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I've been trying to analyse a system consisting of a ball bouncing on a moving table. The system is expected to show chaotic behaviour. To find the time of a particular bounce, I have to solve an equation of the kind:

$$ d(t)=A[1+\sin(\omega t_k + \delta)]+v_k(t-t_k)-\frac{1}{2}g(t-t_k)^2-A[1+\sin(\omega t+\delta)]=0$$

I have to solve this for $t$ given all the other variables. The problem is that I am facing difficulty in doing this numerically. At present I am incrementing $t$ by a fixed step, and seeing if


and taking this to be the solution. But it seems that this is not a good method, as a slight variation in $\epsilon$ seem to affect the solution. So is there a better method which I can use to solve this kind of transcendental equations? Also, $d(t)\geq0$ by the way the parameters are defined.

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You have a one-dimensional root finding problem, as you are looking for $t$ where $d=0$. This is a common subject in numerical analysis. My favorite is chapter 9 of Numerical Recipes. Obsolete versions are free on line.

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