Periodic and chaotic solutions for a model of a bioreactor with cell recycle

A Dynamic Analysis for an Anaerobic Digester: Stability and Bifurcation Branches

“…Notice that this value is independent of in 1 , in 2 , and is the same as the upper bound appearing in (15). Combining (69), (70), (3)-(6), and (59) yields the coordinates of bifurcation FB( 2 ):…”

“…As 0 < eq 2 < /√ 2 according to (16), 2 > 0 if 0 < eq 2 < /√ 2 , < ( 2 max / )( /( + 2√ 2 )) according to (15), and eq 2 > 0 if conditions given by (24) and (25) are satisfied; then, it follows from (42) and (43) that Re( 3 ) < 0 and Re( 4 ) < 0.…”

“…but only the first possibility satisfies condition (15). The graphical analysis of the bifurcations, and the fact that the parameter value (61) leads to a fold bifurcation, will be used to determine the bifurcation type.…”

“…The dependence of desired or undesired operation of biological systems on model parameters is usually analyzed by means of the indirect Lyapunov method (see [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). This method is based on the linearization of the vector field and involves the definition of the following factors: (i) equilibrium points, (ii) parameter constraints that lead to expected intervals of the equilibrium state variables, (iii) Jacobian matrix, eigenvalues, and stability of each equilibrium, (iv) coordinates of local bifurcation points, and (v) parameter regions that lead to either desired or undesired system operation.…”