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Chaotic Dynamics of the Muthuswamy-Chua System

1. Chaotic Dynamics of the Muthuswamy-Chua System. Workshop on Nonlinear Electronics And Its Applications Vellore Institute of Technology, Vellore, India August 31 st 2013. Bharathwaj “Bart” Muthuswamy Assistant Professor of Electrical Engineering Milwaukee School of Engineering

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Chaotic Dynamics of the Muthuswamy-Chua System

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  1. 1 Chaotic Dynamics of the Muthuswamy-Chua System Workshop on Nonlinear Electronics And Its Applications Vellore Institute of Technology, Vellore, India August 31st 2013 Bharathwaj “Bart” Muthuswamy Assistant Professor of Electrical Engineering Milwaukee School of Engineering muthuswamy@msoe.edu http://www.harpgroup.org/muthuswamy/ BS (2002), MS (2005) and PhD (2009) from the University of California, Berkeley PhD Advisor: Dr. Leon O. Chua (co-advised by Dr. Pravin P. Varaiya) 1“On the Integrability of a Muthuswamy-Chua System”, Llibre, J. and Valls, C. Journal of Nonlinear Mathematical Physics, Vol. 19, No. 4, pp. 1250029 – 1250041 December 2012.

  2. What do I work on? • Nonlinear Dynamical Systems and Embedded Systems • - Applications and Mathematical properties of the Muthuswamy-Chua system • - Potential Applications to Turbulence Modeling • (MSOE; University of Western Australia, Perth, Australia) • - Chaotic Hierarchy and Flow Manifolds • (MSOE; University of Western Australia, Perth, Australia; I.U.T. de Toulon, La GardeCedex, France) - Applications of Chaotic Delay Differential Equations using Field Programmable Gate Arrays (FPGAs) (MSOE; Vellore Institute of Technology; University Putra Malaysia, Malaysia; Springer-Verlag) - Pattern Recognition Using Cellular Neural Networks on FPGAs (MSOE; AlteraCorporiation) - Practical Memristors: discharge tubes, PN junctions and Josephson Junctions (MSOE; IIT Chennai; University of Western Australia, Perth, Australia; Vellore Institute of Technology, Vellore, India) Education - Nonlinear Dynamics at the undergraduate level (with folks from all over the world  )

  3. Goal of This Talk Obtain chaos in the circuit [6] below:

  4. Outline I. Prerequisites for understanding this talk: 1. First course in circuit theory 2. First course in differential equations II. Background 1. Introduction to Chaos 2. The fundamental circuit elements III. The Memristor 1. Properties of the memristor 2. Memristive Devices IV. The Muthuswamy-Chua circuit (System) 1. Derivation of Circuit Equations 2. Hardware emulation of the memristor 3. MultiSim (SPICE) simulation 4. An attractor from the circuit 5. Rigorous Mathematical Analysis 6. A Chaotic Hierarchy 7. Potential conservative chaotic systems V. Conclusions, Current (future) work and References

  5. Introduction to Chaos • “Birth” of Chaos: Lorenz Attractor [7] • Edward Lorenz introduced the following nonlinear system of differential equations as a crude model of weather in 1963: (1) • Lorenz discovered that model dynamics were extremely sensitive to initial conditions and the trajectories were aperiodic but bounded. • But, does chaos exist physically? Answer is: YES. For example, chaotic circuits by Sprott [8].

  6. The Fundamental Circuit Elements v q i Φ • Memristors were first postulated by Leon. O Chua in 1971 [2]

  7. Outline I. Prerequisites for understanding this talk: 1. First course in circuit theory 2. First course in differential equations II. Background 1. Introduction to Chaos 2. The fundamental circuit elements III. The Memristor 1. Properties of the memristor 2. Memristive Devices IV. The Muthuswamy-Chua circuit (System) 1. Derivation of Circuit Equations 2. Hardware emulation of the memristor 3. MultiSim (SPICE) simulation 4. An attractor from the circuit 5. Rigorous Mathematical Analysis 6. A Chaotic Hierarchy 7. Potential conservative chaotic systems V. Conclusions, Current (future) work and References

  8. Properties of the Memristor [2] Circuit symbol: • A memristor defines a relation of the from: • If g is a single-valued function of charge (flux), then the memristor is • charge-controlled (flux-controlled) • Memristor i-v relationship: • M(q(t)) is the incremental memristance • Q1: Why is the memristor called “memory resistor”? • Because of the definition of memristance: • Q2: Why is the memristor not relevant in linear circuit theory? • 1. If M(q(t)) is a constant: • 2. Principle of superposition is not* applicable: Slide Number: 8/24

