Nonlinear Dynamics, Magnitude-Period Formula and Forecasts on Earthquake

Based on the geodynamics, an earthquake does not take place until the momentum-energy excess a faulting threshold value of rock due to the movement of the fluid layer under the rock layer and the transport and accumulation of the momentum. From the nonlinear equations of fluid mechanics, a simplified nonlinear solution of momentum corresponding the accumulation of the energy could be derived. Otherwise, a chaos equation could be obtained, in which chaos corresponds to the earthquake, which shows complexity on seismology, and impossibility of exact prediction of earthquakes. But, combining the Carlson-Langer model and the Gutenberg-Richter relation, the magnitude-period formula of the earthquake may be derived approximately, and some results can be calculated quantitatively. For example, we forecast a series of earthquakes of 2004, 2009 and 2014, especially in 2019 in California. Combining the Lorenz model, we discuss the earthquake migration to and fro. Moreover, many external causes for earthquake are merely the initial conditions of this nonlinear system.

💡 Research Summary

The paper attempts to build a unified framework for earthquake generation, forecasting, and spatial migration by invoking nonlinear dynamics. Its central premise is that a fault does not slip until the momentum‑energy accumulated in a rock layer, driven by the motion of a fluid layer beneath it, exceeds a critical threshold. Starting from the Navier‑Stokes equations, the author claims to obtain a simplified nonlinear solution for the momentum (p(t)) that grows exponentially with time and, upon reaching the threshold, undergoes a rapid transition to a chaotic state. This chaotic state is identified with the occurrence of an earthquake, leading to the conclusion that exact prediction is impossible because the system is intrinsically chaotic.

Despite this deterministic‑chaos argument, the author proceeds to derive an approximate “magnitude‑period” (M‑P) relationship that would allow statistical forecasting. The derivation combines three separate ingredients:

1. Carlson‑Langer model – a spring‑block system with velocity‑weakening friction that reproduces stick‑slip cycles and provides a mechanistic link between stress accumulation and slip events.
2. Gutenberg‑Richter law – the empirical relation (\log N = a - bM) describing the frequency‑magnitude distribution of earthquakes.
3. A scaling argument that links the characteristic waiting time (T) between large events to the magnitude (M) through a power‑law of the form (T = k,10^{\alpha M}), where (k) and (\alpha) are constants derived from the previous two models.

Using this M‑P formula, the author claims to have forecast a series of Californian earthquakes in 2004, 2009, 2014, and especially a large event in 2019. The paper also invokes the Lorenz system (originally devised for atmospheric convection) to describe “earthquake migration”. By re‑interpreting the Lorenz variables as “seismic potential”, “stress accumulation”, and “fluid flow intensity”, the author argues that the strange‑attractor dynamics produce a back‑and‑forth motion of the epicenter, analogous to the observed migration of seismicity.

Finally, the paper treats all external triggers—tidal forces, atmospheric pressure changes, anthropogenic activities—as merely different initial conditions of the same nonlinear system, suggesting that they do not constitute independent causes but rather perturb the system’s trajectory within its chaotic basin.

Critical assessment

Physical realism: The reduction of the full Navier‑Stokes equations to a single scalar momentum equation ignores the layered, anisotropic, and visco‑elastic nature of the crust and mantle. Real fault zones involve complex interactions among multiple rock types, fluids, temperature gradients, and chemical processes that cannot be captured by a one‑dimensional flow model.

Mathematical rigor: The “simplified nonlinear solution” is not presented in closed form, nor are the boundary conditions, material parameters, or nondimensional groups specified. Consequently, the claimed threshold behavior lacks reproducibility. Moreover, the transition from a chaotic solution of the fluid equations to a deterministic M‑P scaling is not mathematically justified; the two regimes are treated as if they were seamlessly connected.

Model coupling: The Carlson‑Langer model describes a single block‑spring system, while the Gutenberg‑Richter law is a statistical description of a whole seismic catalog. The paper does not demonstrate how the parameters of the block‑spring system (e.g., stiffness, friction coefficient) map onto the Gutenberg‑Richter (b)-value or the constants (k) and (\alpha) in the M‑P formula. This ad‑hoc coupling undermines the physical meaning of the derived relationship.

Empirical validation: The forecasts cited (2004, 2009, 2014, 2019) are presented without any statistical test of significance. They appear to be post‑hoc matches rather than out‑of‑sample predictions. A robust validation would require a blind forecast, a defined confidence interval, and a comparison against a null hypothesis (e.g., a Poisson or time‑dependent Poisson model).

Lorenz analogy: Re‑assigning Lorenz variables to seismic quantities is conceptually interesting but lacks a mechanistic derivation. The Lorenz equations arise from a truncated set of partial differential equations governing buoyancy‑driven convection; there is no clear physical pathway that reduces fault‑zone dynamics to the same three‑dimensional ordinary differential system.

External triggers: While it is true that initial conditions strongly influence chaotic trajectories, treating all external influences as merely different initial states neglects the possibility of sustained forcing (e.g., seasonal fluid injection, long‑term tectonic loading) that can alter the system’s attractor structure.

Conclusion and outlook

The paper offers a bold, interdisciplinary vision that seeks to place earthquakes within the broader context of nonlinear dynamical systems. However, the current formulation suffers from several critical shortcomings:

1. Oversimplified physics – the fluid‑rock interaction is reduced to an analytically tractable but unrealistic model.
2. Unclear derivations – key equations (the chaotic momentum solution, the M‑P scaling) are not derived rigorously.
3. Lack of quantitative validation – forecasts are not tested against independent data, and uncertainties are not quantified.
4. Speculative analogies – the use of the Lorenz attractor and the reinterpretation of external triggers are not grounded in geophysical evidence.

Future work should focus on integrating high‑resolution geodetic and seismic observations, employing full 3‑D numerical simulations of coupled fluid‑solid dynamics, and establishing statistically robust forecasting protocols. Only with such rigorous testing can the appealing idea that earthquakes are manifestations of deterministic chaos be elevated from a conceptual metaphor to a predictive scientific tool.