EE100 Summer 2007 WRAP UP Alls well that ends well Bharathwaj Muthuswamy NOEL Nonlinear Electronics Lab 151M Cory Hall Department of EECS University of California Berkeley mbharat eecs berkeley edu Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Outline 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 Questions Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Examples of Systems EECS 128 http inst eecs berkeley edu ee128 Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Examples of Systems EECS 128 Linearize Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Examples of Systems EECS 192 http inst eecs berkeley edu ee192 Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Examples of Systems EECS 150 152 http inst eecs berkeley edu cs150 Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Courses you can take after EECS100 http hkn eecs berkeley edu student onlineexams shtml Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Sample Research Applications Field Programmable Gate Array FPGA System On Chip SOC Design for a 802 11 a g p n Reconfigurable Radio Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Introduction to Chaos using Chua s circuit Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Introduction 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 Time domain Phase space Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Chaos 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 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 Berkeley http nonlinear eecs berkeley edu July 30th 2007 Chaos 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 reader may find some comfort in that the accompanying analytic theory predicts that no planet will be ejected before the Sun dies Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Chaos 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 Berkeley http nonlinear eecs berkeley edu July 30th 2007 Hallmarks of Chaos Non periodic behavior in the time domain Time series data from neural nodes implanted in rat cortex Reference 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 1997 Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Hallmarks of Chaos Sensitive dependence on initial conditions Eventually the paths diverge mathematically quantified by the Lyapunov exponent Three initial conditions x10 x20 and x30 they are arbitrarily close to each other Reference http www keldysh ru departments dpt 17 eng ndeng html Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Proving 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 horseshoe Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Proving 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 Berkeley http nonlinear eecs berkeley edu July 30th 2007 Easy Chaos Chua s circuit Designed using systematic nonlinear techniques by Leon O Chua in 1983 circuit Excellent Reference The Genesis of Chua s circuit Chua Leon O Archiv fur Elektronik und Uebertragungstechnik July 1992 vol 46 no 4 250 257 Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Easy Chaos Chua s circuit A consequence of the Poincare Bendixson theorem we need three independent energy storage elements Hence Dr Leon O Chua Director 151M Cory Hall University of California Berkeley http nonlinear eecs berkeley edu July 30th 2007 Easy Chaos
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