperchaos. HD and LD systems behave differently and chaos is persistent (no windows of periodicity) in HD dynamical systems (Albers et al 2005)

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Routes to HDC I Same as routes for LD systems: (a) Period-doubling (b) Quasi-periodicity (c) Intermittency (d) Chaotic transients (e) Crisis First LD chaos by one of the above routes and then to HD chaos. to HD chaos. Harrison & Lai (1999) and Pazo et al (2001)

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Routes to HDC II Quasi-periodicity – torus doubling – torus merging – chaos Venkatesan & Lakshmanan (1998) Quasi-periodicity – torus – 3-period window – chaos Musielak et al (2005) Musielak et al (2005) Sequence of Neimark-Sacker bifurcations Alberts & Sprott (2004)

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SUMMARY High-dimensional (HD) dynamical systems that exhibit chaos can be constructed by adding degrees of freedom to low-dimensional dynamical systems. High dimensional chaos (HDC) is observed in HD nonlinear systems whose strange attractors have dimensions d > 3. High dimensional chaos (HDC) is observed in HD nonlinear systems whose strange attractors have dimensions d cor > 3. Two types of systems with HDC have been identified, those with and without hyperchaos. HD systems may transition to chaos via one of the routes known for LD systems or via new routes; four new known for LD systems or via new routes; four new routes have been identified, others still remain to be routes have been identified, others still remain to be discovered. discovered.

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Acknowledgments Special thanks to Professor Ahmad M. Harb Special thanks to Professor Ahmad M. Harb and the organizers of the IASTED International Workshop on Modern Nonlinear Theory for the invitation to present this tutorial. and the organizers of the IASTED International Workshop on Modern Nonlinear Theory for the invitation to present this tutorial. Support for this work was provided by NASA / MSFC, US Army and The Alexander von Humboldt Foundation in Germany.

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References Albers, D.J., Sprott, J.C. and Crutchfield, J.P., 2005, arxiv.org/abs/nlin.CD/ Albers, D.J. and Sprott, J.C., 2006, Physica D, 223, 194 Argyris, J., Faust, G. and Haase, M., 1993, Phil. Trans. Phys. Sci. & Eng., 344, 207 Benner, J.W., 1997, Ph.D. Dissertation, The University of Alabama in Huntsville Chen, Z.-M.and Price, W.G., 2006, Chaos, Solitons and Fractals, 28, 571 Curry, J.H., 1978, Commun. Math. Phys., 60, 193. Curry, J.H., 1979, SIAM J. Math. Anal., 10, 71. Eckmann, J.P., 1981, Rev. Mod. Phys., 53, 643 Feigenbaum, M.J., 1979, J. Stat. Phys., 21, 669 Feigenbaum, M.J., 1980, Universal Behavior in Nonlinear Systems, Los Alamos Science, 1, 4-27 Gilmore, R., 1998, Rev. Mod. Phys., 70, 1455 Greborgi, C., Ott, E. and Yorke, J.A., 1983, Physica D, 7, 181 Hanon, M., 1969, Q. Appl. Math., 27, 135 Hanon, M., 1976, Comm. Math. Phys., 50, 6977 Harrison, M.A. and Lai, Y.C., 1999, Phys. Rev. E, 59, 3799 Harrison, M.A. and Lai, Y.C., 2000, Int. J. Bifur. Chaos, 10, 1471 R.C. Hilbron, R.C., 1994, Chaos and Nonlinear Dynamics, Oxford, Oxford Uni. Press Howard, L.N. and R.K. Krishnamurti, R.K., 1986, J. Fluid Mech., 170, 385 Humi, M., 2004, arXiv:nlin.CD/ v1 Ivancevic, V.G. and Ivancevic, T.T., 2007, High-Dimensional Chaotic and Attractor Systems, Dordrecht, Springer E.A. Jackson, E.A., 1990, Perspectives of Nonlinear Dynamics, Cambridge, Cambridge Uni. Kennamer, K.S., 1995, M.S. Thesis, The University of Alabama in Huntsville Li, Y., Tang, W.K.S. and Chen, G., 2005, Int. J. Circuit Theor. & Applic., 33, 235 Lorenz, E.N., 1963, J. Atmos. Sci., 20, 130. Moon, F.C., 1987, Chaotic Vibrations, New York, John Wiley & Sons, Inc. Moon, F.C., 1992, Chaotic and Fractal Dynamics, New York, John Wiley & Sons, Inc.

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References (cont’d) Musielak, D.E., Musielak, Z.E. and Kennamer, K.S., 2005, Fractals, 13, 19 Musielak, D.E., Musielak, Z.E. and Benner, J.W., 2005, Chaos, Solitons and Fractals, 24, 907 Newhouse, S.E., Ruelle, D. and Takens, F., 1978, Commun. Math. Phys., 64, 35 Ott, E., 1993, Chaos in Dynamical Systems, Cambridge, Cambridge Uni. Press Pazo, D., Sanchez, E. and Matias, M., 2001, Int. J. Bifur. Chaos, 11, 2683 Poincare, H., 1890, Acta Math., 13,1 Pommeau, Y. and Manneville, P., 1980, Commun. Math. Phys., 74, 189 Reiterer, P., Lainscsek, C. Schurrer, F. Letellier, C. and Maquet, J., 1998, J. Phys. A: Math. Gen., 31, 7121 Roy, D. and Musielak, Z.E., 2007, Chaos, Solitons and Fractals, 31, 747 Roy, D. and Musielak, Z.E., 2007, Chaos, Solitons and Fractals, 32, 1038 Roy, D. and Musielak, Z.E., 2007, Chaos, Solitons and Fractals, 33, 138 Rossler, O., 1979, Phys. Lett., 71, 155 Rossler, O., 1983, Z. Naturforsch., 38, 126 Ruelle, D. and Takens, F., 1971, Commun. Math. Phys., 20, 167 Savi, M.A. and Pacheco, P.M.C.L., 2002, J. Braz. Soc. Mech. Sci., 24, Saltzman, B., 1962, J. Atmos. Sci., 19, 329 Schmutz, M. and Rueff, M., 1984, Physica D, 11, 167 Thamilmaran, K., Lakshmanan, M. and Venkatesan, A., 2004, Int. J. Bifur. Chaos., 14, 221 Thiffeault, J.L., 1995, M.S. Thesis, The University of Texas at Austin Thieffault, J.L. and Horton, W., 1996, Phys. Fluids, 8, 1715

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References (cont’d) Thompson, J.M.T. and H.B. Stewart, H.B., 1986, Nonlinear Dynamics and Chaos, New York, John Wiley & Sons, Inc. Tong, C. and Gluhovsky, A., 2002, Phys. Rev. E, 65, Treve, Y.M. and Manley, O.P., 1982, Physica D, 4, 319 Tucker, W., 1999, C.R. Acad. Sci., 328, 1197 Ueda, Y., 1979, J. Stat. Phys., 20, 181 Ueda, Y., 1980, in New Approaches to Nonlinear Problems in Dynamics, edited by P. J. Holmes (Siam, Philadelphia), p.311

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