1. Chaos in general relativity
Taeyoon Moon
Inje univ.

2. Motivation 2

3. Motivation
Mandelbrot set 3

4. Motivation 4

5. Motivation 5

6. Motivation 6

7. Motivation 7

8. Motivation 8

9. Motivation 9

10. Motivation 10

11. Motivation 11

12. Motivation 12

13. Motivation 13

14. Motivation
This is scale symmetry !
It was hidden ! 14

15. Motivation
This is scale symmetry !
It was hidden ! 15

16. Motivation
This is scale symmetry !
It was hidden ! 16

17. Motivation
This is scale symmetry !
It was hidden ! 17

18. Motivation --> chaosmos (chaos + cosmos) This is scale symmetry !
It was hidden !
--> chaosmos (chaos + cosmos) 18

19. Motivation --> chaosmos (chaos + cosmos) This is scale symmetry !
It was hidden !
--> chaosmos (chaos + cosmos)
Can we describe this
in the field theoretical viewpoint with continuous symmetry ? 19

20. 20

21. Contents Motivation What is chaos? Measuring chaos
-> Poincaré sections
-> Lyapunov exponent
Chaos in general relativity
Chaos in Lifshitz spacetimes
-> motivation
Conclusion
Fractal
Future direction with the motivation

22. What is chaos? 22

23. What is chaos? 23

24. What is chaos? 24

25. What is chaos?

26. What is chaos?

27. Lorenz attractor -> Fixed points:
-> Nonlinearity: the two nonlinearities are xy and xz
-> Symmetry: (x, y) -> (-x, -y)

28. Lorenz attractor
-> Fixed points:  stable point for

29. Lorenz attractor -> Fixed points:  stable point for
 stable points for

30. Lorenz attractor -> Fixed points:  stable point for
 Unstable points for

31. Lorenz attractor -> Fixed points:  stable point for
 Unstable points for

32. Lorenz attractor
 Unstable points for

33. Lorenz attractor
 Unstable points for c

34. Lorenz attractor
 Unstable points for

35. Lorenz attractor
No crossing occur !!!

36. Lorenz attractor
No crossing occur !!!
Strange attractor

37. Lorenz attractor Lorenz system has chaotic solution. Strange attractor
No crossing occur !!!
Lorenz system has chaotic solution.
Strange attractor

38. Logistic Map:
(1)
(2)

39. Logistic Map:
(4)
(8)
(16)

40. Logistic Map:
3.75
Periodic doubling bifurcation

41. Logistic Map: Periodic doubling bifurcation
3.75
Periodic doubling bifurcation
Very sensitive dependence on initial conditions !!!

42. Logistic Map:
Very sensitive dependence on initial conditions !!!

43. Logistic Map:

44. Logistic Map:
Analysis by return map

45. Logistic Map:
Analysis by return map

46. Logistic Map:
Analysis by return map

47. Logistic Map:
Analysis by return map

48. Logistic Map:
Analysis by return map

49. Logistic Map:
Analysis by return map

50. Logistic Map:
3.75
Periodic doubling bifurcation 

51. Logistic Map:
3.75
Periodic doubling bifurcation  route to chaos

52. Lorenz attractor

53. Lorenz attractor -> Fixed points:  stable point for
 stable points for

54. Lorenz attractor -> Fixed points:  stable point for
 stable points for

55. Lorenz attractor

56. Lorenz attractor
Not exact periodic doubling bifurcation

57. Lorenz attractor
Not exact periodic doubling bifurcation  chaos occurs !!

58. Defining Chaos
Not exact periodic doubling bifurcation  chaos occurs !!

59. Defining Chaos
No definition of the term “chaos” is universally accepted !!

60. Defining Chaos
No definition of the term “chaos” is universally accepted !!
Almost everyone would agree on the three ingredients used in the following working definition:

61. Defining Chaos
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions
Aperiodic long-term behaviour:
It implies that there are trajectories which do not settle down to fixed points, periodic or quasi-periodic orbits as t ∞.
(Poincare sections)

62. Defining Chaos
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions
Aperiodic long-term behaviour:
It implies that there are trajectories which do not settle down to fixed points, periodic or quasi-periodic orbits as t ∞.
(Poincare sections)
2. Deterministic system :
It has no random or noisy inputs. Irregular behaviour arises solely from the system’s nonlinearity.

