July 30th2007Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.eduEE100 Summer 2007: WRAP UP“Alls well that ends well”Bharathwaj MuthuswamyNOEL: Nonlinear Electronics Lab151M Cory HallDepartment of EECSUniversity of California, [email protected]. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Outline Examples of systems: Magnetic Levitation: EECS 128 Autonomous R/C cars: EECS 192 Digital Systems Design: EECS150, EECS152 Courses you can take after EECS100/EECS42-43 Sample Research Applications: Reconfigurable Radio: FPGA SOCs Introduction to Chaos using Chua’s circuit What is Chaos? Chaos in Nature Hallmarks of Chaos Non-periodic behavior in time domain Sensitive dependence on initial conditions Proving the existence of Chaos Easy Chaos: Chua’s circuit Building and working with Chua’s circuit References Simulating Chua’s circuit: MultiSim Turning your PC into an oscilloscope: Osqoop Chua’s circuit for high school students Interesting MATLAB experiments QuestionsDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Examples of Systems: EECS 128http://inst.eecs.berkeley.edu/~ee128Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Examples of Systems: EECS 128Linearize!Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Examples of Systems: EECS 192http://inst.eecs.berkeley.edu/~ee192Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Examples of Systems: EECS 150/152http://inst.eecs.berkeley.edu/~cs150Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Courses you can take after EECS100http://hkn.eecs.berkeley.edu/student/onlineexams.shtmlDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Sample Research ApplicationsField Programmable Gate Array (FPGA) System-On-Chip (SOC) Design for a 802.11 a/g/p/nReconfigurable-RadioJuly 30th2007Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.eduIntroduction to Chaos using Chua’s circuitDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Introduction to Chaos: What is Chaos? There is NO universal agreed-upon definition of Chaos Loosely speaking, a Chaotic system is a deterministic system that exhibits random behavior. Example - Chua’s circuit:Set of ordinary differential equations with a simple nonlinearity BUT the system behavior is complex: Phase spaceTime domainDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Chaos in Nature Weather(Reference: “Deterministic Nonperiodic Flow”. Lorenz, Edward N. Journal of Atmospheric Sciences. pp. 130 – 141, 1963)A plot of the Lorenz attractor for r = 28, σ = 10, b = 8/3(Reference: Chaos Theory, Wikipedia Entry. Online at: http://en.wikipedia.org/wiki/Chaos_theory)Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Chaos in Nature Trajectory of planetary orbits:(Reference: “The role of chaotic resonances in the Solar System”. Murray, N. and Holman M. Nature.pp. 773 – 780, vol. 410, 12 April 2001) Irregulary shaped satellites like Hyperion (moon of Saturn) tumble chaotically. Chaos in the orbits of giant planets (Jupiter, Saturn and Uranus) – the location of these planets cannot be predicted on a time scale longer than a few tens of millions of years. Quote from the paper above:“The worried readermay find some comfort in that the accompanying analytic theorypredicts that no planet will be ejected before the Sun dies.”Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Chaos in Nature Brain waves(Reference: Rhythms of the Brain. Buzsaki, Gyorgy. Oxford University Press. 2006)Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Hallmarks of Chaos Non-periodic behavior in the time domainReference: “Taming Chaos: Stabilization of Aperiodic Attractors by Noise”. Freeman, W. et. al. IEEE Trans. On Circuits and Systems – I: Fundamental Theory and Applications.Vol. 44, No. 10, Oct. 1997Time series data from neural nodes implanted in rat cortexDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Hallmarks of Chaos Sensitive dependence on initial conditionsReference: http://www.keldysh.ru/departments/dpt_17/eng/ndeng.htmlEventually, the paths diverge (mathematically quantifiedby the Lyapunov exponent)Three initial conditions x10, x20and x30(they are arbitrarily close to each other)Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Proving the existence of Chaos Mathematically very challenging: Lorenz’s system was proved to be chaotic nearly 30 YEARS after Lorenz’s observations: The Lorenz Attractor Exists. Tucker, Warwick. Ph.D. Thesis, 1998. University of Uppsala. One way to prove chaotic behavior: define a homeomorphism to the Cantor set using a Smale horseshoeDr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Proving the existence of Chaos But for the purposes of EE100: Central concept: Poincare-Bendixson Theorem In a nutshell, the consequence of the theorem is that a continuous time autonomous dynamical system CANNOT be chaotic in the plane (2-dimensions).Dr. Leon O Chua – Director 151M Cory Hall University of California, Berkeleyhttp://nonlinear.eecs.berkeley.edu July 30th2007Easy Chaos: Chua’s circuit Designed using systematic nonlinear circuit techniques by Leon O. Chua in 1983Excellent Reference:“The
View Full Document
Unlocking...