Renormalization Group Theory

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1 Chapter 16 Renormalization Group Theory In the previous chapter a procedure was developed where hiher order 2 n cycles were related to lower order cycles throuh a functional composition and rescalin procedure. Based on this pictorial approach we are led to a formal understandin of the period doublin route to chaos. This renormalization approach was conjectured by Feienbaum [1], based on a similar qualitative analysis, and was later proved riorously by Collet and Eckman [2] for a power law maximum x 1 1+ε 2 with small ε, and usin a riorous numerical proof by Lanford [3] for the quadratic maximum. It is convenient to shift coordinates x x 1 2 so that the maximum is at x = 0. The map is now of the interval 1 2 x 1 2 and is iven by (see fiure 16.1). x n+1 = a ( 1 4 x2 n ), (16.1) 16.1 The fixed point Define an operator T that preforms the functional composition ( iteration ) and rescalin: ( ( T [f ] (x) = αf f x )). (16.2) α Note that T operates on the function f. The universal behavior of the sequence of subharmonic bifurcations is understood from the followin two statements: 1

2 CHAPTER 16. RENORMALIZATION GROUP THEORY 2 Fiure 16.1: Map of the unit interval shifted to have maximum at x = The operation T has a fixed point solution (x) for a particular value of α, i.e. T [] = or ( ( (x) = α x )). (16.3) α To completely define the solution we must fix the nature of the maximum to be quadratic at x = 0, and to set the overall scale (since if (x) is a solution µ(x/µ) is also easily seen to be a solution) we choose (0) = Linearizin about the fixed point of T yields a sinle unstable direction (eienvector) with eienvalue δ, i.e. writin with φ small, and definin the linearized operator f(x)= (x) + φ(x) (16.4) L [φ] = T [ + φ] T [] (16.5) we must have [ L φ (i)] (x) = λ (i) φ (i) (x) (16.6)

3 CHAPTER 16. RENORMALIZATION GROUP THEORY 3 definin the eienvectors φ (i) and the eienvalues λ (i). Then orderin the eienvalues in decreasin sequence we must have λ (1) = δ>0 (and we then write φ (0) as h(x)), and λ (i) < 0 for i>1. Explicitly, we have for the linearization ( ( T [ + φ] = α ( + φ) x ) ( + φ x )). (16.7) α α Then { ( ( L [φ] (x) = α φ x )) ( ( + x )) ( φ x )}. α α α It may be worth emphasizin that we are lookin at the fixed point and linearization of the operation T which acts on the space of functions, and not the fixed point and linearization of the map which acts on a point in the unit interval Evaluation of α and δ The numbers α and δ are defined by this abstract procedure, without any reference to a dynamical system. This can be illustrated with a very crude approximation. First for the fixed point equation we approximate (x) as (x) = 1 + bx 2 + (16.8) and inore all the hiher order terms represented by the. Then ( ( x )) = 1 + b ( 1 + bx 2 /α 2) 2. (16.9) α This is expanded up to O(x 2 ) so that the fixed point equation becomes 1 + bx 2 = α [ 1 + b + (2b 2 /α 2 )x 2]. (16.10) Equatin coefficients then ives b = α/2 and α = (16.11) (choosin the positive root for α since b must be neative).

4 CHAPTER 16. RENORMALIZATION GROUP THEORY 4 The value of δ is evaluated by an even cruder approximation for the eienfunction h(x): we simply take the first term in a Taylor expansion, i.e. h(x) 1, and demand that the linearization equation α { h ( ( x α )) ( + ( x α )) ( h x )} = δh(x) (16.12) α be satisfied at x = 0 (where this approximation is best). This ives α [ ((0)) + 1 ] = δ. (16.13) But (0) = 1 and (1) = 2b = α usin (16.8) for. So δ α 2 α (16.14) These are very crude estimates just to show that the numbers are defined by statements 1 and 2. Better estimates (e.. keep more terms in the power series expansions) ive α = , δ = (16.15) 16.3 Universal map functions As all other directions around the fixed point contract, we can move out from the fixed point alon the unstable direction defined by the eienvector h. Define for lare r. Then r (x) = (x) + δ r h(x) (16.16) T r = T [ + δ r h ] = + δ (r 1) h = r 1, (16.17) where the second equality comes from the linearization near the fixed point and the definition of h as the eienvector with eienvalue δ. Eventually as we repeat this operation the function r for decreasin r moves away from the vicinity of the fixed point and the linearization procedure fails. However we can continue to define r at smaller r throuh the operation of T : r 1 = T r. (16.18)

