We will prove this by constructing a polygonal line with no contact with the flow of the differential equation.
The polygonal line will join the origin to one of the saddle-nodes as in Figure 4.
We explain the construction of the polygonal line when 0 < [s.
As an example of the construction of the polygonal line in the proof of Theorem 1 we present a particular case, done with Maple.
The polygonal line [GAMMA] is determined by 10 points regularly spaced on the circle of center 0 and radius 1.
When the matrix A is real and, assuming that the polygonal line [GAMMA] is symmetric with respect to the real axis and intersects it only in two points, half of the computation can be saved since
In this paper, we have developed a reliable method for counting the eigenvalues in a region surrounded by a user-defined polygonal line.
4; n : order of the matrix; [GAMMA] : definition of the polygonal line (vertices are regularly distributed on a circle C[center,radius], or on an ellipse E[center,horizontal semi-axis,vertical semi-axis], or defined by a rectangle); Neig : number of surrounded eigenvalues; Time: elapsed time (s); Nintv: number of intervals.