Download presentation
Presentation is loading. Please wait.
Published byCharles Knight Modified over 9 years ago
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. 11 1 Period-doubling bif. Pitchfork bif. 11 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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.