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## Quantifying Chaos Introduction Time Series of Dynamical Variables Lyapunov Expon

Milbourn, Mary Ann, Finance Reporter has reference to this Academic Journal, PHwiki organized this Journal Quantifying Chaos Introduction Time Series of Dynamical Variables Lyapunov Exponents Universal Scaling of the Lyapunov Exponents Invariant Measures Kolmogorov-Sinai Entropy Fractal Dimensions Correlation Dimension & a Computational History Comments & Conclusions 1. Introduction Why quantify chaos To distinguish chaos from noise / complexities. To determine active degrees of freedom. To discover universality classes. To relate chaotic parameters to physical quantities. 2. Time Series of Dynamical Variables (Discrete) time series data: x(t0), x(t1), , x(tn) Time-sampled (stroboscopic) measurements Poincare section values Real measurements & calculations are always discrete. Time series of 1 variable of n-D system : If properly chosen, essential features of system can be re-constructed: Bifurcations Chaos on-set Choice of sampling interval is crucial if noise is present (see Chap 10) Quantification of chaos: Dynamical: Lyapunov exponents Kolmogorov-Sinai (K-S) Entropy Geometrical: Fractal dimension Correlation dimension Only 1-D dissipative systems are discussed in this chapter.

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9.3. Lyapunov Exponents Time series: Given i & j, let System is chaotic if with Lyapunov exponent Technical Details: Check exponential dependence. is x dependent = i(xi) / N . N cant be too large as long as bounded systems. = 0 as long as periodic system. i & j shouldnt be too close. Bit- version: dn = d0 2n Logistic Map 9.4. Universal Scaling of the Lyapunov Exponents Period-doubling route to chaos: Logistic map: A = 3.5699 LyapunovExponents.nb < 0 in periodic regime. = 0 at bifurcation point. (period-doubling) > 0 in chaotic regime. tends to increase with A More chaotic as A increases. Huberman & Rudnick: (A > A) is universal as long as periodic-doubling systems: = Feigenbaum 0 = 0.9 ~ order parameter A A ~ T TC

Derivation of the Universal Law as long as Chaotic b in addition to s merge via period-undoubling as long as A > A. Ratio of convergence tends to Feigenbaum . Logistic map Let 2m b in addition to s merge to 2m1 b in addition to s at A = Am . Reminder: 2m b in addition to s bifurcate to 2m+1 b in addition to s at A = Am . Divergence of trajectories in 1 b in addition to : Divergence of trajectories among 2m b in addition to : = effective Lyapunov exponent denoting 2m iterations as one. = Lyapunov exponent as long as If is the same as long as all b in addition to s, then Ex.2.4-1: Assuming n = gives Similarly: i.e.,

9.5. Invariant Measures Definition of Probability Invariant Measures Ergodic Behavior For systems of large DoFs, geometric analysis becomes unwieldy. Alternative approach: Statistical methods. Basic quantity of interest: Probability of trajectory to pass through given region of state space. Definition of Probability Consider an experiment with N possible results (outcomes). After M runs (trials) of the experiment, let there be mi occurrences of the ith outcome. The probability pi of the ith outcome is defined as where ( Normalization ) If the outcomes are described by a set of continuous parameters x, N = . mi are finite M = in addition to pi = 0 i. Remedy: Divide range of x into cells/bins. mi = number of outcomes belonging to the ith cell. Invariant Measures For an attractor in state space: Divide attractor into cells. 1-D case: pi mi / M. Set {pi} is a natural probability measure if it is independent of (almost all) IC. Let then is an invariant probability measure if p(x) dx = probability of trajectory visiting interval [ x, x+dx ] or [ xdx/2 , x+dx/2 ]. = probability of trajectory visiting cell i. Treating M as total mass p(x) = (x)

Example: Logistic Map, A = 4 From § 4.8: For A = 4, logistic map is equivalent to Bernoulli shift. with Numerical: 1024 iterations into 20 bins Ergodic Behavior Time average of B(x): Bt should be independent of t0 as T . Ensemble average of B(x): System is ergodic if Bt = Bp . Comments: Bp is meaningful only as long as invariant probability measures. p(x) may not exist, e.g., strange attractors. Example: Logistic Map, A = 4 Local values of the Lyapunov exponent: Ensemble average value of the Lyapunov exponent: ( same as the Bernoulli shift ) Same as that calculated by time average (c.f. §5.4):

