Real-world systems tend to diverge from the model near singularities. Ceramic supercapacitor investors can tell you about this.
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One simple model, where growth is constant plus square of current value (f'(t) = 1 + f(t)^2), gives growth like the tangent function.
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anything exponential & sustained will get you there, then? difficulty is working out if you're on a sigmoid or exponential curve.
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