"By definition, complex systems—be they financial markets or weather patterns—contain too many moving parts to be reduced to any simple mathematical formula. It’s not just that we haven’t discovered an equation to express the behavior of the stock market; it’s that such an equation does not exist. Instead, researchers like Sornette construct and run computer models in order to gain insight into the potential behavior of these systems.
In developing these models, they have discovered that systems all share some surprisingly simple underlying properties. For instance, systems have the potential to change drastically in very short periods of time and often exhibit early warning signs that indicate when and how these changes will occur. These changes could be stock market crashes, tsunamis, heart attacks, or colony collapses, and in general are known as critical points. The theoretical properties of critical points have some profound—and often alarming—implications for real-world complex systems. In the case of climate change, when a critical level of greenhouse gas emissions is reached, it has been suggested that Earth’s climate may undergo rapid and irreversible changes. Identifying points like this one, and devising smart solutions to avoid the catastrophes they may bring, is critical.
To understand how phenomena like critical points come about in a system, consider a rock concert: A band has just finished its final encore, as a 60,000-plus crowd reacts in rapturous applause. The clapping begins as a cacophonous patter, eventually growing to a loud, chaotic roar. And then something interesting happens. Amid the noise, seemingly without effort or conscious guiding by the audience members, the applause evolves into a synchronized, steady rhythm; the claps become a single beat, with thousands of fans clapping in unison. Finally, it slows to a once again out-of-sync denouement, before abruptly ceasing altogether. The synchronized clapping emerges spontaneously in the crowd and is analogous to what is called a self-organizing property of a complex system, of which critical points are one example.Sornette borrows this metaphor, originally articulated by Phillip Ball, to explain stock-market behavior. “A financial crash is not chaos. It’s when everyone agrees; it’s when everyone is clapping together,” he says. “So you have a synchronization of actions in the same way that clapping becomes synced.” The bubble in the Chinese stock market burst at just such a critical point. This is what Sornette claims to have predicted with his market model. According to him, what indicated that the financial bubble was going to burst was that the behavior of investors began oscillating, more and more wildly, between widespread buying and selling."
This is an interesting article but I'm not sure that I can believe it. Is it possible to know a future state of a complex system?
In general the answer is no. If you have the history of the the system you can look backwards and see how it got here, but you can't look forward. Even if you're observing the same system later and see the same pattern, you can't be sure that in the next instant the system will repeat what it did in the past.
If the system is as simple as the logistic map, the large pattern of behavior can be "predicted" but only if the initial conditions are exactly the same, a condition not possible in the real world.
For an r between about 3.57 and 4, its behavior is chaotic. if you had a system like this one, you might be able to tell when you were approaching that region of chaos.
The mention of heart rhythm is troubling as well. A healthy heart has a chaotic rhythm. An unhealthy heart has a less chaotic rhythm. Breathing has similar characteristics. See secrets of the heart. Mechanical breathing systems work better when chaos is introduced.
I'm just afraid that we keep trying to find simple solutions to complex problems.