1. 1 CHAOTIC CIRCUITS The way the electron bounces

2. 2 TOPICS l Linear Circuits –Inductor –Diode l Non-Linear Circuits –How can we tell if a circuit is behaving non-linearly ? –Diode and inductor –Transistor l Bifurcation Diagrams l The Feigenbaum number l Why are these systems chaotic ? l Attractors

3. 3 MAY POPULATION MODEL

4. 4 Basic Population Model POPULATION SEX FOOD SUPPLY

5. 5 May’s Population Model POPULATION SEX FOOD SUPPLY FEEDBACK

6. 6 Attractor for May model

7. 7

8. 8 LINEAR CIRCUITS

9. 9 INDUCTOR

10. 10 Input Voltage: 5 V

11. 11 Input Voltage: 10 V

12. 12 Input Voltage: 15 V

13. 13 Input Voltage: 20 V

14. 14 Inductor ( Output vs. Input) Input Voltage (peak to Peak) Output Voltage (peak to peak)

15. 15 DIODE

16. 16 Input Voltage: 5 V

17. 17 Input Voltage: 10 V

18. 18 Input Voltage: 15 V

19. 19 Input Voltage: 20 V

20. 20 Diode ( Output vs. Input Voltage) Input Voltage (peak to peak) Output Voltage (peak to peak) y = 0.4949 + 0.7762

21. 21 NON-LINEAR CIRCUITS

22. 22 How can we tell if a circuit is behaving non-linearly?

23. 23 Period 1

24. 24 Period 2

25. 25 Period 4

26. 26 Diode-Inductor Circuit

27. 27 Circuit Schematic Inductor 5.89 mH Diode Variable Voltage Source AC

28. 28 Period 1

29. 29 Period 2

30. 30 Period 4

31. 31 Period 8

32. 32 Period 16

33. 33 Chaos

34. 34 Transistor Circuit

35. 35 Circuit Schematic Inductor A.C. Function Generator Transistor npn type Variable Resistor D.C. Voltage Source Load inductor Load Variable resistor Ground Feedback

36. 36 FEEDBACK SIGNAL AMPLIFIER POWER SUPPLY OUTPUT Simplified Schematic

37. 37 Period 1

38. 38 Period 2

39. 39 Period 4

40. 40 Period 8

41. 41 Period 16

42. 42 Chaos

43. 43 BIFURCATION DIAGRAMS

44. 44 Bifurcation Diagram ( Inductor) Output Voltage (peak to peak) Input Voltage (peak to Peak) y = 1.3717x - 0.0724

45. 45

46. 46 Output Voltage (peak to peak) Input Voltage (peak to peak) Bifurcation Diagram (Diode-inductor)

47. 47

48. 48 Bifurcation Diagram (Transistor) Input Voltage (peak to peak) Output Voltage (peak to peak)

49. 49

50. 50 Mathematical model (May model)

51. 51

52. 52 Input Voltage (peak to peak) Output Voltage (peak to peak) Chaotic Region (diode-inductor)

53. 53

54. 54 Output Voltage (peak to peak) Input Voltage (peak to peak) Periodic Region Amplified (diode)

55. 55

56. 56

57. 57 Feigenbaum Number

58. 58 Output Voltage (peak to peak) Input Voltage (peak to peak) 4.7 V 1.1 V  1 = 4.7 / 1.1 = 4.27 Diode-Inductor Circuit

59. 59 Transistor Circuit 1.2 V 0.28 V  1 = 1.2 / 0.28 = 4.285 Transistor Circuit Input Voltage (peak to peak) Output Voltage (peak to peak)

60. 60 Feigenbaum Number Theoretical value:   ∞  = 4.669201 Experimental value:   = 4.27 (diode-inductor) Experimental value:   = 4.285 (transistor) Other experimental values recorded: Electrical circuit (varactor)   = 4.257 Fluid Mechanics (Convection)   = 4.4

61. 61 Why are these systems chaotic ?

62. 62 Water gate analogy of a diode

63. 63 Water rate corresponding to time constant

64. 64 Simple Linear Amplifier SIGNAL AMPLIFIER POWER SUPPLY

65. 65 Chaotic Amplifier Circuit SIGNAL AMPLIFIER POWER SUPPLY FEEDBACK

66. 66 ATTRACTORS

67. 67

68. 68

69. 69 Inductor

70. 70 Input Voltage: 3.23 V

71. 71 Input Voltage: 4.44 V

72. 72 Input Voltage: 5.74 V

73. 73 Input Voltage: 7.11 V

74. 74 Input Voltage: 9.18 V

75. 75 Input Voltage: 11.0 V

76. 76 Input Voltage: 13.0 V

77. 77 Diode-Inductor

78. 78 Period 1 Input Voltage: 2.16 V

79. 79 Butterfly Diagram ( Period 2) Input Voltage: 2.75 V

80. 80 Period 4 Input Voltage: 7.45 V

81. 81 Period 8 Input Voltage: 8.55 V

82. 82 CHAOS Input Voltage: 8.5 V

83. 83 Transistor

84. 84 Input Voltage: 0.259 V

85. 85 Input Voltage: 0.709 V

86. 86 Input Voltage: 1.63 V

87. 87 Begin shifting phase Input Voltage: 1.86 V

88. 88 Phase shift complete Input Voltage: 1.98 V

89. 89 Butterfly Diagram ( Period 2) Input Voltage: 3.06 V

90. 90 Period 2 (anomaly) Input Voltage: 3.34 V

91. 91 Period 4 Input Voltage: 4.19 V

92. 92 Period 8 Input Voltage: 4.38 V

93. 93 Period 16 ? Input Voltage: 4.54 V

94. 94 Approaching Chaos Input Voltage: 4.96 V

95. 95 CHAOS Input Voltage: 7.34 V

96. 96 More Chaos

97. 97 Acknowledgments l M.J. Murdock Charitable Trust l Fluke Corporation l Thomas J. Holthaus l Pacific Lutheran University l Dr. Keith Clay l Lori Briggs l Jana Steiner l Christian Dilley