1. 1 Universal Critical Behavior of Period Doublings in Symmetrically Coupled Systems Sang-Yoon Kim Department of Physics Kangwon National University Korea  Low-dimensional Dynamical Systems (1D Maps, Forced Nonlinear Oscillators) Universal Routes to Chaos via Period Doublings, Intermittency, and Quasiperiodicity: Well Understood  Coupled High-dimensional Systems (Coupled 1D Maps, Coupled Oscillators) Coupled Systems: used to model many physical, chemical, and biological systems such as Josephson junction arrays, chemical reaction-diffusion systems, and biological-oscillation systems  Purpose To investigate critical scaling behavior of period doubling in coupled systems and to extend the results of low-dimensional systems to coupled high-dimensional systems.

2. 2 Period-doubling Route to Chaos in The 1D Map  1D Map with A Single Quadratic Maximum  An infinite sequence of period doubling bifurcations ends at a finite accumulation point  When exceeds, a chaotic attractor with positive  appears.

3. 3 Critical Scaling Behavior near A=A   Parameter Scaling:  Orbital Scaling:  Self-similarity in The Bifurcation Diagram A Sequence of Close-ups (Horizontal and Vertical Magnification Factors:  and  ) 1st Close-up 2nd Close-up

4. 4 Renormalization-Group (RG) Analysis of The Critical Behavior  RG operator (f n : n-times renormalized map)  Squaring operator Looking at the system on the doubled time scale  Rescaling operator Making the new (renormalized) system as similar to the old system as possible  Attraction of the critical map to the fixed map with one relevant eigenvalue    ’

5. 5 Critical Behavior of Period Doublings in Two Coupled 1D Maps  Two Symmetrically Coupled 1D Maps g(x,y): coupling function satisfying a condition g(x,x) = 0 for any x  Exchange Symmetry  Invariant Synchronization Line y = x Synchronous (in-phase) orbits on the y = x line Asynchronous (out-of-phase) orbits  Concern Critical scaling behavior of period doublings of synchronous orbits

6. 6 Stability Analysis of Synchronous Periodic Orbits  Two stability multipliers for a synchronous orbit of period q: Longitudinal Stability Multiplier | | Determining stability against the longitudinal perturbation along the diagonal : Same as the 1D stability multiplier Transverse Stability Multiplier  Determining stability against the transverse perturbation across the diagonal (Reduced coupling function) ’ ’   Period-doubling bif. Saddle-node bif. 11 1 Period-doubling bif. Pitchfork bif. 11 1

7. 7 Renormalization-Group (RG) Analysis for Period Doublings  Period-doubling RG operator for the symmetrically 1D maps T RG Eqs. for the uncoupled part f and the coupling part g: n-times renormalized map, (RG Eq. for the 1D case),  Reduced period-doubling RG operator Def: Reduced Coupling Function ’ ’ Note that keeps all the essential informations contained in. [It’s not easy to directly solve the Eq. for the coupling fixed function g * (x, y).]

8. 8 Fixed Points of and Their Relevant Eigenvalues  Three fixed points (f *,G * ) of (f *,G * ) = (f *,G * ): fixed-point Eq. f * (x): 1D fixed function, G * (x): Reduced coupling fixed function G*(x)G*(x) c (CE) (CTSM) I0 II 21 III Nonexistent0  Relevant eigenvalues of fixed points Reduced Linearized Operator Note the reducibility of into a semi-block form  Critical stability multipliers (SMs)  One relevant eigenvalue  (=4.669…) (1D case): Common Eigenvalue  c : Coupling eigenvalue (CE) For the critical case, : SMs of an orbit of period 2 n : Critical SMs (1D critical SM): Common SM ’ ’ ’ ’

9. 9 Critical Scaling Behaviors of Period Doublings 1. Linearly-coupled case with g(x, y) = c(y  x) Stability Diagram for The Synchronous Orbits Asymptotic Rule for The Tree Structure 1. U branching Occurrence only at the zero c-side (containing the zero-coupling point) 2. Growth like a “chimney” Growth of the other side without any further branchings  Bifurcation Routes 1. U-route converging to the zero-coupling critical point 2. C-routes converging to the critical line segments  Critical set Zero-coupling Critical Point + an Infinity of Critical Line Segments

10. 10 A. Scaling Behavior near The Zero-Coupling Critical Point Governed by the 1st fixed point G I = 0 with two relevant CE’s 1 =  (-2.502 …) and 2 =2. CTSM: | | =  = * (=  1.601…)  Scaling of The Nonlinearity and Coupling Parameters for large n;  Scaling of The Slopes of The Transverse SM ,n (A , c) * * *  ~ q(period) = 2 n 