5 CHAPTER 16. RENORMALIZATION GROUP THEORY 5 f R R=R Stable Manifold Fixed Point Fiure 16.2: Flows of under T. The solid circles show successive r alon the unstable direction iven by operations of T. The empty circles show successive T n f R. This defines the sequence of function 0, 1... approachin the limit. This is illustrated in fiure Now thinkin of these functions as one dimensional maps f, remember that if f has a stable 2 n cycle then f(f(x))has a stable 2 n 1 cycle, and since T ives just this functional composition (toether with rescalin) the operation (16.18) acts to decrease the period order by a factor of 2. Thus we can choose to label the r such that r hasa2 r cycle (and then itself shows a 2 cycle). These r, defined only throuh referrin to the fixed point structure, are universal versions of the sequence of maps studied in chapter 15. We have not fixed the normalization of h(x), which affects the functional form of the r for small r. This choice determines what type of 2 r cycle r appears (superstable, marinally unstable etc.). The choice of normalization defined by the requirement 0 (0) = 0 selects the superstable cycles. (This is because the period 1 superstable cycle corresponds to the situation where the unit slope diaonal throuh the oriin intersects the map curve at the

6 CHAPTER 16. RENORMALIZATION GROUP THEORY 6 maximum and this occurs when then maximum is at y = 0, see fiure 16.1) Bifurcations in the physical map The fixed point in function space has one unstable direction: all other directions are contractin and we can construct a hypersurface in function space (the stable manifold) such that functions f ( ) on this hypersurface evolve towards the fixed point under the operation T. Since the stable manifold is codimension 1 we miht expect a function f R (x) parameterized by a sinle parameter R to intersect the stable manifold for some value of R which we will call R. Then under the operation of T T n f R as n. (16.19) What about f R for R near R? Initially (for small n) T n f R will follow T n f R, since R R is small, i.e. it will flow towards the fixed point. If R R is sufficiently small, T n f R will approach very close to the fixed point for some rane of n (fiure 16.2). Since we understand the behavior here in terms of the fixed point and linearization we learn properties of the physical map from the properties of the fixed point and the universal maps r. However, since the fixed point has one unstable direction, eventually T n f R will bein to flow away from the fixed point alon a path close to the unstable direction h(x), and we need to proceed carefully. We split the n operations of T into a number of sements. First we operate some finite number q times, with q sufficiently lare to brin T q f R that the components alon the unstable directions have decayed, i.e. to brin the function into the vicinity of the fixed point. The number q will not depend on R R for small values of this quantity, and is rouhly the number of operations needed to brin f R into the vicinity of the fixed point. Taylor expansion then ives the amplitude alon the unstable direction to be linear in the deviation R R for small enouh R R, so that T q f R = (x) + c (R R) h(x) (16.20) with c some number. The initial flow away from the fixed point is iven by the linearization. So next we operate a number p times with p lare (tendin to as R R 0), but small enouh so that function remains in the linear reime

7 CHAPTER 16. RENORMALIZATION GROUP THEORY 7 alon the unstable direction. This ives T p+q f R = T p [(x) + c (R R) h(x)] = (x) + c (R R) δ p h(x). (16.21) And then we complete T n with a further n p q actions, which may take the function into the nonlinear reime. Finally we may write T n f R = T n p q [ (x) + c (R R) δ p h(x) ] = T n p [ (x) + c (R R) δ p h(x) ] (16.22) with c = cδ q. Now if we make the special choices of R = R m defined by we have c (R R m ) = δ m (16.23) T n f Rm = T n p [ (x) + δ p m h(x) ] = m n, (16.24) with m n the universal superstable 2 m n cycle. But each operation of T decreases the order of the cycle by a factor of 2, and so we see that f Rm must have a superstable 2 m cycle. Thus we have shown that if R R = c 1 δ m then f R will show a superstable 2 m cycle, and as m increases to infinity we will find a cascade of period doublin bifurcations, with superstable orbits at values of R with separation ratios iven by the universal constant δ. We have shown the existence of the period doublin cascade with universal properties based on the existence of the fixed point of T with the assumed properties, and the assumption that f R will cross the unstable manifold of the fixed point for some R. Since the stable manifold is of codimension 1 this latter should be a common occurrence Scalin of the map function Puttin in the scalin factors we have f 2n (x) = ( α) n T n f(( α) n x) (16.25) for any function f. In particular usin (16.24) for f Rm we have f 2n R m = ( α) n m n (( α) n x) (16.26)

