Nonlinear multi-variable systems

In this final section we no longer distinguish between boxes and variables. We just consider the total number of system variables. For instance, a system with four variables (that is, a four-dimensional system) can either describe four chemical species in one box, two chemical species in two boxes, or one species in four boxes. Only linear systems are discussed multi-dimensional nonlinear systems can be extremely complex and do not allow for a short and concise systematic discussion. 

An optimization problem is a mathematical model which in addition to the aforementioned elements contains one or multiple performance criteria. The performance criterion is denoted as objective function, and it can be the minimization of cost, the maximization of profit or yield of a process for instance. If we have multiple performance criteria then the problem is classified as multi-objective optimization problem. A well defined optimization problem features a number of variables greater than the number of equality constraints, which implies that there exist degrees of freedom upon which we optimize. If the number of variables equals the number of equality constraints, then the optimization problem reduces to a solution of nonlinear systems of equations with additional inequality constraints. 

A simple two-variable theoretical model is proposed. It exhibits most of the temporal dynamical behaviours reported for nonlinear chemical systems [1] oscillations, excitability, multi stability. This autocatalytic model, part of our nonlinear model of calcium metabolism [2], has been associated with bone calcification processes (nucleation and crystal growth). It is described by the differential system ... 

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