# Does Lyapunov exponent equate to exponential inflation?

Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors and attractors are simple exponential expansion or contraction and are represented by a Lyapunov exponent. My question is how could one prove or disprove that exponential inflation is just an example of a repellor with a Lyapunov exponent?

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Do you have something particular in mind for exponential inflation? – Wrzlprmft Apr 2 '14 at 7:17
I can't say I fully understand this question, but there are several papers which use a Lyapunov exponent to describe inflation: ms.mcmaster.ca/craig/BrandenbergerCraig08c.pdf and sbfisica.org.br/bjp/files/v31_131.pdf – DavePhD Apr 8 '14 at 16:30
you might want to change your title, as 'equating' is not the same as 'is just an example of'. – chris Apr 9 '14 at 8:35
@DavePhD your reply qualifies as an answer as the citations are a useful starting place for me. – Daniel Geisler Apr 9 '14 at 9:38
@DavePhD it is worth noting that neither of the papers you cited describe inflation with a Lyapunov exponent. They use the exponent to describe the exponential growth in the equations of motion of low-density matter and they both explicitly state that this description is done in a non-expanding universe – Jim Apr 9 '14 at 19:05

If you think that the whole universe as a single state in state space, then there is no way you can compare with any other state. You may think that there might be a phase space with (macroscopic) parameters $(V,\{\alpha_i\})$, so we can have a maximum Lyapunov exponents $\xi$ and a vector $\delta Z$ deviated from a state $Z_0$. The resulting state might be described by $Z_0+e^{\xi t}\delta Z$.
However, why do you need a phase space description? The other phase space states do not matter to us at all since we can't observe their dynamic. An exponential inflating universe can be simply described by $\dot{V}=kV$. In comparison, most dynamic system can actually be reproduced by repeating the experiments and this will give us the dynamic of the whole phase space.