Parameter-dependent transitions and the optimal control of dynamical diseases

Many theoretical studies in biological and physical sciences consider the dynamical behavior of an n-dimensional ordinary differential equation that contains a large number of independent parameters. A frequently asked question is, are there permissible parameter sets that result in periodic or chaotic behavior? The large number of distinct parameters often limits the feasibility of trial and error calculations. The large dimension and nonlinearity of the system make application of analytic methods at best difficult and at worst effectively impossible. It is shown here that a computational search for parameter-dependent transitions of attractor topology can be effected by constrained optimization of quantitative measures of dynamical behavior (Hurwitz polynomials, Floquet coefficients, Lyapunov exponents and correlation dimension). As an example, we examine a three-dimensional nonlinear ordinary differential equation containing seven parameters that was constructed by Goldbeter and Segel to model periodic synthesis of cyclic AMP in Dictyostelium. A search for bifurcations to periodic solutions is made by minimizing Hurwitz coefficients subject to parameter constraints. By comparing four optimization algorithms, the defects and advantages of the procedure are identified. It is also argued that it may be possible to use this characterization of dynamics to construct optimal responses to dynamical diseases (those disorders that result from parameter-dependent bifurcations in physiological control systems).