BLACK RABBITS EVERYWHERE
This chart shows a simple discrete dynamical system in which each value is the sum of the two preceding values. It follows the Fibonacci sequence, but begins with 1 and −0.618033 instead of two ones.
This chart shows the actual exchange rate of the cryptocurrency Zcash against the euro in August 2026.
After some fluctuations, both systems enter a stable regime. Then, suddenly, they exhibit explosive growth. In this sense, both systems were “pregnant with crisis” long before the visible breakout occurred.
Similar transitions from apparent stability to rapid destabilization can be observed in many fields. In medicine, physiological homeostasis may be followed by a sudden rise in blood pressure or a dangerous disruption of cardiac rhythm. Engineering systems provide many examples of loss of stability: unstable oscillations caused crashes of Saab JAS 39 Gripen prototypes, the forward-swept-wing X-29 operated close to the limits of controllability, and the Chernobyl reactor entered an uncontrollable power surge that ended in catastrophe.*
These phenomena cannot all be explained by a single mathematical model or attributed to the same physical mechanism. Nevertheless, in some systems, an approaching loss of stability can be detected in advance by identifying hidden hyperbolic points in their dynamics. By analogy with Nassim Nicholas Taleb’s Black Swan, we call this effect the Black Rabbits of Fibonacci.
The intelligent systems developed by our laboratory are capable of detecting such tipping points, as well as applying robust stabilization methods where appropriate to preserve or restore system stability.
We develop intelligent systems built on AI models, equipping them with domain knowledge, specialized tools, and the ability to act. Our systems are designed to protect your data and privacy. We design products that remain reliable, stable, and behave predictably.
Gunter Stein, “Respect the Unstable”, IEEE Control Systems Magazine, vol. 23, no. 4, pp. 12–25, August 2003.
Dr. Standalone
A private desktop application for maintaining a personal medical history, daily observations, health measurements, documents, and images. A key feature of the application is its ability to help identify early warning signs of developing medical conditions from longitudinal health data.
Crypto Tracer
Crypto Tracer demonstrates how transactions associated with a known public address can be reconstructed and examined. It illustrates why privacy cannot be assumed in conventional public blockchains—and why zero-knowledge technologies matter.
For a quick visualization, use the sample address below—or any other address you want to explore. Copy it, open Crypto Tracer, and paste it into the address field.
Sophia
An intelligent system for students that transforms unstructured assignments into clear plans, stages, deadlines, and calendar events. Sophia organizes source materials and supports the academic writing process, from coursework and essays to advanced research.
BLACK SWANS, DRAGON-KINGS & BLACK RABBITS
“In an environment where shocks can change direction so quickly, it would not be wise to anchor our decisions too firmly…”
Before the laboratory was founded, members of our group worked on discrete dynamical systems, identifying early signs of instability and developing effective methods of robust stabilization.
The idea that extreme events can radically change the world is not new. Nassim Nicholas Taleb described a Black Swan as a rare event with an extreme impact that appears explicable only after it has occurred. From the perspective of an observer looking forward, such an event lies outside the realm of reliable prediction.[1]
Didier Sornette and his co-authors developed a different perspective through the concept of Dragon-Kings. Unlike ordinary extreme events belonging to the tail of the same statistical distribution, Dragon-Kings are meaningful outliers generated by distinct internal mechanisms. Such mechanisms may include positive feedback, collective synchronization, phase transitions, and bifurcations. When their development produces identifiable precursors, the approaching extreme event may be diagnosed—and in some cases predicted—before it occurs.[2]
Black Rabbits represent a slightly different phenomenon. They exhibit the external characteristics of a Black Swan, yet they are entirely determined by the initial state and internal dynamics of the system itself. No external shock or change in the system’s parameters is required. Even a simple linear system may appear to stabilize before suddenly entering explosive growth, collapse, or another form of instability. In this sense, the system is pregnant with crisis from the very beginning.[3]
REFERENCES
- Nassim Nicholas Taleb, The Black Swan: The Impact of the Highly Improbable. New York: Random House, 2007.
- Didier Sornette, “Dragon-Kings, Black Swans and the Prediction of Crises”. International Journal of Terraspace Science and Engineering, 2(1), 1–18, 2009.
- Alexey Solyanik, The Black Rabbits of Fibonacci. arXiv:2412.20222, 2024.
GET IN TOUCH
If you are interested in collaboration, academic research, product development, or consulting in intelligent systems, robust stabilization, zero-knowledge technologies, cryptocurrency, or crypto taxation, feel free to reach out.