Dynamical system theory of periodically collapsing bubbles

TL;DR

This paper models asset and bond prices with coupled equations, showing how nonlinear dynamics can lead to periodic bubbles and crashes through bifurcations, with critical behavior similar to phase transitions.

Contribution

It introduces a minimal dynamical system capturing bubble formation via bifurcations, linking financial crashes to phase transition analogies.

Findings

Periodic bubbles emerge from bifurcations in the model.

Bubble amplitude diverges with critical exponent 1.

Waiting times between bubbles diverge with critical exponent 1/2.

Abstract

We propose a reduced form set of two coupled continuous time equations linking the price of a representative asset and the price of a bond, the later quantifying the cost of borrowing. The feedbacks between asset prices and bonds are mediated by the dependence of their "fundamental values" on past asset prices and bond themselves. The obtained nonlinear self-referencing price dynamics can induce, in a completely objective deterministic way, the appearance of periodically exploding bubbles ending in crashes. Technically, the periodically explosive bubbles arise due to the proximity of two types of bifurcations as a function of the two key control parameters $b$ and $g$, which represent, respectively, the sensitivity of the fundamental asset price on past asset and bond prices and of the fundamental bond price on past asset prices. One is a Hopf bifurcation, when a stable focus transforms into an unstable focus and a limit cycle appears. The other is a rather unusual bifurcation, when a stable node and a saddle merge together and disappear, while an unstable focus survives and a limit cycle develops. The lines, where the periodic bubbles arise, are analogous to the critical lines of phase transitions in statistical physics. The amplitude of bubbles and waiting times between them respectively diverge with the critical exponents $\gamma = 1$ and $\nu = 1/2$, as the critical lines are approached.