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Demonstrating Chaos with “Sprott Circuits”

Demonstrating Chaos with “Sprott Circuits”. Michael Braunstein Central Washington University PNACP, April 15 and 16, 2005 University of Portland. Non-Linear Dynamics and Chaos is rapidly becoming a standard component of the undergraduate physics curriculum, e.g.:

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Demonstrating Chaos with “Sprott Circuits”

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  1. Demonstrating Chaos with “Sprott Circuits” Michael Braunstein Central Washington University PNACP, April 15 and 16, 2005 University of Portland.

  2. Non-Linear Dynamics and Chaos is rapidly becoming a standard component of the undergraduate physics curriculum, e.g.: • Analytical Mechanics, Fowles and Cassiday, 6th Edition; Classical Dynamics of Particles and Systems, Marion and Thornton, 4th edition. • Classes: Computational Physics, Nonlinear Dynamics and Chaos • Laws, P. W. (2004). "A unit on oscillations, determinism and chaos for introductory physics students." American Journal of Physics72(4): 446-452. • Physics is an experimental science – hence instruction requires: • Demonstrations • Hands on activities • Laboratory Exercises

  3. “If it’s chaotic, then how can we measure anything?” Physics Student

  4. Mechanical Chaotic systems Kinesthetic - less abstract; Typically slower evolving; Transducers to measure; Mechanical skill level/expense for assembly Berdahl, J. P. and K. V. Lugt (2001). "Magnetically driven chaotic pendulum." American Journal of Physics69(9): 1016-1019. Nunes, J. E. B. G. and Jr. (1997). "A mechanical Duffing oscillator for the undergraduate laboratory." American Journal of Physics65(9): 841-846. VERSUS Electronic Chaotic systems More abstract; Audio frequencies or higher; Voltage output, Electronic skill level for assembly; Components are inexpensive Roy, P. K. and A. Basuray (2003). "A high frequency chaotic signal generator: A demonstration experiment." American Journal of Physics71(1): 34-37. Jones, B. K. and G. Trefan (2001). "The Duffing oscillator: A precise electronic analog chaos demonstrator for the undergraduate laboratory." American Journal of Physics69(4): 464-469. Wiener, R (2005). Controlling Chaos in a Simple Electronic Circuit, PNACP (unpublished).

  5. Publications introducing “Sprott” systems: • Sprott, J. C. (2000), "Simple chaotic systems and circuits," American Journal of Physics 68(8): 758-763 • Kiers, K., D. Schmidt, J.C. Sprott (2004), "Precision measurements of a simple chaotic circuit," American Journal of Physics 72(4): 503-509.

  6. C R Vin - Vout + LM741

  7. R V1 R1 V2 R - + LM741 Vout

  8. Sprott, J. C. (2000), "Simple chaotic systems and circuits," American Journal of Physics 68(8): 758-763

  9. Sprott, J. C. (2000), "Simple chaotic systems and circuits," American Journal of Physics 68(8): 758-763

  10. Circuit Analysis Rv BTW: Can be expressed as three coupled first order ODE’s Notes: Scaling of time; Rv is the control parameter (don’t use voltage divider)

  11. D(x) X=0 X=0 V Diode = Braunstein (Wiener, et. al)

  12. Demonstrating Chaos Audio Amplifier and Speaker

  13. Investigating Chaos Oscilloscope

  14. Time Series and Phase Space

  15. Time Series and Phase Space

  16. Investigating Chaos Computer Data Acquisition and Analysis NI-DAQ LabVIEW

  17. Time Series

  18. Time Series

  19. Return or Lorenz Map

  20. NI-DAQ LabVIEW Investigating Chaos Computer Data Acquisition and Analysis with computer controlled (voltage controlled) resistor -LM13700

  21. Bifurcation Diagram

  22. What Else ? • Full Characterization of the strange attractor • Other Sprott circuits and other chaotic circuits • Synchronization of chaos (tuning/encryption) • Control of Chaos

  23. Thanks to the following Central Washington University students: Sam Rowswell David Cross Erika Beam Ryan Tervo Alfredo Meza

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