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
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
Difference decreased?
n Rounded Orbit "True" Orbit Difference 1 0.333 0.3330000000 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
dx/dt = -x, 0 ≤ t ≤ Tmax , x(0) = 1 Largest error of 0.019 at t = 1 Error decreases from 1< t < Tmax Stable case
Shadowed orbit is a true orbit with a different initial condition that follows very closely to numerical orbit.
dy/dt = y, 0 ≤ t ≤ Tmax , y(0) = 1 Euler’s method: yn+1 = (1+∆t)*yn Exact solution: y = et Shadowing solution: y = 117.4*e-t
Def: Pn is a pseudo orbit if pn+1 = f(pn) + ε , where ε accounts for roundoff errors. Assuming that there is a maximum error d such that |pn+1- f(pn)| ≤ d , then pn 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)
n Rounded Orbit "True" Orbit Shadowed 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
Assume B(x0) and x0 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 x0
From Chaos: An introduction to dynamical systems
Theorem 5.19 - Let B denote the skinny baker map, and let d > 0. Assume that there is a set of points {x0, x1, … , xk-1, xk = x0} s.t. each coordinate of B(xi) and xi+1 differ by less than d for i = 0,1,...,k-1. Then there is a periodic orbit {z0,...zk-1} s.t. |xi-zi|<2d (1) Further if {x0, x1, … , xk-1, xk} are a set of points s.t. each coordinate of B(xi) and xi+1 differ by less than d for i = 0,1,...,k-1, then there exists a true orbit within 2d of xi (1)
Calculate the axes of the ellipse by taking the square roots of the eigenvalues of An(An)T
(1)
If N0 is a disk around a point x0 with radius r then An(N0) is the ellipse centered at An(x0) with axes re1
n and re2 n (1)
For the Cat Map e1 ≈ 2.6 and e2 ≈ 0.4 (1) True orbit within 1.3d Are computer pictures of cat map orbits accurate assuming the maximum error of 10-6? x0 Ax0 N0 AN0 stable unstable
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?
For the logistic map f(x) = 1-2x2, 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.
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