Oscillatory finite-time singularities in finance, population and rupture

Abstract

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity;(3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. 7 Overall dynamics for n > 1 and m > 2 with α > 0: |y 1 (t → t c )| < +∞ (except for isolated initial conditions) and |y 2 (t → t c )| = +∞ Recall from Section 5.1 for the sub-dynamical system with only the oscillatory element that the case n > 1 corresponds to highly nonlinear oscillations with a monotonically decreasing period as the amplitude of the oscillations increases ( Figure 2). From Section 5.2 for the sub-dynamical system with only the singular element, the case m > 2 corresponds to finite-time singularity with finite increment in y 1 and infinite increment in y 2 (Figure 4). Furthermore, Section 6.3 on the phase space of the full dynamical system showed the following results:1. any trajectory y(t; y 0 , t 0 ) starting away from the origin connects the origin in backward time and |y 2 | → ∞ in forward time;

Renormalization group analysis of the 2000–2002 anti-bubble in the US S&P500 index: explanation of the hierarchy of five crashes and prediction

“…Positive feedbacks, i.e., self-reinforcement, refer to the fact that, conditioned on the observation that the market has recently moved up (respectively down), this makes it more probable to keep it moving up (respectively down), so that a large cumulative move may ensue. In a dynamical model, positive feedbacks with nonlinear amplifications have been shown to give rise to finite-time singularities which are good mathematical representations of the market price trajectory on the approach to crashes [20,7]. However, such dynamical formulation describes only the price trajectory up to its singularity.…”

Nonlinear dynamical model of regime switching between conventions and business cycles

“…The general weak point of nonlinear equations with polynomial nonlinearities is the appearance of unstable solutions, when either the market price, or the price rate, or both become divergent, often even at a finite moment of time (Ide and Sornette, 2002). Actual divergences in markets are, of course, unrealistic.…”

“…But, p f (t) is not directly observable, so that we define the mispricing variable as the difference between the logarithm of the market price and the logarithm of the fundamental price (Ide and Sornette, 2002) x ≡ log p − log p f .…”

“…This creates a threshold-like behavior which is adequately represented by a nonlinear term such as Ax 3 . Ide and Sornette (2002) considered other powers x n with n > 1.…”

“…which is a variant of the Ide-Sornette (2002) model. Then, the basins of attraction disappear completely.…”