# Periodic sequence with exponentially increasing period?

I have to develop a physical model for a certain type of biological oscillation that can be built upon periodic sequences.

From earlier questions I know that any periodic sequence (containing $0$s and $1$s) can be developed based on the form of such a Discrete Inverse Fourier:

$$\psi(n)=\sum_{k=1}^T\frac{1}{T} \cos \left( \frac{-2 \pi(k-1)n}{T} \right) \qquad; \;T,k,n \in \Bbb N \qquad(1)$$

Also I got an elegant answer yesterday how to describe an oscillation which period exponentially increases along the abscissa by applying $\log x$ (in the example below visible on the zeros):

$$\varphi(x)=\sin(\pi\log_T x) \qquad;T \in \Bbb N \;;x \in \Bbb R \qquad (2)$$

What I am trying to do is to modify the equation $(1)$ in a way that it would describe a periodic sequence similar $\psi(x)$ (containing $0$s and $1$s) which period of occurance of $1$ (that is $T$) increases exponentially. That would help me to describe the behaviour of my biological oscillator.

What would such a function similar $\psi(x)$ be?

@Nathaniel the case is far more primitive than approaching by heteroclinic cycles. However, I will keep your wise link in mind, thanks. By the way, the question to equation $1$ was solved by physicists at the end and none of the addressed math forums could make it. That is I brought this question to here and not math.STE. Also equation $2$ was a great solution out of this forum. – al-Hwarizmi Jun 24 '13 at 10:36
Just work out the periodic function with the required shape, then in the resulting function replace $t$ by log($t$). – John Rennie Jun 24 '13 at 11:51
@JohnRennie thanks. would you help with details what you exactly mind. Indeed I substituted in $(1)$ before submitting this question $n$ (guess your $t$) with $\log (n)$ (guess your $\log (t)$). However this is exactly what does not work, you will not get a precise sequence as is for instance $\psi$. Would be glad to know your specific answer. – al-Hwarizmi Jun 24 '13 at 12:16