Davidenko differential equation

If Hx x, s) is regular we get to the so-called Davidenko differential equation [Dav53] ... 

For the global homotopy the Davidenko differential equation reads... 

The direction x s) given by the right hand side of the Davidenko differential equation is just the tangent at the solution curve x in s. 

Having just seen that if no turning points are present, the path x(s) is just the solution of an ordinary differential equation, the Davidenko differential equation, we could conclude this chapter by referring to numerical methods for ODEs. Unfortunately, this differential equation cannot be solved in a stable way. Small errors are not damped out. This differential equation belongs to the class of differential equations with invariants as it has H x s), s) = 0 as a first integral. This class will... 

This is just a step of the explicit Euler method for ODEs applied to the Davidenko differential equation, cf. Sec. 4.1.1. 

The system of differential equations (E3.13) can be reduced to a single differential equation for the determinant of J h ) through the application of Davidenko s (1960) trace theorem let (s) be an n x matrix whose elements are differentiable functions in s. Let (s) be the characteristic polynomial associated with (s). Then, for all s for which (s) 0,... 

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See also in sourсe #XX -- [ ]

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