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We present a mechanism for Arnold diffusion in energy in a model of the elliptic Hill four-body problem. Our model is expressed as a small perturbation of the circular Hill four-body problem, with the small parameter being the eccentricity of the orbits of the primaries. The mechanism relies on the existence of two normally hyperbolic invariant manifolds (NHIM's), and on the corresponding homoclinic and heteroclinic connections. The dynamics along homoclinic/heteroclinic orbits is encoded via scattering maps, which we compute numerically. Having several scattering maps, at each point we select the scattering map that gives the largest gain in energy or the scattering map that gives the smallest loss in energy. Using Birkhoff's Ergodic Theorem we show that there are pseudo-orbits generated by the selected scattering maps along which, on average, the energy grows by an amount independent of the small parameter. A shadowing lemma yields the existence of diffusing orbits. 

We present a heuristic argument for the propensity of Topological Data Analysis (TDA) to detect early warning signals of critical transitions in financial time series. Our argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) of an asset price superimposed with oscillations increasing in frequency and decreasing in amplitude when approaching a critical transition (tipping point). We show that whenever the LPPLS model is fitting with the data, TDA generates early warning signals. As an application, we illustrate this approach on a sample of positive and negative bubbles in the Bitcoin historical price.