Consider a nonadiabatic tubular reactor with negligible mass and heat dispersions and recycle where a first-order exothermic reaction takes place . This system is governed by the following two dimensionless partial differential equations and boundary conditions BC1 and BC2:
Here is concentration and is temperature, is the activation energy, is the heat evolution parameter, is the dimensionless cooling temperature, is the dimensionless cooling parameter, is the recycle ratio, is the Damköhler number, and is the Lewis number.
As can be seen from snapshot 1, periodic solutions are obtained for and . In the figure, and are plotted versus in blue and green, respectively. Also, you can see from snapshot 1 that a limit cycle is obtained for the above values of the parameters.
 S. Subramanian and V. Balakotaiah, "Classification of Steady-State and Dynamic Behavior of Distributed Reactor Models," Chemical Engineering Science, 51(3), 1996 pp. 401–421. doi:10.1016/0009-2509(95)00261-8.