INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS by Bruce

INTRODUCTION TO NONLINEAR DYNAMICS AND CHAOS by Bruce A. Mork Michigan Technological University Presented for EE 5320 Course Michigan Technological University March 27, 2001

NONLINEARITIES - EXAMPLES • CHEMISTRY AND PHYSICS – – CHEMICAL REACTIONS (KINETICS) THIN FILM DEPOSITION (LASER DEPOSITION) TURBULENT FLUID FLOWS CRYSTALLINE GROWTH • MEDICAL & BIOLOGICAL – – – HEART DISORDERS (ARRHYTHMIA, FIBRILLATION) BRAIN AND NERVOUS SYSTEM POPULATION DYNAMICS EPIDEMIOLOGY FORECASTING (FLU, DISEASES) FOOD SUPPLY (VS. WEATHER & POPULATION) • ECONOMY – STOCK MARKET – WORLD ECONOMY

NONLINEARITIES - EXAMPLES • ENGINEERING – STRUCTURES: BUILDINGS, BRIDGES – HIGH VOLTAGE TRANSMISSION LINES - WIND-INDUCED VIBRATIONS - GALLOPING – MAGNETIC CIRCUITS – COMMUNICATIONS SCRAMBLING - CHAOS GENERATOR – POWER SYSTEMS LOAD FLOW - VOLTAGE COLLAPSE – NONLINEAR CONTROLS – SIMPLE PENDULUM - LINEAR FOR SMALL SWINGS, BECOMES NONLINEAR FOR LARGE ANGLE OSCILLATIONS • KEY REALIZATIONS – NONLINEARITY IS THE RULE, NOT THE EXCEPTION ! – LINEARIZATION OR REDUCED-ORDER MODELING MAY ONLY BE VALID FOR A SMALL RANGE OF SYSTEM OPERATION (I. E. SMALL-SIGNAL RESPONSE) ……… BE CAREFUL !

DUFFING’S OSCILLATOR A SIMPLE NONLINEAR SYSTEM

FERRORESONANCE IN WYE-CONNECTED SYSTEMS VC A H 1 B H 2 X 1 VA VB C H 3 X 2 X 3 X 0

BACKFED VOLTAGE DEPENDS ON CORE CONFIGURATION TRIPLEX WOUND OR STACKED 3 -LEG STACKED CORE SHELL FORM 5 -LEG STACKED CORE 5 -LEG WOUND CORE 4 -LEG STACKED CORE

NONLINEAR DYNAMICAL SYSTEMS: BASIC CHARACTERISTICS • MULTIPLE MODES OF RESPONSE POSSIBLE FOR IDENTICAL SYSTEM PARAMETERS. • STEADY STATE RESPONSES MAY BE OF DIFFERENT PERIOD THAN FORCING FUNCTION, OR NONPERIODIC (CHAOTIC). • STEADY STATE RESPONSE MAY BE EXTREMELY SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS. • BEHAVIORS CANNOT PROPERLY BE PREDICTED BY LINEARIZED OR REDUCED ORDER MODELS. • THEORY MATURED IN LATE 70 s, EARLY 80 s. • PRACTICAL APPLICATIONS FROM LATE 80 s.

EXPERIMENTAL PROCEDURE: CATEGORIZATION OF MODES OF FERRORESONANCE BEHAVIOR • FULL SCALE LABORATORY TESTS. • 5 -LEG WOUND CORE, RATED 75 -k. VA, WINDINGS: 12, 470 GY/7200 - 480 GY/277 (TYPICAL IN 80% OF U. S. SYSTEMS). • RATED VOLTAGE APPLIED. • ONE OR TWO PHASES OPEN-CIRCUITED. • CAPACITANCE(S) CONNECTED TO OPEN PHASE(S) TO SIMULATE CABLE. • VOLTAGE WAVEFORMS ON OPEN PHASE(S) RECORDED AS CAPACITANCE IS VARIED.

VOLTAGE X 1 -X 0 C = 9 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD ONE ” PHASE PLANE DIAGRAM FOR VX 1

VOLTAGE X 1 -X 0 C = 9 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD ONE ” DFT FOR VX 1 ONLY ODD HARMONICS

VOLTAGE X 1 -X 0 C = 10 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD TWO ” PHASE PLANE DIAGRAM FOR VX 1

VOLTAGE X 1 -X 0 C = 10 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD TWO ” DFT FOR VX 1 HARMONICS AT MULTIPLES OF 30 Hz.

VOLTAGE X 1 -X 0 C = 15 F X 2, X 3 ENERGIZED X 1 OPEN “ TRANSITIONAL CHAOS ” PHASE PLANE DIAGRAM FOR VX 1 TRAJECTORY DOES NOT REPEAT.

VOLTAGE X 1 -X 0 C = 15 F X 2, X 3 ENERGIZED X 1 OPEN “ TRANSITIONAL CHAOS ” DFT FOR VX 1 NOTE: DISTRIBUTED SPECTRUM.

VOLTAGE X 1 -X 0 C = 17 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD FIVE ” PHASE PLANE DIAGRAM FOR VX 1

VOLTAGE X 1 -X 0 C = 17 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD FIVE ” DFT FOR VX 1 HARMONICS AT “ODD ONE-FIFTH” SPACINGS. i. e. 12, 36, 60, 84. . .

VOLTAGE X 1 -X 0 C = 18 F X 2, X 3 ENERGIZED X 1 OPEN “ TRANSITIONAL CHAOS ” PHASE PLANE DIAGRAM FOR VX 1 NOTE: TRAJECTORY DOES NOT REPEAT.

