An antidote for hawkmoths: on the prevalence of structural chaos in non-linear modeling

Abstract

This paper deals with the question of whether uncertainty regarding model structure, especially in climate modeling, exhibits a kind of “chaos.” Do small changes in model structure, in other words, lead to large variations in ensemble predictions? More specifically, does model error destroy forecast skill faster than the ordinary or “classical” chaos inherent in the real-world attractor? In some cases, the answer to this question seems to be “yes.” But how common is this state of affairs? Are there precise mathematical results that can help us answer this question? And is dependence on model structure “sensitive” in that arbitrarily small errors can destroy forecast skill? We examine some efforts in the literature to answer this last question in the affirmative and find them to be unconvincing.

Appendices

Appendix A: Proof of Theorem 1

Theorem 1

Suppose we are given a difference equation of the form

$$ x_{n + 1} = f(x_{n}), $$

(13)

where $x_{i} \in \mathbb {R}$ and f : A → B is an arbitrary function from the bounded interval $A \subset \mathbb {R}$ into the bounded interval $B \subset \mathbb {R}$. Then given any function g : A → B and 𝜖 > 0, there exists δ > 0 and η > 0 such that the maximum one-step error of

$$ x^{\prime}_{n + 1} = \eta f(x^{\prime}_{n}) + \delta g(x^{\prime}_{n}), $$

(14)

from Eq. 9 is at most 𝜖 and x′_n+ 1 ∈ B.

Proof

Set δ and η such that

$$|\delta| \leqslant \left|\frac{\epsilon}{2\sup\{g(x_{n})\}}\right| \quad \text{and} \quad |\eta-1| \leqslant \left|\frac{\epsilon}{2\sup\{f(x_{n})\}}\right|,$$

where sup{f(x)} denotes the supremum²⁷ of f over all $x \in \mathbb {R}$. Note that the suprema exist²⁸ because f and g are bounded. Since the one-step error is

$$\begin{array}{@{}rcl@{}} |x^{\prime}_{n + 1} - x_{n + 1}| &=& |\eta f(x_{n}) + \delta g(x_{n}) - f(x_{n})| \\ &=& |(\eta - 1)f(x_{n}) + \delta g(x_{n})|, \end{array} $$

by the triangle inequality, the maximum one-step error is

$$\begin{array}{@{}rcl@{}} \sup\{|x'_{n + 1} - x_{n + 1}|\} \!&=&\! \sup\{|(\eta - 1)f(x_{n}) + \delta g(x_{n})|\} \\ \!&\leqslant&\! \sup\{|\eta - 1||f(x_{n})| + |\delta||g(x_{n})|\} \\ \!&\leqslant&\! \sup\left\{\left|\frac{\epsilon}{2\sup\{f(x_{n})\}}\right||f(x_{n})|\right\} + \sup\left\{\left|\frac{\epsilon}{2\sup\{g(x_{n})\}}\right||g(x_{n})|\right\}\\ \!&=&\! \left|\frac{\epsilon\sup\{f(x_{n})\}}{2\sup\{f(x_{n})\}}\right| + \left|\frac{\epsilon \sup\{g(x_{n})\}}{2\sup\{g(x_{n})\}}\right| \\ \!&=&\! \frac{\epsilon}{2} + \frac{\epsilon}{2} \\ \!&=&\! \epsilon, \end{array} $$

as desired. □

Appendix B: Proof of Theorem 2

Theorem 2

For all 𝜖 > 0 and1 > δ > 0, there exists an infinite set of polynomials g : [0, 1] → [0, 1], written

$$g(x) = \alpha_{n}x^{n} + \alpha_{n-1}x^{n-1} + {\cdots} + \alpha_{k + 1}x^{k + 1} + \alpha_{k}x^{k},$$

such that

$$ \min\{|\alpha_{k}|, |\alpha_{k + 1}|, \ldots, |\alpha_{n}|\} \geqslant 1 - \delta \quad \text{and} \quad \sup|g(x)| < \epsilon. $$

(15)

Proof

Let 0 ≤ e²(n − k)/𝜖 < k < n ≤ 1/δ, where e = 2.71828… is Euler’s constant, and define α_k,…,α_n by setting

$$ \sum\limits_{i=k}^{n}\alpha_{i} = 0, \quad |1 - |a_{i}|| < \delta, \quad \text{and} \quad |\alpha_{i + 1} + \alpha_{i}| < {\epsilon \over n-k} $$

(16)

for all k ≤ i ≤ n. Then since

$$\begin{array}{@{}rcl@{}} \sup|g(x)| &=& \sup|\alpha_{n}x^{n} + \alpha_{n-1}x^{n-1} + {\cdots} + \alpha_{k + 1}x^{k + 1} + \alpha_{k}x^{k}| \\ &\leqslant& \sup|\alpha_{n}x^{n} + \alpha_{n-1}x^{n-1}| + {\cdots} + \sup|\alpha_{k + 1}x^{k + 1} + \alpha_{k}x^{k}|, \end{array} $$

it suffices to determine the extrema of g_i(x) := α_i+ 1x^i+ 1 + α_ixⁱ for all k ≤ i ≤ n − 1. In that direction, note that the extrema of a function occurs either where that function’s first derivative vanishes or at the end points. Thus, the possible maxima are g_i(0) = 0,

$$|g_{i}(1)| = |\alpha_{i + 1} + \alpha_{i}| < {\epsilon \over n-k},$$

by Eq. 16, and, since

$$\begin{array}{@{}rcl@{}} 0 = {dg_{i}(x) \over dx} &=& x^{i}(\alpha_{i + 1}x + \alpha_{i}) \\ &=& ix^{i}(\alpha_{i + 1}x + \alpha_{i}) + \alpha_{i + 1}x^{i} \\ &=& i\alpha_{i + 1}x^{i + 1} + (i\alpha_{i} + \alpha_{i + 1})x^{i} \\ &=& x^{i}(i\alpha_{i + 1}x + i\alpha_{i} + \alpha_{i + 1}) \end{array} $$

has solutions at x = 0 and x = −(iα_i + α_i+ 1)/iα_i+ 1,

$$\begin{array}{@{}rcl@{}} \left| g_{i}\left( -{i\alpha_{i} + \alpha_{i + 1}\over i\alpha_{i + 1}}\right) \right| &=& \left|\alpha_{i + 1}\left( -{i\alpha_{i} + \alpha_{i + 1}\over i\alpha_{i + 1}} \right)^{i + 1} + \alpha_{i}\left( -{i\alpha_{i} + \alpha_{i + 1}\over i\alpha_{i + 1}} \right)^{i} \right| \\ &=& \left|(-1)^{i + 1}{\alpha_{i + 1}(i\alpha_{i} + \alpha_{i + 1})^{i + 1}\over (i\alpha_{i + 1})^{i + 1}} + (-1)^{i}{\alpha_{i}(i\alpha_{i} + \alpha_{i + 1})^{i}\over (i\alpha_{i + 1})^{i}} \right| \\ &=& \left|{\alpha_{i + 1}(i\alpha_{i} + \alpha_{i + 1})^{i + 1} - i\alpha_{i}\alpha_{i + 1}(i\alpha_{i} + \alpha_{i + 1})^{i}\over (i\alpha_{i + 1})^{i + 1}} \right| \\ &=& \left| {\alpha_{i + 1}^{2}(i\alpha_{i} + \alpha_{i + 1})^{i}\over (i\alpha_{i + 1})^{i + 1}} \right|. \end{array} $$

Applying (16), the definition of k and n, and the well-known inequality (1 + 1/i)ⁱ ≤ e, we have

$$\left| g_{i}\left( -{i\alpha_{i} + \alpha_{i + 1}\over i\alpha_{i + 1}}\right) \right| < {i^{i}(1 + \delta)^{i} \over i^{i + 1}\alpha_{i + 1}^{i-1}} = {(1 + \delta)^{i} \over i(1 - \delta)^{i-1}} \leqslant {(1 + 1/i)^{i} \over i(1 - 1/i)^{i-1}} \leqslant {e^{2} \over k} < {\epsilon \over n-k}.$$

Hence, after summing over all i, we conclude that sup |g(x)| < 𝜖. Since the first inequality in Eq. 15 holds by definition, and there exist an uncountable infinity of coefficients α_k,…,α_n satisfying (15), this yields the desired result. □