restart;

ddesys := {diff(x__1(t), t, t)+x__1(t-tau__1) = 0, diff(x__2(t), t, t)+x__2(t-tau__2) = 0, x__1(0) = 0, x__2(0) = 0, (D(x__1))(0) = 1, (D(x__2))(0) = 1};

$\mathrm{ddesys}≔\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{x\_\_1}\left(t\right)+\mathrm{x\_\_1}\left(t-\mathrm{\τ\_\_1}\right)=0\,\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{x\_\_2}\left(t\right)+\mathrm{x\_\_2}\left(t-\mathrm{\τ\_\_2}\right)=0\,\mathrm{x\_\_1}\left(0\right)=0\,\mathrm{x\_\_2}\left(0\right)=0\,\mathrm{D}\left(\mathrm{x\_\_1}\right)\left(0\right)=1\,\mathrm{D}\left(\mathrm{x\_\_2}\right)\left(0\right)=1\right\}$

dsn := dsolve(eval(ddesys, {tau__1 = 0., tau__2 = .1}), numeric):

plots[odeplot](dsn, [[t, x__1(t), color = blue], [t, x__2(t), color = red]], 0 .. 20, labels = [t, ""]);

ddesys := {diff(x__2(t), t, t)+x__2(t-tau__2) = 0, diff(x__1(t), t, t)-tau__1*(diff(x__1(t), t))+x__1(t) = 0, x__1(0) = 0, x__2(0) = 0, (D(x__1))(0) = 1, (D(x__2))(0) = 1};

$\mathrm{ddesys}≔\left\{\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{x\_\_2}\left(t\right)+\mathrm{x\_\_2}\left(t-\mathrm{\τ\_\_2}\right)=0\,\frac{{\ⅆ}^2}{\ⅆt^2}\mathrm{x\_\_1}\left(t\right)-\mathrm{\τ\_\_1}\left(\frac{\ⅆ}{\ⅆt}\mathrm{x\_\_1}\left(t\right)\right)+\mathrm{x\_\_1}\left(t\right)=0\,\mathrm{x\_\_1}\left(0\right)=0\,\mathrm{x\_\_2}\left(0\right)=0\,\mathrm{D}\left(\mathrm{x\_\_1}\right)\left(0\right)=1\,\mathrm{D}\left(\mathrm{x\_\_2}\right)\left(0\right)=1\right\}$

dsn := dsolve(eval(ddesys, {tau__1 = .1, tau__2 = .1}), numeric):

plots[odeplot](dsn, [[t, x__1(t), color = blue], [t, x__2(t), color = red]], 0 .. 20, labels = [t, ""]);

restart;

dsys_var := {diff(x(t), t) = -x(t-1/2-(1/2)*exp(-t)), x(0) = 1};

$\mathrm{dsys\_var}≔\left\{\frac{\ⅆ}{\ⅆt}x\left(t\right)=-x\left(t-\frac{1}{2}-\frac{{\ⅇ}^{-t}}{2}\right)\,x\left(0\right)=1\right\}$

max_delay := fsolve(t = 1/2+(1/2)*exp(-t), t);

$\mathrm{max\_delay}≔0.7388350311$

dsn_var := dsolve(dsys_var, numeric, delaymax = .74):

plots:-odeplot(dsn_var, 0 .. 5, size = [600, "golden"]);

restart;

ddesys := {diff(pred(t), t) = 10*pred(t-tau)*prey(t-tau)-(1/2)*pred(t)-(1/10)*pred(t)^2,
          diff(prey(t), t) = prey(t)*(1-prey(t))-pred(t)*prey(t), pred(0) = 1,
          prey(0) = 1};

$\mathrm{ddesys}≔\left\{\frac{\ⅆ}{\ⅆt}\mathrm{pred}\left(t\right)=10\mathrm{pred}\left(t-\mathrm{\tau }\right)\mathrm{prey}\left(t-\mathrm{\tau }\right)-\frac{\mathrm{pred}\left(t\right)}{2}-\frac{{\mathrm{pred}\left(t\right)}^2}{10}\,\frac{\ⅆ}{\ⅆt}\mathrm{prey}\left(t\right)=\mathrm{prey}\left(t\right)\left(1-\mathrm{prey}\left(t\right)\right)-\mathrm{pred}\left(t\right)\mathrm{prey}\left(t\right)\,\mathrm{pred}\left(0\right)=1\,\mathrm{prey}\left(0\right)=1\right\}$

dsn := dsolve(eval(ddesys, tau = 0), numeric):

plots[odeplot](dsn, [[t, prey(t), color = green], [t, pred(t), color = red]], 0 .. 300, legend = [ prey, pred ], labels = [t,""] );

dsn := dsolve(eval(ddesys, tau = .25), numeric):

plots[odeplot](dsn, [[t, prey(t), color = green], [t, pred(t), color = red]], 0 .. 300, legend = [ prey, pred ], labels = [t,""] );

dsn := dsolve(eval(ddesys, tau = 5), numeric, maxfun = 0):

plots[odeplot](dsn, [[t, prey(t), color = green], [t, pred(t), color = red]], 0 .. 300, legend = [ prey, pred ], labels = [t,""] );

dsn := dsolve(eval(ddesys, tau = 25), numeric):

plots[odeplot](dsn, [[t, prey(t), color = green], [t, pred(t), color = red]], 0 .. 300, legend = [ prey, pred ], labels = [t,""] );

restart;

dsn := dsolve({diff(y(t), t) = 2*y(t-2)/(1+y(t-2)^9.65)-y(t), y(0) = .5, z(t) = y(t-2)}, numeric):

plots[odeplot](dsn, [y(t), z(t)], 2 .. 100, numpoints = 15000);

restart;

ddesys := {diff(y(t), t) = -y(t)-5*piecewise(t-1 < 0, sin(t-1), y(t-1))-2*piecewise(t-2 < 0, sin(t-2), y(t-2)), y(0) = sin(0), z(t) = diff(y(t), t)};

$\mathrm{ddesys}≔\left\{\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)=-y\left(t\right)-5\left(\left\{\begin{array}{cc}\sin \left(t-1\right)& t<1\\ y\left(t-1\right)& \mathrm{otherwise}\end{array}\right.\right)-2\left(\left\{\begin{array}{cc}\sin \left(t-2\right)& t<2\\ y\left(t-2\right)& \mathrm{otherwise}\end{array}\right.\right)\&comma;y\left(0\right)=0\&comma;z\left(t\right)=\frac{\&DifferentialD;}{\&DifferentialD;t}y\left(t\right)\right\}$

dsn := dsolve(ddesys, numeric):

plots[odeplot](dsn, [[t, y(t), color = red], [t, z(t), color = blue]], 0 .. 5, legend = [y, z], labels = [t,""] );