1. Emergent complexity Chaos and fractals

2. Uncertain Dynamical Systems c-plane

5. Julia sets

8. Overcoming computational complexity

9. What does this have to do with complex systems? This classic computational problem illustrates an important idea, but in an easily visualized way. Most computational problems involve uncertain dynamical systems, from protein folding to complex network analysis. Not easily visualized. Natural questions are typically computationally intractable, and conventional methods provide little encouragement that this can be systematically overcome.

10. Main idea e.g. the boundary moves.

11. Main idea Points near the boundary are “fragile.” Merely stating the obvious in this case. But illustrates general principle that can be exploited by the right algorithms.

12. 5 10 15 20 25 30 # iterations Points not in M.

13. 5 10 15 20 25 30 # iterations Color indicates number of iterations of simulation to show point is not in M.

14. But simulation cannot show that points are in M.

15. But simulation is fundamentally limited Gridding is not scalable Finite simulation inconclusive

16. It’s easy to prove that this disk is in M. Other points in M are fragile to the definition of the map. Merely stating the obvious. Main idea

17. Sufficient condition

19. Trivial to prove that these points are in Mandelbrot set.

20. Main idea The longer the proof, the more fragile the remaining regions. The proof of this region is a bit longer.

21. Main idea And so on… Proof even longer.

23. Easy to prove these points are in Mset.

24. Easy to prove these points are not in Mset.

25. Proofs get harder. (But all still “easy.”) What’s left gets more fragile.

26. Complexity  Chaos  Fractals Emergent complexity.

27. Complexity implies fragility What matters to organized complexity.

28. Emergent complexity

29. How might this help with organized complexity and “robust yet fragile”? Long proofs indicate a fragility. Either a true fragility (a useful answer) or an artifact of the model (which must then be rectified) Potentially fundamentally changes computational complexity for organized complexity Brings back together two research areas that have been separated for decades: Numerical analysis and ill-conditioning Computational complexity (P, NP/coNP, undecidable)

30. Proof? New proof methods that is scalable and systematic (can be automated).

31. Breaking hard problems SOSTOOLS proof theory and software Nested family of (dual) proof algorithms Each family is polynomial time Recovers most “gold standard” algorithms as special cases, and immediately improves No a priori polynomial bound on depth (otherwise P=NP=coNP) Conjecture: Complexity implies fragility

32. Safety Verification and Reachability Analysis Safety critical applications. Exhaustive simulation is not exact. Set propagation is computationally expensive. Find a barrier certificate B(x) Initial set Unsafe set B(x) = 0 Scalable computation using SOS machinery.

33. Parametric Memoryless Dynamic (IQC) Hybrid, Uncertain, Stochastic Hybrid systems can be handled easily, even for systems with uncertainty: Use supermartingales as certificates. Get guaranteed bound on reach probability. Also stochastic hybrid systems: (Prajna, Jadbabaie – HSCC ’04) (Prajna, Jadbabaie, Pappas – CDC ’04)

34. Feedback control Variable supply/demand Physical network Components Functional requirements Hardware constraints “Horizontal” Decompositions “Vertical” layering Unifying role of dual proofs and decomp- ositions

35. Main idea Think of this as a robustness problem.

36. How robust is stability to perturbations in c? Globally stable.

37. Region of convergence. How robust is stability to perturbations in c?

39. Simulation is fundamentally limited

41. 5 10 15 20 25 30 # iterations

42. 10 20 30 40 50 60 iterations

43. 10 20 30 40 50 60 iterations -2

44. 10 20 30 40 50 60 iterations -2

45. 10 20 30 40 50 60 iterations

46. 10 20 30 40 50 60 iterations

47. 70 120 iterations

48. 120 iterations 60

52. realc

53. -3-20123 -2 0 1 2 3 4 Fixed points Stable Unstable Stable

54. -3-20123 -2 0 1 2 3 4 Unstable Stable Fixed points

55. -3-20123 -2 0 1 2 3 4 Unstable Stable Equilibria

56. -3-20123 -2 0 1 2 3 4

57. Sufficient condition

59. Trivial to prove that these points are in Mandelbrot set.

60. -3-20123 -2 0 1 2 3 4 Unstable Equilibria

62. Bifurcations for c “last 200 x”

63. c Zoom-in

64. “last 200 x” c Zoom-in

65. -3-20123 -2 0 stable Bifurcations to chaos

66. -2 0 -3-20123 -2 0 1

67. -2 0 -3-20123 -2 0 1

68. -3-20123 -2 0 1 2 3 4

69. -3-20123 -2 0 1 2 3 4

70. -3-20123 -2 0 stable Bifurcations to chaos

71. Invariant set? Invariance

72. -3-20123 -2 0 Bifurcations to chaos

73. Special case of SOS

75. Contradiction!

77. -3-20123 -2 0 What is the shortest proof possible? Can prove the whole yellow region using SOSTOOLS!

78. Lyapunov argument

79. How to prove membership? 2-period lobes

80. 1 2 3 3 4 4 4 Proof lengths

81. Prove membership of 2-period lobe: Using a stability argument of the 2-period map. Using an invariance argument.

82. Formulate the invariance Problem as the emptiness of a semialgebraic set. Then use SOSTOOLS to construct the certificate.

83. Discrete → Continuous

84. Lyapunov’s theorem

86. Can we test these conditions algorithmically? Use the Sum of Squares decomposition!

87. is SOS Find such that SOSTOOLS Then equilibrium is asymptotically stable.

88. Robust Stability? Use SOSTOOLS to construct V(x,p). www.cds.caltech.edu/sostools

89. Chemical oscillator Nondimensional state equations

90. 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 a b Can be computed analytically, which is not scalable.

91. 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 a b 00.10.20.30.40.50.6 1 1.5 2 2.5 3 a = 0.1, b = 0.13 Numerical simulation.

92. a = 1, b = 2 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 a b a = 0.6, b = 1.1 (1.1, 0.6) (2, 1)

93. 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 a b

94. equilibrium

95. 00.20.40.60.811.21.4 0 0.2 0.4 0.6 0.8 1 a b

96. ModelingAnalysis Set of possible system behaviors Set of bad system behaviors Proof of robustness More on Saturday