de-CH
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Stationary Solutions: Convergence
dgl-05-02
multiple
1000
randRangeExclude(-8, 8, [-1,0,1] ) randRangeExclude(-8, 8, [-1,0,1] ) randRange(2, 5)
randRange(2, 9) randRange(2, C) randRangeExclude(A, D, [A, B, D] )

Let us be given the ODE y'= (Cy- D*C) (Fy^2 - (A+B)*F y + A*B*F) with initial value y_0 = Y and solution t \mapsto y(t).

Determine \displaystyle \color{teal}\lim_{t \to \infty} y(t).

x \displaystyle \color{teal}\lim_{t \to \infty} y(t) = B

Stationary solutions {\color{orange}y_{\infty}} with {\color{orange}y'_{\infty}} = 0 are candidates for asymptotes.

To find stationary solutions, we search for zeros of the right-hand side of the ODE (Cy- D*C) (Fy^2 - (A+B)*F y + A*B*F).

This simplifies to C*F(y- D) (y - A) (y - B), i.e. {\color{orange}y_{\infty}} \in \{ A, B, D\}.

For convergence, we consider the graph of the function F with F(y)=C*F(y- D) (y - A) (y - B).

Its qualitative behavior can be understood from the following graph.

graphInit({ range: [[ -10, 10 ] ,[ -17.5, 17.5 ]], scale: [ 10, 10 ], gridStep: [ 2, 2 ], tickStep: [ 10, 10 ], labelStep: [ 10, 10 ], unityLabels: true, }); // draw curve style({ stroke: BLUE, strokeWidth: 2.2 }, function() { plot( function( x ) { return 0.1 * (x - A) * (x - B) * (x - D) ; }, [ A-1, D+1 ] ); }); style({ fill: BLACK, stroke: BLACK }); circle( [A, 0], 0.25 ); style({ fill: PURPLE, stroke: RED }); circle( [B, 0], 0.25 ); style({ fill: ORANGE, stroke: ORANGE }); circle( [D, 0], 0.25 );

In the graph we also see the initial value {\color{blue}y_0= Y}.

style({ fill: BLUE, stroke: BLUE }); circle( [Y, 0], 0.25 );

Since F(y_0) < 0 and because y'(t_0) = F(y_0) < 0 , the solution function t \mapsto y(t) is at the initial value {\color{blue}y_0= Y} strictly monotonically decreasing. In fact, it is monotonically decreasing as long as F(y) < 0. Therefore, we must have \displaystyle \color{teal}\lim_{t \to \infty} y(t) = B.

Since F(y_0) > 0 and because y'(t_0) = F(y_0) > 0 , the solution function t \mapsto y(t) is at the initial value {\color{blue}y_0= Y} strictly monotonically increasing. In fact, it is monotonically increasing as long as F(y) > 0. Therefore, we must have \displaystyle \color{teal}\lim_{t \to \infty} y(t) = B.