VOLTAGE X 1 -X 0 C = 18 F X 2, X 3 ENERGIZED X 1 OPEN “ TRANSITIONAL CHAOS ” DFT FOR VX 1 NOTE: DISTRIBUTED SPECTRUM.

VOLTAGE X 1 -X 0 C = 25 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD THREE ” PHASE PLANE DIAGRAM FOR VX 1

VOLTAGE X 1 -X 0 C = 25 F X 2, X 3 ENERGIZED X 1 OPEN “ PERIOD THREE ” DFT FOR VX 1 HARMONICS AT “ODD ONE-THIRD” SPACINGS. i. e. 20, 60, 100. . .

VOLTAGE X 1 -X 0 C = 40 F X 2, X 3 ENERGIZED X 1 OPEN “ CHAOS ” POINCARÉ SECTION FOR VX 1 ONE POINT PER CYCLE SAMPLED FROM PHASE PLANE TRAJECTORY.

VOLTAGE X 1 -X 0 C = 40 F X 2, X 3 ENERGIZED X 1 OPEN “ CHAOS ” DFT FOR VX 1 NOTE: DISTRIBUTED FREQUENCY SPECTRUM.

SPONTANEOUS TRANSITION BETWEEN MODES OF PERIOD ONE. C = 20 F X 1, X 3 ENERGIZED X 2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

SPONTANEOUS TRANSITION BETWEEN MODES OF PERIOD ONE. C = 14 F X 1 ENERGIZED X 2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

SPONTANEOUS TRANSITION BETWEEN MODES OF PERIOD ONE. C = 14 F X 1 ENERGIZED X 2 OPEN BLURRED AREAS SHOW MODE TRANSITION.

INTERMITTENCY X 2 & X 3 ENERGIZED, X 1 OPEN, C = 45 F

GLOBAL PREDICTION OF FERRORESONANCE • PREDICTION APPEARS DIFFICULT DUE TO WIDE RANGE OF POSSIBLE BEHAVIORS. • A TYPE OF BIFURCATION DIAGRAM, AS USED TO STUDY NONLINEAR SYSTEMS, IS INTRODUCED FOR THIS PURPOSE. • MAGNITUDES OF VOLTAGES FROM SIMULATED POINCARÉ SECTIONS ARE PLOTTED AS THE CAPACITANCE IS SLOWLY VARIED (BOTH UP AND DOWN). • POINTS ARE SAMPLED ONCE EACH 60 -Hz CYCLE. • AN “ADEQUATE ” MODEL IS REQUIRED.

CAPACITANCE VARIED 0 - 30 F MODES: 1 -2 -C-5 -C-3 -C BIFURCATION DIAGRAMS: ENERGIZE X 2, X 3. X 1 LEFT OPEN. CAPACITANCE VARIED 30 - 0 F

CONCLUSIONS • FERRORESONANT BEHAVIOR IS TYPICAL OF NONLINEAR DYNAMICAL SYSTEMS. • RESPONSES MAY BE PERIODIC OR CHAOTIC. • MULTIPLE MODES OF RESPONSE ARE POSSIBLE FOR THE SAME PARAMETERS. • STEADY STATE RESPONSES CAN BE SENSITIVE TO INITIAL CONDITIONS OR PERTURBATIONS. • SPONTANEOUS TRANSITIONS FROM ONE MODE TO ANOTHER ARE POSSIBLE. • WHEN SIMULATING, THERE MAY NOT BE “ONE CORRECT” RESPONSE.

CONCLUSIONS (CONT’D) • BIFURCATIONS OCCUR AS CAPACITANCE IS VARIED UPWARD OR DOWNWARD. • PLOTTING Vpeak vs. CAPACITANCE OR OTHER VARIABLES GIVES DISCONTINUOUS OR MULTI-VALUED FUNCTIONS. • THEREFORE, SUPPOSITION OF TRENDS BASED ON LINEARIZING A LIMITED SET OF DATA IS PARTICULARLY PRONE TO ERROR. • BIFURCATION DIAGRAMS PROVIDE A ROAD MAP, AVOIDING NEED TO DO SEPARATE SIMULATIONS AT DISCRETE VALUES OF CAPACITANCE AND INITIAL CONDITIONS.

CONCLUSIONS (CONT’D) • DFTs ARE USEFUL FOR CATEGORIZING THE DIFFERENT MODES OF FERRORESONANCE. • PHASE PLANE DIAGRAMS PROVIDE A UNIQUE SIGNATURE OF PERIODIC RESPONSES. • PHASE PLANE TRAJECTORIES PROVIDE THE DATA FOR POINCARÉ SECTIONS, FRACTAL DIMENSION, AND LYAPUNOV EXPONENTS. • CATEGORIZATION OF RESPONSES CAN BE MORE CLEARLY DONE USING THE SYNTAX OF NONLINEAR DYNAMICS.

CONTINUING WORK • THE AUTHORS ARE CONTINUING THIS WORK UNDER SEPARATE FUNDING. – IMPROVEMENT OF LUMPED PARAMETER TRANSFORMER MODELING. – CONTINUED APPLICATION OF NONLINEAR DYNAMICS AND CHAOS TO LOW-FREQUENCY TRANSIENTS. • ACKNOWLEDGEMENTS – National Science Foundation RIA Grant – BONNEVILLE POWER ADMINISTRATION. – AREA UTILITIES, NRECA, AND CONSULTANTS.

COMMENTS? QUESTIONS?