63. Defining Chaos
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions
Aperiodic long-term behaviour:
It implies that there are trajectories which do not settle down to fixed points, periodic or quasi-periodic orbits as t ∞.
(Poincare sections)
2. Deterministic system :
It has no random or noisy inputs. Irregular behaviour arises solely from the system’s nonlinearity.
3. Sensitive dependence on initial conditions :
Nearby trajectories diverge exponentially fast
(Lyapunov exponent)

64. Defining Chaos
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions
Aperiodic long-term behaviour:
It implies that there are trajectories which do not settle down to fixed points, periodic or quasi-periodic orbits as t ∞.
(Poincare sections)
2. Deterministic system :
It has no random or noisy inputs. Irregular behaviour arises solely from the system’s nonlinearity.
3. Sensitive dependence on initial conditions :
Nearby trajectories diverge exponentially fast
(Lyapunov exponent)

65. Defining Chaos
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions
Aperiodic long-term behaviour:
It implies that there are trajectories which do not settle down to fixed points, periodic or quasi-periodic orbits as t ∞.
(Poincare sections)
2. Deterministic system :
It has no random or noisy inputs. Irregular behaviour arises solely from the system’s nonlinearity.
3. Sensitive dependence on initial conditions :
Nearby trajectories diverge exponentially fast
(Lyapunov exponent)

66. Lyapunov exponent

67. Lyapunov exponent
Lorenz attractor

68. Poincaré section

69. Poincaré map (recurrence map)
Poincaré section
: Poincaré invented a technique to “simplify” representations of complicated phase space diagrams
 2d representations of 3d phase space diagram plots
Poincaré map (recurrence map)

70. Poincaré section
: Poincaré invented a technique to “simplify” representations of complicated phase space diagrams
 2d representations of 3d phase space diagram plots
The points of intersection are labeled, x1, x2, x3, etc.
The resulting set of points {xi} forms a pattern.

71. Poincaré section
: Poincaré invented a technique to “simplify” representations of complicated phase space diagrams
 2d representations of 3d phase space diagram plots
The points of intersection are labeled, x1, x2, x3, etc.
The resulting set of points {xi} forms a pattern.
Sometimes, the pattern is regular or irregular.

72. Poincaré section Irregularity of the pattern can be a sign of chaos !!
: Poincaré invented a technique to “simplify” representations of complicated phase space diagrams
 2d representations of 3d phase space diagram plots
The points of intersection are labeled, x1, x2, x3, etc.
The resulting set of points {xi} forms a pattern.
Sometimes, the pattern is regular or irregular.
Irregularity of the pattern can be a sign of chaos !!

73. J. Taylor, Classical Mechanics(2nd),
Poincaré section
J. Taylor, Classical Mechanics(2nd),
Ch 12. Nonlinear Mechanics and Chaos, p.464
J. Bevivino, The Path From the Simple Pendulum to Chaos

74. J. Taylor, Classical Mechanics(2nd),
Poincaré section
J. Taylor, Classical Mechanics(2nd),
Ch 12. Nonlinear Mechanics and Chaos, p.464
J. Bevivino, The Path From the Simple Pendulum to Chaos

75. Poincaré section

76. Poincaré section

77. Poincaré section

78. Poincaré section

79. Poincaré section

80. KAM theorem

81. KAM theorem
KAM (Kolmogorov–Arnold–Moser) theorem is a result in dynamical systems
  about the persistence of quasi-periodic motions under small perturbations.

82. KAM theorem
KAM (Kolmogorov–Arnold–Moser) theorem is a result in dynamical systems
  about the persistence of quasi-periodic motions under small perturbations.
The issue was…
whether or not a small perturbation of a conservative dynamical system
results in a lasting quasi-periodic orbit.

83. KAM theorem (conserving) KAM torus
KAM (Kolmogorov–Arnold–Moser) theorem is a result in dynamical systems
  about the persistence of quasi-periodic motions under small perturbations.
The issue was…
whether or not a small perturbation of a conservative dynamical system
results in a lasting quasi-periodic orbit.
(conserving) KAM torus

84. KAM theorem
For a system with the strong nonlinearity…
KAM torus

85. KAM theorem
For a system with the strong nonlinearity…

86. KAM theorem For a system with the strong nonlinearity…
The breaking of the KAM torus in Poincaré sections
can be one of the strongest indicators of chaotic behavior !!

87. Chaos in General Relativity
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

88. Chaos in General Relativity
The study of chaotic oscillations in the early stage of the universe near the initial singularity
The study of chaotic motion of particles around black holes
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

89. Chaos in General Relativity
The study of chaotic oscillations in the early stage of the universe near the initial singularity
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

