Exploring Parameter Space

So far we have concentrated on the particular parameter values 7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. 

There is a vast three-dimensional parameter space to be explored, and much remains to be discovered. To simplify matters, many investigators have kept 7 = 10 and b = y while varying r. In this section we give a glimpse of some of the phenomena observed in numerical experiments. See Sparrow (1982) for the definitive treatment. 

Much of this picture is familiar. The origin is globally stable for r 1. At r = 1 the origin loses stability by a supercritical pitchfork bifurcation, and a symmetric pair 

At = 24.74 the fixed points lose stability by absorbing an unstable limit cycle in 

Now for the new results. As we decrease r from, the unstable limit cycles expand and pass precariously close to the saddle point at the origin. At r 13.926 the cycles touch the saddle point and become homoclinic orbits hence we have a homoclinic bifurcation. (See Section 8.4 for the much simpler homoclinic bifurcations that occur in two-dimensional systems.) Below r= 13.926 there are no limit cycles. Viewed in the other direction, we could say that a pair of unstable limit cycles are created as r increases through r = 13.926. 

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A set of compounds substituted in the 5 -position was designed to explore the physiochemical parameter space represented by 7C, F, R and MR. This initial compound set was restricted to cover these minimum parameters in the interest of efficiency. Later, if the level of activity warranted it, the initial set could be expanded to more fully explore parameter space. In order to adequately cover the chosen parameter space, a 2 factorial design was utilized. ( 5) The 16 required compounds were selected via cluster analysis from our substituent physical-chemical database. ( ) Marker points which represented the factorial design were included in the data set prior to clustering. (2) In this way substituents which best represented the factorial design were those that were... 

A full discussion of the model results obtained by exploring the whole parameter space will be presented elsewhere (Romano, Tosi Matteucci, in preparation). Here we want to address a few hot questions and focus on a couple of points. 

Normally it is hardly feasible nor necessary to explore a certain parameter space to the full theoretically possible extent. Rather certain areas of parameter space are examined. These areas can be determined by intuition, synthetic accessibility or by structure-activity-relationships and structure-activity-hypotheses. Consider for example structure 6 to, be a lead compound which needs to be optimized with respect to an activity profile consisting of two components, i.e. potency and oral effectivity (gastro-intestinal absorption). Let us for simplicity assume that we want to confine ourselves to variations of the substituent in 4-position (7) and that we... 

We will use the model to explore the complex static/dynamic bifurcation behavior of this system in the two-dimensional D — Cso parameter space and show the implications of bifurcation phenomena on substrate conversion and ethanol yield and productivity. The system parameters for the specific fermentation unit under consideration are given below. 

A closer exploration of the parameter space showed that symmetry breaking is sensitive to the thermodynamic stabilities of the dimers but also to the rate of their formation, predicting that the heterochiral dimers are formed faster and are more stable than the homochiral ones [69]. The fitting of the experimentally obtained data (Fig. 5) resulted in rate parameters that confirm these conclusions and that simultaneously give rise to mirror-symmetry breaking. 

The research of Roy Jackson combines theory and experiment in a distinctive fashion. First, the theory incorporates, in a simple manner, inertial collisions through relations based on kinetic theory, contact friction via the classical treatment of Coulomb, and, in some cases, momentum exchange with the gas. The critical feature is a conservation equation for the pseudo-thermal temperature, the microscopic variable characterizing the state of the particle phase. Second, each of the basic flows relevant to processes or laboratory tests, such as plane shear, chutes, standpipes, hoppers, and transport lines, is addressed and the flow regimes and multiple steady states arising from the nonlinearities (Fig. 6) are explored in detail. Third, the experiments are scaled to explore appropriate ranges of parameter space and observe the multiple steady states (Fig. 7). One of the more striking results is the... 

Gough, W.A. and Welch, W.J. (1994). Parameter space exploration of an ocean general circulation model using an isopycnal mixing parameterization. Journal of Marine Research, 52, 773-796. 

From numerous results achieved using combinatorial and high-throughput methods, the most successful have been in the areas of molecular imprinting, polymeric compositions, catalytic metals for field-effect devices, and metal oxides for conductometric sensors. In those materials, the desired selectivity and sensitivity have been achieved by the exploration of multidimensional chemical composition and process parameters space at a previously unavailable level of detail at a fraction of time required for conventional one-at-a-time experiments. These new tools provided the opportunity for the more challenging, yet more rewarding explorations that previously were too time consuming to pursue. 

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