Shadowing

Figure 1: An \(\alpha\)-pseudotrajectory

Figure 2: A \(\beta\)-pseudotrajectory

Shadowing refers to the relationship between the mathematical solutions of a differential equation (or map) and approximate solutions obtained in the presence of noise or round-off error. A mathematical solution is said to shadow a noisy solution if it stays close to the noisy solution for some amount of time.

[edit] Round-off errors

Consider a mathematical solution to a deterministic chaotic system with an initial condition \(x.\) Because numbers are represented in computers with finite precision, there will typically be a small difference between \(x\) and how \(x\) is registered on a computer. As the system is evolved forward in time, this difference will be amplified exponentially due to the chaotic nature of the system. Round-off error at each step in a numerical calculation further compounds the problem. The question therefore arises as to whether or not computer-generated solutions to chaotic systems are related to their true mathematical solutions.

The existence of arbitrarily long shadowing solutions has been proven for invertible hyperbolic maps by Anosov (1967) and Bowen (1970, 1978). For non-hyperbolic systems, it is often possible to estimate the length of time for which a noisy trajectory is shadowed by a mathematical solution (Grebogi, Hammel, Yorke and Sauer (1990), Sauer, Grebogi and Yorke (1997)).

[edit] Shadowing lemma

The following lemma is due to Bowen (1970, 1978).

Definition: A sequence \(\{x_i\}_{i=a}^b\) is an \(\alpha\)-pseudo-orbit for \(f\) if \(d(x_{i+1},f(x_i))<\alpha\) for all \(a \leq i \leq b\ .\) See Figure 1.

Definition: The point \(y\) \(\beta\)-shadows \(\{x_i\}_{i=a}^b\) if \(d(f^i(y),x_i)|<\beta\) for \(a \leq i \leq b\ .\) See Figure 2.

Shadowing Lemma: Let \(\Lambda\) be a hyperbolic invariant set. Then for every \(\beta>0\ ,\) there is an \(\alpha>0\) such that every \(\alpha\)-pseudo-orbit \(\{x_i\}_{i=a}^b\) in \(\Lambda\) is \(\beta\)-shadowed by a point \(y \in \Lambda\ .\)

See also the article "Shadowing lemma for flows".

[edit] Alternative definitions

The following definitions are also encountered in the literature.

Definition\[\{p_n\}_{n=a}^b\] is a \(\delta_f\)-pseudotrajectory for \(f\) if \(|p_{n+1} - f(p_n)|<\delta_f\) for \(a \leq n \leq b\ ,\) where \(\delta_f\) is the noise amplitude and \(n\) is an integer.

Definition\[\{x_n\}_{n=a}^b\] is a true trajectory if it satisfies \(x_{n+1}=f(x_n)\) for \(a \leq n \leq b\ .\)

Definition of shadowing: The true trajectory \(\{x_n\}_{n=a}^b\) \(\delta_x\)-shadows the pseudotrajectory \(\{p_n\}_{n=a}^b\) on \(a \leq n \leq b\) if \(|x_n-p_n|<\delta_x\) for \(a \leq n \leq b\ .\)

[edit] Estimating shadowing times

In some systems, shadowing times (meaning the amount of time a mathematical solution shadows a noisy one) can be very short. Sauer, Grebogi, and Yorke (1997) estimate shadowing times for chaotic systems that exhibit unstable dimension variability as follows: \[\langle \tau \rangle \sim \delta^{-2m/\sigma^2}\] where \(\delta\) is the one-step error (noise), and \(m\) and \(\sigma\) are the mean and standard deviation of the finite-time Lyapunov exponent closest to zero, respectively.

[edit] References

D.V. Anosov, Proc. Steklov Inst. Math. 90, 1 (1967).
R. Bowen, American Journal of Mathematics 92, 725 (1970).
R. Bowen, CBMS Regional Conference Series in Mathematics, Vol. 35. AMS Publications: Providence.
C. Grebogi, S.M. Hammel, J.A. Yorke, and T. Sauer, Physical Review Letters 65, 1527 (1990).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
T. Sauer, C. Grebogi, and J.A. Yorke, Physical Review Letters 79, 59 (1997).

Internal references

James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.

Kendall E. Atkinson (2007) Numerical analysis. Scholarpedia, 2(8):3163.