In this Letter we introduce a generalization of the Lorenz dynamical system using fractional derivatives. Thus, the system can have an effective noninteger dimension $\Sigma$ defined as a sum of the orders of all involved derivatives. We found that the system with $\Sigma <3$ can exhibit chaotic behavior. A striking finding is that there is a critical value of the effective dimension ${\Sigma }_{\mathrm{c}\mathrm{r}}$, under which the system undergoes a transition from chaotic dynamics to regular one.

• Received 8 April 2003

DOI:https://doi.org/10.1103/PhysRevLett.91.034101