11. 11 Hyperchaotic Attractors near The Zero-Coupling Critical Point    c,1 = 

12. 12 B. Scaling Behavior near The Critical Line Segments Consider the leftmost critical line segment with both ends c L and c R on the A = A  line Governed by the 2nd fixed point G II (x)=  [f * (x)-1] with one relevant CE = 2. CTSM:  = 1 * *  Scaling of The Nonlinearity and Coupling Parameters for large n;  Scaling of The Slopes of The transverse SM ,n (A , c) ( = 2) c L (=  1.457 727 …) c R (=  1.013 402 …) 1212 ’ At both ends, (1) Scaling Behavior near The Both Ends  q(period) = 2 n

13. 13 (2) Scaling Behavior inside The Critical Line Governed by the 3rd fixed point G III (x)=  f * (x) with no relevant CE’s and  = 0 * 1212 ’ * Scaling Behavior: Same as that for the 1D case [Det = 1  2 = 0  1D] Transverse Lyapunov exponents near the leftmost critical line segment Inside the critical line, Synchronous Feigenbaum Attractor with   < 0 on the diagonal  1D-like Scaling Behavior When crossing both ends, Synchronous Feigenbaum State: Transversely unstable (   > 0)

14. 14 Synchronous Chaotic Attractors near The Left End of The Leftmost Critical Line    c = 2

15. 15 2. Dissipatively-coupled case with g(x, y) = c(y 2  x 2 ) One critical line with both ends c 0 = 0 and c 0 =  A  on the A = A  line ’ Stability Diagram for The Synchronous Orbits c0c0 c0c0 ’

16. 16 A. Scaling Behavior near Both Ends c 0 and c 0 Governed by the 1st fixed point G I = 0 with two relevant CE’s 1 =  (-2.502 …) and 2 =2. * (no constant term)  There is no component in the direction of with c = 1  Only 2 becomes a relevant one!  Scaling of The Nonlinearity and Coupling Parameters for large n  Scaling of The Slopes of The transverse SM ,n (A , c) At both ends, B. Scaling Behavior inside The Critical Line Governed by the 3rd fixed point G III (x)=  f * (x) with no relevant CE’s and  = 0 * 1212 ’ * (The scaling behavior is the same as that for the 1D case.) ’ ( 2 = 2)  q(period) = 2 n

17. 17 Hyperchaotic Attractors near The Zero-Coupling Critical Point   

18. 18 Period Doublings in Coupled Parametrically Forced Pendulums  Parametrically Forced Pendulum (PFP) Normalized Eq. of Motion:  Symmetrically Coupled PFPs coupling function O S  l m  = 0: Normal Stationary State  =  : Inverted Stationary State  Dynamic Stabilization Inverted Pendulum (Kapitza)

19. 19 Stability Diagram of The Synchronous Orbits  Same structure as in the coupled 1D maps Critical set = zero-coupling critical point + an infinity of critical lines  Same critical behaviors as those of the coupled 1D maps

20. 20 Scaling Behaviors near The Zero-Coupling Critical Point    c,1 = 

21. 21 Scaling Behaviors near The Right End of The Rightmost Critical Line    = 2

22. 22 Summary  Three Kinds of Universal Critical Behaviors Governed by the Three Fixed Points of the Reduced RG Operator (Reduced RG method: useful tool for analyzing the critical behaviors) RG results: Confirmed both in coupled 1D maps and in coupled oscillators. [ S.-Y. Kim and H. Kook, Phys. Rev. E 46, R4467 (1992); Phys. Lett. A 178, 258 (1993); Phys. Rev. E 48, 785 (1993). S.-Y. Kim and K. Lee, Phys. Rev E 54, 1237 (1996). S.-Y. Kim and B. Hu, Phys. Rev. E 58, 7231 (1998). ]  Remarks on other relevant works 1. Extension to the even maximum-order case f (x) = 1 – A x z (z = 2, 4, 6, …) The relevant CE’s of G I (x) = 0 vary depending on z [ S.-Y. Kim, Phys. Rev. E 49, 1745 (1994). ] 2. Extension to arbitrary period p-tuplings (p = 2, 3, 4, …) cases (e.g. period triplings, period quadruplings) Three fixed points for even p; Five fixed points for odd p [ S.-Y. Kim, Phys. Rev. E 52, 1206 (1995); Phys. Rev. E 54, 3393 (1996). ] 3. Intermittency in coupled 1D maps [ S.-Y. Kim, Phys. Rev. E 59, 2887 (1999). Int. J. Mod. Phys. B 13, 283 (1999). ] 4. Quasiperiodicity in coupled circle maps (unpublished) * 

23. 23 Effect of Asymmetry on The Scaling Behavior  : asymmetry parameter, 0    1  = 0: symmetric coupling   0: asymmetric coupling,  = 1: unidirectional coupling Pitchfork Bifurcation (  = 0)  Transcritical Bifurcation (   0) Structure of The Phase Diagram and Scaling Behavior for all   Same as those for  = 0 