Different roads to chaos in chemical reactors.

*(English)*Zbl 0788.92027
Lions, Jacques-Louis (ed.) et al., Boundary value problems for partial differential equations and applications. Dedicated to Enrico Magenes on the occasion of his 70th birthday. Paris: Masson. Res. Notes Appl. Math. 29, 279-286 (1993).

Modelling chemical reactors requires, in general, the use of highly nonlinear equations: the nonlinearity comes from the generation term that can present different forms according to the specific reacting systems. For this reason they are, typically, the most studied systems in the field of chemical engineering for detecting unusual and pathological behaviours.

We try in this paper to outline the main lines of the path we have followed, starting from simple systems described by ordinary differential equations to arrive at more complex reactor models described by partial differential equations. Analyzing reactor dynamical behaviours, we focused the attention on those particular conditions that lead to chaos and on the different roads followed by the system to reach the chaos itself. It has to be pointed out that we have detected chaotic situations only for closed loop reactors, i.e., when a proportional-integral (PI) feedback controller is applied: absurdly, it is just the integral controller, whose action should stabilize the system at the set point, to induce chaos occurrence. It is important to outline that the generic Arrhenius type reaction (exponential term) we have considered is assumed to be exothermic, i.e. the reaction takes place with heat generation, that, if uncontrolled, can give rise to substantial instability of the system.

For the entire collection see [Zbl 0782.00097].

We try in this paper to outline the main lines of the path we have followed, starting from simple systems described by ordinary differential equations to arrive at more complex reactor models described by partial differential equations. Analyzing reactor dynamical behaviours, we focused the attention on those particular conditions that lead to chaos and on the different roads followed by the system to reach the chaos itself. It has to be pointed out that we have detected chaotic situations only for closed loop reactors, i.e., when a proportional-integral (PI) feedback controller is applied: absurdly, it is just the integral controller, whose action should stabilize the system at the set point, to induce chaos occurrence. It is important to outline that the generic Arrhenius type reaction (exponential term) we have considered is assumed to be exothermic, i.e. the reaction takes place with heat generation, that, if uncontrolled, can give rise to substantial instability of the system.

For the entire collection see [Zbl 0782.00097].

##### MSC:

92E20 | Classical flows, reactions, etc. in chemistry |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

80A30 | Chemical kinetics in thermodynamics and heat transfer |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37N99 | Applications of dynamical systems |