A structured model of bioreactor for an activated sludge process was presented. The stability and bifurcation characteristics of the model are investigated, the bifurcation analysis of the model shows static and complex dynamic behavior (periodic and complex) over a wide range of the model parameters. The model exhibits a new interesting behavior (in some range of parameters) including four static limit points (turning points) and two Hopf points, that cause different kinds and rich of stability characteristics ranging from asymptotically stable and hysteresis to periodic and complex behavior

In this paper, we study the bifurcation of a cancer model with completely unknown parameters. The bifurcation analysis of the biologically feasible steady-states of this model will be discussed. It is proved that the system appears to exhibit many cases of bifurcation for some ranges of system parameters. Numerical analysis and extensive numerical examples of the bifurcation for some ranges were carried out for various system parameter values and different initial densities.

Optimal control is one of the most popular decision-making tools recently in many researches and in many areas. The Lorenz-Rössler model is one of the interesting models because of the idea of consolidation of the two models: Lorenz and Rössler. This paper discusses the Lorenz-Rössler model from the bifurcation phenomena and the optimal control problem (OCP). The bifurcation property at the system equilibrium ( )found that saddle-node and Hopf bifurcations can be holed under some conditions on the parameters. Also, the problem of the optimal control of Lorenz-Rössler model is discussed and it uses the Pontryagin's Maximum Principle (PMP) to derive the optimal control inputs that achieve the optimal trajectory. Numerical examples and solutions for bifurcation cases and the optimal controlled system are carried out and shown graphically to show the effectiveness of the used procedure.

A dynamical model of an activated sludge process is considered. A mathematical analysis of the model equations with general rate functions is given. The dissipativity, boundedness, invariance of non-negativity, persistence, stability, bifurcation and periodicity of solutions are discussed. It was also shown that the rate functions have no influence for stability in the special case disinfected feed.