90. Chaos in General Relativity
The study of chaotic oscillations in the early stage of the universe near the initial singularity
[Adv. Phys. 19, 525 (1970), V. Belinskii, I. Khalatinikov, E. Lifshits]
 BKL instability in anisotropic universe
[Class. Quan. Grav. 1, 417 (1984), D. Page]
 FRW universe: (uncountably infinite) bouncing aperiodic solutions
[Gen. Rel. Gra. 22, 349 (1990), A. Burd, N. Buric, G. Ellis]
 Bianchi IX universe: chaotic solution by analyzing Lyapunov exponent
[Class. Quan. Grav. 10, 1825 (1993), E. Calzetta, C. Hasi]
 Chaotic solution in FRW universe through Lyapunov E. and Poincare S.
[Phys. Rev. D47, 5336 (1993), A. Burd, R. Tavakol]
There is difficulty in defining coordinate invariant measures of chaos
[Phys. Rev. Lett. 102, (2009), A. Motter, A. Saa]
“Relativistic invariance of Lyapunov exponents”
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

91. Chaos in General Relativity
Magnetized spacetime ernst spacetime.

92. Chaos in General Relativity
The study of chaotic motion of particles around black holes
Magnetized spacetime ernst spacetime.

93. Chaos in General Relativity
The study of chaotic motion of particles around black holes
Point particle is completely integrable in the general Kerr-Newman background
[Phys. Rev. 174, 1559 (1968), B. Carter]
 Point particle around a gravitationally perturbed black hole
[Class, Quan, Grav. 9, 2573 (1992), L. Bombell, E. Calzetta]
 Point particle around a magnetized black hole
[Gen. Rel. Grav. 24, 729 (1992), V. Karas, D. Vokrouhlicky]
To obtain chaotic point-particle dynamics, we need to consider quite complicated
multi-black-hole spacetimes (Majumdar-Papapetrou type)
[Phys. Rev. D50, 618 (1994), C. Dettmann, N. Frankel, N. Cornish]
Magnetized spacetime ernst spacetime.
Spinning-particle can be chaotic in Schwarzchild spacetime
[Phys. Rev. D55, 4848 (1997), S. Suzuki, K. Maeda]
 Point particle around a rotating black ring
[Phys. Rev. D83, (2011), T. Igata, H. Ishihara, Y. Takamori]

94. Chaos in General Relativity
The study of chaotic motion of particles around black holes
Point particle is completely integrable in the general Kerr-Newman background
[Phys. Rev. 174, 1559 (1968), B. Carter]
Magnetized spacetime ernst spacetime.

95. Chaos in General Relativity
The study of chaotic motion of particles around black holes
Point particle is completely integrable in the general Kerr-Newman background
[Phys. Rev. 174, 1559 (1968), B. Carter]
 Test circular string can be chaotic in the Schwarzchild spacetime
[Class. Quan. Grav. 16, 3717 (1999) A. Frolov, A. Larsen]
Magnetized spacetime ernst spacetime.

96. Motivations

97. Motivations
Question:

98. Motivations Question:
Can we find chaotic behavior of test string in other geometries?

99. Motivations -> chaotic behavior of test string in AdS soliton
Question:
Can we find chaotic behavior of test string in other geometries?
-> chaotic behavior of test string in AdS soliton
[Phys. Lett. B699, 388 (2011), P. Basu, D. Das, A. Ghosh]
-> chaotic behavior of test string in AdS5×T1,1
[Phys. Lett. B700, 243 (2011), P. Basu, A. Zayas]

100. Motivations -> chaotic behavior of test string in AdS soliton
Question:
Can we find chaotic behavior of test string in other geometries?
-> chaotic behavior of test string in AdS soliton
[Phys. Lett. B699, 388 (2011), P. Basu, D. Das, A. Ghosh]
-> chaotic behavior of test string in AdS5×T1,1
[Phys. Lett. B700, 243 (2011), P. Basu, A. Zayas]
Can we find chaotic behavior of test string in 4 Dim’l geometries?

101. Motivations -> chaotic behavior of test string in AdS soliton
Question:
Can we find chaotic behavior of test string in other geometries?
-> chaotic behavior of test string in AdS soliton
[Phys. Lett. B699, 388 (2011), P. Basu, D. Das, A. Ghosh]
-> chaotic behavior of test string in AdS5×T1,1
[Phys. Lett. B700, 243 (2011), P. Basu, A. Zayas]
Can we find chaotic behavior of test string in 4 Dim’l geometries?
AdS spacetimes

102. Motivations -> chaotic behavior of test string in AdS soliton
Question:
Can we find chaotic behavior of test string in other geometries?
-> chaotic behavior of test string in AdS soliton
[Phys. Lett. B699, 388 (2011), P. Basu, D. Das, A. Ghosh]
-> chaotic behavior of test string in AdS5×T1,1
[Phys. Lett. B700, 243 (2011), P. Basu, A. Zayas]
Can we find chaotic behavior of test string in 4 Dim’l geometries?
AdS spacetimes
 It seems that the behavior of test string is regular (our guess)

