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I have a vector of sample data that describes a divergent oscillation like in this image.

Divergent Oscillation

I want to transform this data (just the data, not the system or anything), so that the data does converge to zero. I also want to keep the peaks of data at the same sample point, and the graph smooth. Simply multiplying the data by decreasing values moves the peaks of the data which is not desirable.

What sort of method should I use?

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    This is why I wrote "I believe you can **force** it to converge to zero". If your function goes up and down, and you scale the derivatives, then every time it goes up a little bit slower, it goes down slower even more--you add a constant term. If it was the same speed for both up and down, then it would stay at zero, but now they differ. Not too much, but still. It's like doing three steps forward, but only two back. How to fix it: you can't add something that changes(that would move your peaks), but you can add a constant. There surely are some adaptive solutions, but I have no time for this.2012-03-18

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I did not get it. Do you have a function that generates this graphic? If so, you can multiply for a positive real function (say, $e^{-\alpha x}$, where you can adjust the value of $alpha >0$ for faster zero convergence). That should do if I got your question right.

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    Wouldn't that change the position of peaks? For example $\sin' x = \cos x$, but $\left(e^{-\alpha x}\sin x\right)' = e^{-\alpha x} (-\alpha\sin x + \cos x)$ will have zeros in different places--the local extremes will move.2012-03-17
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If your function is $f(x)$, try $g(x) = -\int_x^\infty f'(t) h(t)\ dt$ where $h$ is some function that is always positive but decreases rapidly enough that $\int_0^\infty |f'(t)| h(t)\ dt < \infty$.

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I assume you want your data to look like the graph on the left of the $Y$-axis, where $x <= 0$

enter image description here

To do that, simply multiply each $x$ value by $-1$.

FYI: The graph below is not your your data but looks somewhat close, it is the graph of $xsin(0.025x)$.