ЗАКЛЮЧЕНИЕ.. СПИСОК ИСПОЛЬЗОВАННОЙ ЛИТЕРАТУРЫ.. ПРИЛОЖЕНИЕ.

ЗАКЛЮЧЕНИЕ.

В данной курсовой работе была рассмотрена тема «Элементы качественной теории дифференциальных уравнений и теории колебаний», а именно решение практических заданий по этой теме.

Благодаря написанию данной работы мной были получены более углубленные знания по данной теме работы, а также были получены практические навыки применения методов качественного исследования нелинейных дифференциальных уравнений и систем уравнений.

СПИСОК ИСПОЛЬЗОВАННОЙ ЛИТЕРАТУРЫ.

1. Буркин И.М. Обыкновенные дифференциальные уравнения. Методы интегрирования. Теория устойчивости. Теория колебаний: учеб.пособие/ И.М.Буркин. – ТулГУ, 2004. – 207 с.

2. Филиппов А.Ф. Сборник задач по дифференциальным уравнениям. – Москва: Наука, 1979

ПРИЛОЖЕНИЕ.

Задача 1.

Для точки M₁:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)^2-x(t)^2, diff(y(t), t) = ln((1-x(t)+x(t)^2)*(1/3))], [x(t), y(t)], t = -20 .. 20, [[x(0) = -1.03, y(0) = -1.03], [x(0) = -1.1, y(0) = -1.1], [x(0) = -.9, y(0) = -.9], [x(0) = -1.1, y(0) = -.9], [x(0) = -.9, y(0) = -1.1], [x(0) = -.95, y(0) = -.95], [x(0) = -.95, y(0) = -1.05], [x(0) = -1.05, y(0) = -.95], [x(0) = -.99, y(0) = -.99], [x(0) = -.99, y(0) = -1.01], [x(0) = -1.01, y(0) = -.99], [x(0) = -1.01, y(0) = -1.01]], x = -1.2 .. -.8, y = -1.2 .. -.8, stepsize = .1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Для точки M₂:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)^2-x(t)^2, diff(y(t), t) = ln((1-x(t)+x(t)^2)*(1/3))], [x(t), y(t)], t = -20 .. 20, [[x(0) = -1.03, y(0) = 1.03], [x(0) = -1.1, y(0) = 1.1], [x(0) = -.9, y(0) = .9], [x(0) = -1.1, y(0) = .9], [x(0) = -.9, y(0) = 1.1], [x(0) = -.95, y(0) = .95], [x(0) = -.95, y(0) = 1.05], [x(0) = -1.05, y(0) = .95], [x(0) = -.99, y(0) = .99], [x(0) = -.99, y(0) = 1.01], [x(0) = -1.01, y(0) = .99], [x(0) = -1.01, y(0) = 1.01]], x = -1.2 .. -.8, y = .8 .. 1.2, stepsize = .1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Для точки M₃:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)^2-x(t)^2, diff(y(t), t) = ln((1-x(t)+x(t)^2)*(1/3))], [x(t), y(t)], t = -20 .. 20, [[x(0) = 1.93, y(0) = 1.93], [x(0) = 2.1, y(0) = 2.1], [x(0) = 1.9, y(0) = 1.9], [x(0) = 2.1, y(0) = 1.9], [x(0) = 1.9, y(0) = 2.1], [x(0) = 1.95, y(0) = 1.95], [x(0) = 1.95, y(0) = 2.05], [x(0) = 2.05, y(0) = 1.95], [x(0) = 1.99, y(0) = 1.99], [x(0) = 1.99, y(0) = 2.01], [x(0) = 2.01, y(0) = 1.99], [x(0) = 2.01, y(0) = 2.01]], x = 1.8 .. 2.2, y = 1.8 .. 2.2, stepsize = .1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3);

Фазовый портрет системы:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)^2-x(t)^2, diff(y(t), t) = ln((1-x(t)+x(t)^2)*(1/3))], [x(t), y(t)], t = -20 .. 20, [[x(0) = 1.7, y(0) = 1.8], [x(0) = 2.4, y(0) = 2.4], [x(0) = 1.5, y(0) = 1.5], [x(0) = 2.4, y(0) = 1.5], [x(0) = 1.5, y(0) = 2.4], [x(0) = 1.7, y(0) = 1.7], [x(0) = 1.7, y(0) = 2.2], [x(0) = 2.2, y(0) = 1.8], [x(0) = 1.9, y(0) = 1.9], [x(0) = 1.9, y(0) = 2.1], [x(0) = 2.4, y(0) = 1.9], [x(0) = 2.1, y(0) = 2.1], [x(0) = -1.3, y(0) = 1.3], [x(0) = -1.5, y(0) = 1.5], [x(0) = -.4, y(0) = .4], [x(0) = -1.5, y(0) = .4], [x(0) = -.4, y(0) = 1.5], [x(0) = -.95, y(0) = .95], [x(0) = -.9, y(0) = 1.5], [x(0) = -1.05, y(0) = .95], [x(0) = -.9, y(0) = .9], [x(0) = -.9, y(0) = 1.1], [x(0) = -1.1, y(0) = .9], [x(0) = -1.1, y(0) = 1.1], [x(0) = -1.3, y(0) = -1.3], [x(0) = -1.5, y(0) = -1.5], [x(0) = -.4, y(0) = -.4], [x(0) = -1.5, y(0) = -.4], [x(0) = -.4, y(0) = -1.5], [x(0) = -.95, y(0) = -.95], [x(0) = -.95, y(0) = -1.5], [x(0) = -1.5, y(0) = -.9], [x(0) = -.9, y(0) = -.9], [x(0) = -.9, y(0) = -1.1], [x(0) = -1.1, y(0) = -.9], [x(0) = -1.1, y(0) = -1.1]], x = -2 .. 3, y = -2 .. 3, stepsize = .1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Задача 2:

