de-CH
utf-8
math math-format graphie
With Direction Field: Convergence
dgl-09-04
radio
1053
randRange(1,2) randRange(-4,4) randRange(2,4) randRange(2,4) A+T A-D randRange(-6,6) randRangeExclude(C-1,B+1,[A,B,C])

Below we see the slope field of an ODE y' = F(y) with three stationary solutions.

Which convergence behavior does the solution \color{orange}y with \color{orange}y(X) = Y exhibit?

style({ stroke: "black", strokeWidth: 0.05 }); graphInit({ range: [[-8, 8], [C-3,B+3]], scale: [22, 22], axisArrows: "->", tickStep: 10, }); label([0,A], "\\color{red}A", "left"); label([0,B], "\\color{blue}B", "left"); label([0,C], "\\color{black}C", "left"); // Vektorfeld for(var i = -8; i <= 8; i+=1) { for(var j = C-2; j <= B+2; j+=.75) { var dy = pow(-1,P)*(j - A)*( j - B)*( j - C)/45; line([i - 0.25, j - dy*.25], [i + 0.25, j + dy*0.25], { arrows: "", strokeWidth: 1, stroke: "gray" }); } }

\lim\limits_{t \to + \infty} = + \infty \lim\limits_{t \to + \infty} = - \infty \lim\limits_{t \to + \infty} = A \lim\limits_{t \to + \infty} = C \lim\limits_{t \to + \infty} = B

At a given point (x_0,y_0), the slope field shows a tiny piece of the tangent of a solution of the ODE, on which (x_0,y_0) lies.

We observe that y_{\infty, 1}=C, y_{\infty, 2}=A, y_{\infty, 3}=B are the three stationary solutions.

for(var i = -8; i <= 8; i+=1) { line([i - 0.15, C ], [i + 0.15, C], { arrows: "", strokeWidth: 2.75, stroke: "orange" }); line([i - 0.15, B ], [i + 0.15, B], { arrows: "", strokeWidth: 2.75, stroke: "orange" }); line([i - 0.15, A ], [i + 0.15, A], { arrows: "", strokeWidth: 2.75, stroke: "orange" }); } label( [X, Y], "\\color{teal} (X, Y)", "below" ); circle( [X, Y], 3 / 15, { fill: GREEN, stroke: "none" });

Those provide candidates for the limit \lim\limits_{t \to + \infty}.

Since {\color{teal}y(X) = Y} > B, each tangent piece has positive slope, i.e.

\lim\limits_{t \to + \infty} = + \infty.

Since {\color{teal}y(X) = Y} < C, each tangent piece has negative slope, i.e.

\lim\limits_{t \to + \infty} = + \infty.

The points lie in the strip C < {\color{teal}y(X) = Y} < B.

The tangents have a negative slope for {\color{teal}y(t)} > A and a positive slope for {\color{teal}y(t)} < A .

Therefore y_{\infty, 2}=A attracts the solution and \lim\limits_{t \to + \infty} = A.

The tangent pieces have a positive slope for {\color{teal}y(t)} > A and a negative slope for {\color{teal}y(t)} < A .

Therefore y_{\infty, 2}=A repels the solution and it is not a limit.

The tangent pieces have a negative slope for {\color{teal}y(t)} > B and a positive slope for {\color{teal}y(t)} < B .

Therefore y_{\infty, 1}=B attracts the solution and \lim\limits_{t \to + \infty} = B.

The tangent pieces have a negative slope for {\color{teal}y(t)} > C and a positive slope for {\color{teal}y(t)} < C .

Therefore y_{\infty, 3}=C attracts the solution and \lim\limits_{t \to + \infty} = C.