  9. Memristive Devices [3] The functions R and f are defined as:

  10. Outline I. Prerequisites for understanding this talk: 1. First course in circuit theory 2. First course in differential equations II. Background 1. Introduction to Chaos 2. The fundamental circuit elements III. The Memristor 1. Properties of the memristor 2. Memristive Devices IV. The Muthuswamy-Chua circuit (System) 1. Derivation of Circuit Equations 2. Hardware emulation of the memristor 3. MultiSim (SPICE) simulation 4. An attractor from the circuit 5. Rigorous Mathematical Analysis 6. A Chaotic Hierarchy 7. Potential conservative chaotic systems V. Conclusions, Current (future) work and References

  11. Derivation of Circuit Equations [6] Circuit equations: System equations: Specifically: Parameters:

  12. Hardware emulation of the Memristor [6]

  13. MultiSim (SPICE simulation)

  14. An attractor from the circuit [6]

  15. Rigorous Mathematical Analysis Paper by Ginoux et. al. “Topological Analysis of Chaotic Solution of Three- Element Memristive Circuit”. International Journal of Bifurcation and Chaos, Vol. 20, No. 11., pp. 3819 – 3829, Nov. 2010.

  16. A Chaotic Hierarchy [5] System 1 phase portrait System 2 phase portrait Blue: System 1 (differentiable) Red: System 2 (non-differentiable)

  17. Potential Conservative Chaos

  18. Potential Conservative Chaos (contd.) Now, a live demo of the POTENTIAL physical implementation…

  19. Outline I. Prerequisites for understanding this talk: 1. First course in circuit theory 2. First course in differential equations II. Background 1. Introduction to Chaos 2. The fundamental circuit elements III. The Memristor 1. Properties of the memristor 2. Memristive Devices IV. The Muthuswamy-Chua circuit (System) 1. Derivation of Circuit Equations 2. Hardware emulation of the memristor 3. MultiSim (SPICE) simulation 4. An attractor from the circuit 5. Rigorous Mathematical Analysis 6. A Chaotic Hierarchy 7. Potential conservative chaotic systems V. Conclusions, Current (future) work and References

  20. Conclusions and Current (future) Work I. Conclusions: 1. We obtained a circuit that uses only three fundamental circuit elements (only one active) to obtain chaos. 2. We can pick our choice of nonlinearity, we discussed a few choices II. Current (future) work: 1. Work with colleagues at the University of Western Australia (VIT and IIT Chennai, related to the ideal memristor: cos(phi) term in the Josephson Junction) . 2. Is the Muthuswamy-Chua circuit canonical, like Chua’s circuit? *Utilizing physical memristive devices like discharge tubes. Note that circuit needs a nonlinear, active element

  21. Current (future) Work

  22. Current (future) Work contd.

  23. A Canonical Muthuswamy-Chua circuit? The Muthuswamy-Chua SYSTEM: • Can the system above exhibit canonical behavior like the global unfolding in Chua’s circuit? • Utilizing a Physical Memristive Device like the discharge tube?

  24. References Alligood, K. T., Sauer, T. and Yorke, J. A. Chaos: An Introduction to Dynamical Systems. Springer, 1997. Chua, L. O. “Memristor-The Missing Circuit Element”. IEEE Transactions on Circuit Theory, Vol. CT-18, No. 5, pp. 507- 519. September 1971. Chua, L. O. and Kang, S. M. “Memristive Devices and Systems”. Proceedings of the IEEE, Vol. 64, No. 2, pp. 209- 223. February 1976. Hirsch, M. W., Smale S. and Devaney, R. Differential Equations, Dynamical Systems and An Introduction To Chaos. 2nd Edition, Elsevier, 2004. McCullough, M. C., Iu, H. and Muthuswamy, B. “Chaotic Behaviour in a Three Element Memristor Based Circuit using Fourth Order Polynomial and PWL Nonlinearity”. To appear in the Proceedings of the 2013 IEEE International Symposium on Circuits and Systems. Muthuswamy, B. and Chua, L. O. “Simplest Chaotic Circuit”. International Journal of Bifurcation and Chaos, vol. 20, No. 5, pp. 1567-1580. May 2010. Lorenz, E. N. “Deterministic Nonperiodic Flow”. Journal of Atmospheric Sciences, vol. 20, pp/ 130-141, 1963. Sprott, J. C. “Simple Chaotic Systems and Circuits”. American Journal of Physics, vol. 68, pp. 758-763, 2000. Strogatz, S. H. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994. Strukov, D. B., Snider, G. S., Steward, D. R. and Williams, S. R. “The missing memristor found”. Nature, vol. 453, pp. 81-83, 1st May 2008. Questions?

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