К рисунку 2.2:

restart; with(DEtools); with(plots); u := (1/3)*x^3-2*x^2+3*x; plot(u, x = -1 .. 4, y = -1 .. 2)

К рисунку 2.3:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t), diff(y(t), t) = -x(t)^2+4*x(t)-3], [x(t), y(t)], t = -20 .. 20, [[x(0) = -1, y(0) = -1], [x(0) = 0, y(0) = 0], [x(0) = 1, y(0) = 1], [x(0) = 2, y(0) = 2], [x(0) = 2.5, y(0) = 2.5], [x(0) = 2.5, y(0) = .5], [x(0) = 2, y(0) = 1], [x(0) = .9, y(0) = .5], [x(0) = .5, y(0) = .5]], x = -2 .. 5, y = -5 .. 5, stepsize = .5, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 2)

К рисунку 2.4:

restart; with(DEtools); with(plots);

plots[pointplot]([[3.739, .1], [3.792, .2], [3.820, .3]], color = black, style = line)

Задача 4:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)-x(t)^5, diff(y(t), t) = -x(t)-y(t)^5], [x(t), y(t)], t = 0 .. 20, x = -.1 .. .1, y = -.1 .. .1, [[x(0) = 0.5e-1, y(0) = 0.4e-1]], stepsize = .1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Задача 5:

К рисунку 5.1:

restart; with(DEtools); DEplot([diff(x(t), t) = exp(y(t))-exp(x(t)), diff(y(t), t) = sqrt(3*x(t)+y(t)^2)-2], [x(t), y(t)], t = 0 .. 40, [[x(1) = 1.02, y(1) = 1.02], [x(1) = 1.02, y(1) = .98], [x(1) = .98, y(1) = 1.02], [x(1) = .98, y(1) = .98]], x = .9 .. 1.1, y = .9 .. 1.1, stepsize = 0.1e-1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

К рисунку 5.2:

restart; with(DEtools); DEplot([diff(x(t), t) = exp(y(t))-exp(x(t)), diff(y(t), t) = sqrt(3*x(t)+y(t)^2)-2], [x(t), y(t)], t = 0 .. 400, [[x(0) = -4.004, y(0) = -4.005], [x(0) = -4.003, y(0) = -3.995], [x(0) = -3.998, y(0) = -4.002], [x(0) = -3.996, y(0) = -3.995], [x(0) = -3.995, y(0) = -4.003], [x(0) = -4.001, y(0) = -3.995], [x(0) = -3.998, y(0) = -4.005]], x = -4.005 .. -3.995, y = -4.005 .. -3.995, stepsize = 0.1e-1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Задача 6:

restart; with(DEtools); DEplot([diff(x(t), t) = y(t)-x(t)*(4-sqrt(x(t)^2+y(t)^2)), diff(y(t), t) = -x(t)-y(t)*(4-sqrt(x(t)^2+y(t)^2))], [x(t), y(t)], t = -20 .. 0, [[x(0) = 4, y(0) = 4], [x(0) = 5, y(0) = 5], [x(0) = .1, y(0) = .1]], x = -6 .. 6, y = -6 .. 6, stepsize = 0.1e-1, method = rkf45, arrows = SLIM, linecolour = magenta, color = x, thickness = 3)

Задача 7:

К рисунку 7.1:

restart;

with(DEtools);

with(plots);

μ := 0.5;

y(t) := (1/2)*μ*sin(t)+(1/128)*μ^3*(sin(3*t)-3*sin(t));

d(t) := diff(y(t), t);

l₁ := DEplot(diff(x(t), t, t) = -5*x(t)-x(t)^3+2*μ*sin(t), x(t), t = 0 .. 2*Pi, x = -0.5 .. 0.5, [[x(0) = subs(t = 0, y(t)), (D(x))(0) = subs(t = 0, d(t))]], linecolour = yellow);

l₂ := plot(y(t), t = 0 .. 2*Pi);

display(l₁, l₂);

К рисунку 7.2: