A concept of antifragility for dynamical systems

Abstract

This paper defines antifragility for dynamical systems as convexity of a newly introduced "logarithmic rate". It shows how to compute this rate for positive linear systems, and it interprets antifragility in terms of pulsed alternations of extreme strategies in comparison to average uniform strategies.

Figures (3)

• Figure 1: Left: A convex function $\rho$, with dosages $u$, $v$, and $w = \frac{u+v}{2}$. Here $\rho (w)$ is smaller than $\overline{\rho }$. Right: Concave $\rho$; now $\rho (w)$ is larger than $\overline{\rho }$.
• Figure 2: A pulsed protocol: iteration of high dose $u$ (green) followed by low dose $v$ (red). Shown also is a uniform protocol with average dose $w = \frac{u+v}{2}$ (blue).
• Figure 3: Population growth under pulsed protocol. Iteration of dose $u$ (green, with negative growth rate) followed by $v$ (red, positive growth rate). The resulting equivalent growth rate $\overline{\rho }$ is the average of the two rates; blue dashed line shows the equivalent growth under this rate, $e^{\overline{\rho } N}\, x_0$. A uniform protocol (solid blue lines) will provide a lower growth rate (perhaps negative) or a higher one, depending on the convexity of the rate function $\rho$.

Theorems & Definitions (8)

• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Theorem 1
• Theorem 2
• Remark 3
• Theorem 3