9.6. Kolmogorov-Sinai Entropy Brief Review of Entropy: Microcanonical ensemble (closed, isolated system in thermal equilibrium): S = k ln N = k ln p p = 1/N Canonical ensemble (small closed subsystem): S = k i pi ln pi i pi = 1 2nd law: S 0 as long as spontaneous processes in closed isolated system. S is maximum at thermodynamic equilibrium Issue: No natural way to count states in classical mechanics. S is defined only up to an constant ( only S physically meaningful ) Quantum mechanics: phase space volume of each state = hn , n = DoF. Entropy as long as State Space Dynamics Divide state space into cells (e.g., hypercubes of volume LDof ). For dissipative systems, replace state space with attractors. Start evolution as long as an ensemble of I.C.s (usually all located in 1 cell). After n time steps, count number of states in each cell. Note: Non-chaotic motion: Number of cells visited (& hence S ) is independent of t & M on the macroscopic time-scale. Chaotic motion: Number of cells visited (& hence S ) increases with t but independent of M. R in addition to om motion: Number of cells visited (& hence S ) increases with both t & M k = Boltzmann constant Only S is physically significant. Kolmogorov-Sinai entropy rate = K-S entropy = K is defined as For iterated maps or Poincare sections, = 1 so that E.g., if the number of occupied cells Nn is given by in addition to all occupied cells have the same probability then Pesin identity: i = positive average Lyapunov exponents

Alternative Definition of the K-S Entropy See Schuster Map out attractor by running a single trajectory as long as a long time. Divide attractor into cells. Start a trajectory of N steps & mark the cell its in at t = nas b(n). Do the same as long as a series of other slightly different trajectories starting from the same initial cell. Calculate the fraction p(i) of trajectories described by the ith cell sequence. Then where Exercise: Show that both definitions of K give roughly the same result as long as all 3 types of motions discussed earlier. 9.7. Fractal Dimensions Geometric aspects of attractors Distribution of state space points of a long time series Dimension of attractor Importance of dimensionality: Determines range of possible dynamical behavior. Dictates long-term dynamics. Reveals active degrees of freedom. For a dissipative system : D < d, D dimension of attractor, d dimension of state space. D < D, D = dimension of attractor on Poincare section. For a Hamiltonian system, D d 1, D = dimension of points generated by one trajectory ( trajectory is confined on constant energy surface ) D < D, D = dimension of points on Poincare section. Dimension is further reduced if there are other constants of motion. Example: 3-D state space attractor must shrink to a point or a curve system cant be quasi-periodic ( no torus ) no q.p. solutions as long as the Lorenz system. Dissipative system: Strange attractor = Attractor with fractional dimensions (fractals) Caution: Therere many inequivalent definitions of fractal dimension. See J.D.Farmer, E.Ott, J.A.Yorke, Physica D7, 153-80 (1983) Capacity ( Box-Counting ) Dimension Db Easy to underst in addition to . Not good as long as high d systems. 1st used by Komogorov N(R) = Number of boxes of side R that covers the object Example 1: Points in 2-D space A single point: Box = square of sides R. Set of N isolated points: Box = square of sides R. R = ½ (minimal distance between points). Example 2: Line segment of length L in 2-D space Box = square of sides R. Example 3: Cantor Set Starting with a line segment of length 1, take out repeatedly the middle third of each remaining segment. Caution: Given M finite, set consists of 2M line segments Db = 1. Given M infinite, set consists of discrete points Db = 0. Limits M in addition to R 0 must be taken simultaneously. At step M, there remain 2M segments, each of length 1/3M.

Measure of the Cantor set: Length of set Ex. 9.7-5: Fat Fractal Example 4: Koch Curve Start with a line segment of length 1. a) Construct an equilateral triangle with the middle third segment as base. b) Discard base segment. Repeat a) in addition to b) as long as each remaining segment. At step M, there exists 4M segments of length 1/3M each. Types of Fractals Fractals with self-similarity: small section of object, when magnified, is identical with the whole. Fractals with self-affinity: same as self-similarity, but with anisotropic magnification. Deterministic fractals: Fixed construction rules. R in addition to om fractals: Stochastic construction rules (see Chap 11).

Fractal Dimensions of State Space Attractors Difficulty: R 0 not achievable due to finite precision of data. Remedy: Alternate definition of fractal dimension (see §9.8) Logistic map at A , renormalization method: Db = 0.5388 (universal) Elementary estimates: Consider A A+ ( from above ). Sarkovskiis theorem chaotic b in addition to s undergo doubling-splits as A A+ . Feigenbaum universality splitted b in addition to s are narrower by 1/ in addition to 1/2 . Assume points in each b in addition to distributed uni as long as mly splitting is Cantor-set like. 1st estimate: R decreases by factor 1/ at each splitting. 2nd estimate: Db procedure dependent. An infinity of dimensional measures needed to characterize object (see Chap 10) The Similarity Dimensions as long as Nonuni as long as m Fractals

9.8. Correlation Dimension & a Computational History 9.9. Comments & Conclusions

## Milbourn, Mary Ann Finance Reporter

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