103. Motivations -> chaotic behavior of test string in AdS soliton
Question:
Can we find chaotic behavior of test string in other geometries?
-> chaotic behavior of test string in AdS soliton
[Phys. Lett. B699, 388 (2011), P. Basu, D. Das, A. Ghosh]
-> chaotic behavior of test string in AdS5×T1,1
[Phys. Lett. B700, 243 (2011), P. Basu, A. Zayas]
Can we find chaotic behavior of test string in 4 Dim’l geometries?
AdS spacetimes
 It seems that the behavior of test string is regular (our guess)
 Only integrable solution is in the case with z = 1, corresponding to the AdS
[JHEP 1406 (2014) 018, D. Giataganas, K. Sfetsos]

104. Motivations

105. Motivations
Infalling observer, the spacetime is geodesically incomplete
Singularity is reached in finite proper tiem be infalling observer.

106. Motivations AdS spacetimes (z=1)  Lifshitz spacetimes (if not z=1)
Infalling observer, the spacetime is geodesically incomplete
Singularity is reached in finite proper tiem be infalling observer.
AdS spacetimes (z=1)  Lifshitz spacetimes (if not z=1)

107. Motivations In particular, let’s try to analyze this system
by regarding the critical exponent z as a control parameter!!
Infalling observer, the spacetime is geodesically incomplete
Singularity is reached in finite proper tiem be infalling observer.
AdS spacetimes (z=1)  Lifshitz spacetimes

108. Equations for test circular string

109. Equations for test circular string

110. Equations for test circular string

111. Equations for test circular string

112. Equations for test circular string

113. Equations for test circular string

114. Equations for test circular string

115. Equations for test circular string

116. Poincaré section (results)

117. Poincaré section (results)

118. Poincaré section (results)

119. Poincaré section (results)

120. Lyapunov exponent (result)
0.15

121. Lyapunov exponent (result)
~10-3

122. Conclusion Two primary tools to observe chaos –
the Poincaré section and Lyapunov Exponent indicate that
if z = 1, the motion of the string is regular,
while in the case slightly off z = 1, its behavior can be chaotic.
To generalize this result,
we need to explore the chaoticity of the system
given in other Lifshitz spacetimes.

123. Fractal
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

124. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

125. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]
Enlarge this area !

126. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

127. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]
Enlarge this area !

128. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

129. Fractal in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

130. Fractal in Logistic Map
self similarity
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

131. Fractal in Logistic Map
self similarity
whole in the part
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

132. Fractal in Logistic Map
self similarity
whole in the part
the behavior is universal
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

133. Fractal in Logistic Map
self similarity
whole in the part
the behavior is universal
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]
We can say that there is a FRACTAL hidden in here.

134. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

135. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

136. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

137. Universality in Logistic Map
Feigenbaum constant
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

138. Universality in Logistic Map
Feigenbaum constant
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

139. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

140. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

141. Universality in Logistic Map
Feigenbaum constant
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

142. Universality in Logistic Map
Feigenbaum constant
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

143. Universality in Logistic Map
Feigenbaum constant
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

144. Universality in Logistic Map
Feigenbaum constant
Universality
in Logistic Map!!
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

145. Universality in Logistic Map
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

146. Universality in Logistic Map
Mandelbrot set
BKL instability라고 해서 anisotropy 를 고려하면, chaotic solution 얻을 수 있다는 것…
Page paper:For a Friedman-Robertson-Walker universe minimally coupled to a massive scalar field, Hawking (1983) has shown that there is a countably infinite discrete set of periodic solutions which bounce without a singularity. Here it is suggested that there is also an uncountably infinite but still discrete set of perpetually bouncing aperiodic solutions. The latter set appears to form a fractal with positive Hausdorff-Besicovitch dimension.
Hasi et al paper: spatially closed frw metric coupled massive scalar…[Lya]
Prd paper: Bianchi type Ix universe[Lya]

147. Mandelbrot set
147

148. Mandelbrot set
148

149. Logistic map
Mandelbrot set
149

150. Future direction

151. gauge symmetry

152. gauge symmetry

153. gauge symmetry

154. gauge symmetry
global gauge symmetry

155. gauge symmetry

156. gauge symmetry

157. gauge symmetry

158. gauge symmetry
local gauge symmetry

159. gauge symmetry
local gauge symmetry

160. scale symmetry

161. scale symmetry

162. scale symmetry
global scale symmetry

163. local scale symmetry
Weyl (1918)

164. local scale symmetry
Weyl (1918)

165. local scale symmetry Weyl (1918) local scale symmetry
(conformal symmetry)

166. Future direction with motivation
This is scale symmetry !
It was hidden !
--> chaosmos (chaos + cosmos)
local scale symmetry
166

167. Future direction with motivation
“Chaos in fundamental interactions”
by G. Mandelbaum
167

168. Thank you 감사합니다.
168