Chaos in general relativity 2015. 11. 27 Taeyoon Moon Inje univ.

Motivation 2

Motivation Mandelbrot set 3

Motivation 4

Motivation 5

Motivation 6

Motivation 7

Motivation 8

Motivation 9

Motivation 10

Motivation 11

Motivation 12

Motivation 13

Motivation This is scale symmetry ! It was hidden ! 14

Motivation This is scale symmetry ! It was hidden ! 15

Motivation This is scale symmetry ! It was hidden ! 16

Motivation This is scale symmetry ! It was hidden ! 17

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

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

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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

What is chaos? 22

What is chaos? 23

What is chaos? 24

What is chaos?

What is chaos?

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

Lorenz attractor -> Fixed points:  stable point for

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

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

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

Lorenz attractor  Unstable points for

Lorenz attractor  Unstable points for c

Lorenz attractor  Unstable points for

Lorenz attractor No crossing occur !!!

Lorenz attractor No crossing occur !!! Strange attractor

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

Logistic Map: (1) (2)

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

Logistic Map: 3.75 Periodic doubling bifurcation

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

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

Logistic Map:

Logistic Map: Analysis by return map

Logistic Map: Analysis by return map

Logistic Map: Analysis by return map

Logistic Map: Analysis by return map

Logistic Map: Analysis by return map

Logistic Map: Analysis by return map

Logistic Map: 3.75 Periodic doubling bifurcation 

Logistic Map: 3.75 Periodic doubling bifurcation  route to chaos

Lorenz attractor

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

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

Lorenz attractor

Lorenz attractor Not exact periodic doubling bifurcation

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

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

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

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:

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)

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.

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)

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)

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)

Lyapunov exponent

Lyapunov exponent Lorenz attractor

Poincaré section

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)

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.

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.

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 !!

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

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

Poincaré section

Poincaré section

Poincaré section

Poincaré section

Poincaré section

KAM theorem

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

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.

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

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

KAM theorem For a system with the strong nonlinearity…

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 !!

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]

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]

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]

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, 184101 (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]

Chaos in General Relativity Magnetized spacetime ernst spacetime.

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

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, 047501 (2011), T. Igata, H. Ishihara, Y. Takamori]

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.

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.

Motivations

Motivations Question:

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

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]

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?

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

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)

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]

Motivations

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

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)

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

Equations for test circular string

Equations for test circular string

Equations for test circular string

Equations for test circular string

Equations for test circular string

Equations for test circular string

Equations for test circular string

Equations for test circular string

Poincaré section (results)

Poincaré section (results)

Poincaré section (results)

Poincaré section (results)

Lyapunov exponent (result) 0.15

Lyapunov exponent (result) ~10-3

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.

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]

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]

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 !

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]

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 !

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]

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]

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]

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]

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]

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.

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

Mandelbrot set 147

Mandelbrot set 148

Logistic map Mandelbrot set 149

Future direction

gauge symmetry

gauge symmetry

gauge symmetry

gauge symmetry global gauge symmetry

gauge symmetry

gauge symmetry

gauge symmetry

gauge symmetry local gauge symmetry

gauge symmetry local gauge symmetry

scale symmetry

scale symmetry

scale symmetry global scale symmetry

local scale symmetry Weyl (1918)

local scale symmetry Weyl (1918)

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

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

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

Thank you 감사합니다. 168