8 CHAPTER 16. RENORMALIZATION GROUP THEORY 8 for lare m, n with m n. In particular, for example, if we take m = n then lim f 2n n R n = ( α) n 0 (( α) n x) (16.27) This says that the rescaled (by α n ) version of the 2 n th order functional composition of any map f (with a quadratic maximum) at the value of R ivin a superstable 2 n cycle will tend to the universal function 0 for lare n. This operation was performed in chapter 15, demonstration 11. Similarly takin n = m 1 will yield 1, etc. Alternatively if we first set R = R then lim f 2n n R = ( α) n (( α) n x) (16.28) i.e. we approach the universal fixed point function, which we see to be the universal onset of chaos function. This limit was approached in chapter 15, demonstration Applications - the Lyapunov exponent Suppose we want to calculate some property P [f ] of the map function f, such as the Lyapunov exponent. Since accordin to the development in the previous chapter T n f Rm = m n (16.29) for m, n lare and n m, if we can relate P [f ] to P [Tf], then by repeated operation of T we can relate the desired property P [f ] to a property of the universal map i.e. to P [ n ], which of course is universal. Usin this approach we can show that certain properties of the physical map f are universal, and we can sometimes calculate the universal property precisely. Some of the calculations, such as for the Lyapunov exponent, are quite simple. Others, such as the power spectrum and the scalin of the separation of points in the orbit on which this depends, can be quite intricate. The map f = f Rm has a stable 2 m cycle, and the Lyapunov exponent is iven by the slopes of the map at the points x i in the cycle: λ [f ] = 1 2 m 1 2 m lo f (x i ). (16.30) i=0

9 CHAPTER 16. RENORMALIZATION GROUP THEORY 9 Writin this sum instead as the sum of successive pairs of points λ [f ] = 1 2 m m lo ( f (x 2j ) f (x 2j+1 ) ) (16.31) j=0 and usin the chain rule for differentiatin f 2 = f(f(x))we can write this as λ [f ] = m 1 1 ( ) f 2 2 m 1 lo 2 (x 2j ) = 1 2 λ [ f 2]. (16.32) j=0 It is easy to check that puttin in the scalin factors in the definition of T does not chane this result i.e. λ [f ] = 1 2 λ [ f 2]. (16.33) This simply is the fact that we et the same diverence of orbits iteratin f 2 half as many times. Repeatin this many times we have λ [ ] 1 f Rm = 2 n λ [ T n ] 1 f Rm = 2 n λ [ ] m n. (16.34) Now we choose n to be comparable to m. Let us for example choose n = m, so that λ [ ] 1 f Rm = 2 m λ [ 0]. (16.35) But λ [ 0 ] is some number independent of the nature of f and of the index m. This ives us the important scalin result λ [ ] f Rm 2 m. It is often convenient to rewrite the scalin result in terms of the evolution with the map parameter R (e.. as in the Lyapunov plots of chapter 14). Thus we write λ [ ] f Rm = c 1 2 m = R R m = c 1 δ m = ce m lo 2 c e m lo δ. (16.36) Eliminatin m we can write this as λ R R β with β = lo lo δ (16.37)

10 CHAPTER 16. RENORMALIZATION GROUP THEORY 10 This functional dependence ives the shape of the envelope of the Lyapunov exponent for a fixed stability type (since in (16.36) this is what the values R m refer to). The same result applies above R, where positive Lyapunov exponents indicate chaotic dynamics (aain for a fixed type of behavior, e.. the band merin points see chapter 17). February 4, 2000

11 Biblioraphy [1] M.J. Feienbaum, J. Stat. Phys. 21, 669 (1979) [2] P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, (Birkhauser, Boston) [3] O.E. Lanford, Bull. Am. Math. Soc. 6, 427 (1982) 11

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