Shadowing James Stewart, Marty DeWitt Computer Errors Numbers - PowerPoint PPT Presentation

Shadowing James Stewart, Marty DeWitt

Computer Errors Numbers stored using finite bits Floating point operations create round off errors True value can diverge from calculated value f(x,y) = (x+d, y+d) where d = 10 -7 , if computer rounds at 10 -6 then the computer will view every point as a fixed point (1) From How float or double values are stored in memory? Retrieved from : https://www.log2base2.com/storage/how-float-values-are-stored-in-memory.html

Logistic Map 4x(1-x) with initial value 0.333 n Rounded Orbit "True" Orbit Difference 1 0.333 0.3330000000 0 2 0.888 0.8884440000 0.000444 3 0.398 0.3964450355 0.0015549645 4 0.958 0.9571054773 0.0008945227 5 0.161 0.1642183306 0.0032183306 6 0.540 0.5490026819 0.0090026819 7 0.994 0.9903949490 0.003605051 8 0.024 0.0380511770 0.014051177 9 0.094 0.1464131410 0.052413141 10 0.341 0.4999053330 0.158905333 Difference decreased?

Basics of Shadowing dx/dt = -x, 0 ≤ t ≤ T max , x(0) = 1 Largest error of 0.019 at t = 1 Error decreases from 1< t < T max Stable case

Basics of Shadowing Shadowed orbit is a true orbit with a different initial condition that follows very closely to numerical orbit. dy/dt = y, 0 ≤ t ≤ T max , y(0) = 1 Euler’s method: y n+1 = (1+∆t)*y n Exact solution: y = e t Shadowing solution: y = 117.4*e -t

Shadowing Lemma Def: P n is a pseudo orbit if p n+1 = f(p n ) + ε , where ε accounts for roundoff errors. Assuming that there is a maximum error d such that |p n+1 - f(p n )| ≤ d , then p n is a d-pseudo orbit (3) Shadowing Lemma: If f is a hyperbolic diffeomorphism, then for every ε > 0 there is a d > 0 such that every d-pseudo orbit can be ε-shadowed (3)

Difference between rounded orbit and shadowed orbit for Logistic Map Shadowed n Rounded Orbit "True" Orbit Orbit Difference (True) Difference (Shadowed) 1 0.333 0.333 0.332702322 0.00000 0.00030 2 0.888 0.888444000 0.888045948 0.00044 0.00005 3 0.398 0.396445035 0.397681369 0.00155 0.00032 4 0.958 0.957105477 0.958123591 0.00089 0.00012 5 0.161 0.164218331 0.160491100 0.00322 0.00051 6 0.540 0.549002682 0.538934828 0.00900 0.00107 7 0.994 0.990394949 0.993936317 0.00361 0.00006 8 0.024 0.038051177 0.024107661 0.01405 0.00011 9 0.094 0.146413141 0.094105925 0.05241 0.00011 10 0.341 0.499905333 0.341 0.15891 0.00000

Challenge 5 - Shadowing the Skinny Baker Map Assume B(x 0 ) and x 0 are within d distance in both coordinates and above/below y=1/2 By looking at forward and backward iterations of the region, it is found that there exists a fixed point within 2d of x 0 From Chaos: An introduction to dynamical systems

Challenge 5 - Shadowing the Skinny Baker Map Theorem 5.19 - Let B denote the skinny baker map, and let d > 0. Assume that there is a set of points {x 0 , x 1 , … , x k-1 , x k = x 0 } s.t. each coordinate of B(x i ) and x i+1 differ by less than d for i = 0,1,...,k-1. Then there is a periodic orbit {z 0 ,...z k-1 } s.t. |x i -z i |<2d (1) Further if {x 0 , x 1 , … , x k-1 , x k } are a set of points s.t. each coordinate of B(x i ) and x i+1 differ by less than d for i = 0,1,...,k-1, then there exists a true orbit within 2d of x i (1)

Challenge 5 - Extending to the Cat Map Calculate the axes of the ellipse by taking the square roots of the eigenvalues of A n (A n ) T (1) If N 0 is a disk around a point x 0 with radius r then AN 0 A n (N 0 ) is the ellipse centered at A n (x 0 ) with axes n and re 2 n re 1 (1) N 0 For the Cat Map e 1 ≈ 2.6 and e 2 ≈ 0.4 (1) Ax 0 x 0 True orbit within 1.3d stable Are computer pictures of cat map orbits accurate assuming the maximum error of 10 -6 ? unstable

Cat Map and Beyond Cat map has the same Jacobian Matrix (Df(v)) everywhere, with stable and unstable direction (hyperbolic) For nonlinear maps, the Df(v) can be calculated at every point. As long as there is a stable and unstable direction, shadowing is possible Hyperbolic along orbit What is wrong with f(x,y) = (x+d,y+d) from earlier?

When shadowing fails For the logistic map f(x) = 1-2x 2 , the true orbit with i.c. x=0 stays within [-1,1]. However, with an error in the first iteration, the orbit goes outside [-1,1], eventually to -∞. No true orbit near x=0 will follow the errored orbit.

Questions?

References 1) Alligood, K. T., Sauer, T. D., & Yorke, J. A. (2010). Chaos: An introduction to dynamical systems . New York, NY: Springer. 2) How float or double values are stored in memory? (n.d.). Retrieved from https://www.log2base2.com/storage/how-float-values-are-stored-in-memory.html 3) Sanz-Serna, J., & Larsson, S. (1993). Shadows, chaos, and saddles. Applied Numerical Mathematics,13 (1-3), 181-190. doi:10.1016/0168-9274(93)90141-d 4) Sauer, T., & Yorke, J. A. (1991). Rigorous verification of trajectories for the computer simulation of dynamical systems. Nonlinearity, 4 (3), 961-979. doi:10.1088/0951-7715/4/3/018