Universal Critical Behavior of Period Doublings in Symmetrically Coupled Systems

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

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: d and a) 2nd Close-up 1st Close-up

4. ’ Renormalization-Group (RG) Analysis of The Critical Behavior  RG operator (fn: 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. 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. 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 ’ Period-doubling bif. Saddle-node bif.  1 -1 Transverse Stability Multiplier  Determining stability against the transverse perturbation across the diagonal  ’ (Reduced coupling function) Period-doubling bif. Pitchfork bif.  1 -1

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

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 ’ ’  Relevant eigenvalues of fixed points Reduced Linearized Operator Note the reducibility of into a semi-block form • One relevant eigenvalue  (=4.669…) (1D case): Common Eigenvalue c: Coupling eigenvalue (CE)  Critical stability multipliers (SMs) For the critical case, : SMs of an orbit of period 2n : Critical SMs ’ (1D critical SM): Common SM ’

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. ~ - A. Scaling Behavior near The Zero-Coupling Critical Point Governed by the 1st fixed point GI = 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) = 2n 

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

12. B. Scaling Behavior near The Critical Line Segments Consider the leftmost critical line segment with both ends cL and cR on the A = A line cL (= -1.457 727 …) cR (= -1.013 402 …) (1) Scaling Behavior near The Both Ends 1 2 Governed by the 2nd fixed point GII (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) q(period) = 2n At both ends,  ( = 2)

13. (2) Scaling Behavior inside The Critical Line 1 2 Governed by the 3rd fixed point GIII (x)= -f* (x) with no relevant CE’s and  =0 ’ * * 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. Synchronous Chaotic Attractors near The Left End of The Leftmost Critical Line   c = 2 

15. 2. Dissipatively-coupled case with g(x, y) = c(y2-x2) Stability Diagram for The Synchronous Orbits One critical line with both ends c0 = 0 and c0 = -A on the A = A line ’ ’ c0 c0

16. A. Scaling Behavior near Both Ends c0 and c0 ’ Governed by the 1st fixed point GI = 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 q(period) = 2n  • Scaling of The Slopes of The transverse SM ,n(A, c) At both ends, (2 = 2) B. Scaling Behavior inside The Critical Line 1 2 ’ Governed by the 3rd fixed point GIII (x)= -f* (x) with no relevant CE’s and  =0 * * (The scaling behavior is the same as that for the 1D case.)

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

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

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

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

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 GI (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. 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