Global analysis of steady points for systems of differential equations with sigmoid interactions

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Abstract
A new method to investigate asymptotic properties of linear differential equations with strong threshold and switching effects is presented. The method is applied to systems of equations of the form dx/dt = F(x) - yx, where y = constant and the dependence of F on x is mediated by sigmoid functions. Using a special sigmoid function called a logoid, which rises monotonically from zero to one in a narrow interval surrounding the threshold value, exact analytical expressions for the limiting value of all steady points can be found in the limit when the logoid approaches a step function. The limiting values are independent of the shape of the logoid for a large class of logoids. Relations between steady points and limit cycles of the equations with logoids, their step function limit and the corresponding piecewise linear equations are derived. It is found that the approximation of sigmoids by the step function idealization is not always warranted. The results strongly suggest the use of logoids instead of other sigmoids hitherto employed

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