Synchronization, critical points, and relaxation oscillators

In Chapter 18 of the book, we discuss the behavior around critical points in the context of reliability, both at the small-scale in terms of component breakdown, and in the large-scale in the context of earthquake triggering which was introduced in Chapter 13. The connection is that things break at all scales, with the common mechanism of a varying rate of progression to the critical point:


As indicated in the figure caption, the failure rate is generally probabilistic but with known external forcings, there is the potential for a better deterministic prediction of the breakdown point, which is reviewed below:

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Forced Natural Responses to LTE Solution

In Chapter 12 of the book, we describe in detail the solution to Laplace’s Tidal Equations (LTE), which were introduced in Chapter 11.  Like the solution to the linear wave equation, where there are even (cosine) and odd (sine) natural responses, there are also  even and odd responses for nonlinear wave equations such as the Mathieu equation, where the natural response solutions are identified as MathieuC and MathieuS.  So we find that in general the mix of even and odd solutions for any modeled problem is governed by the initial conditions of the behavior along with any continuing forcing. We will describe how that applies to the LTE system next:

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Lunisolar Forcing of the Chandler Wobble

In Chapter 13 of the book, we have a description of the mechanism forcing the Chandler Wobble in the Earth’s rotation. Even though there is not yet a research consensus on the mechanism, the prescribed lunisolar forcing seemed plausible enough that we included a detailed analysis in the text.  Recently we have found a recent reference to a supporting argument to our conjecture, which is presented below …

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