O. Perron's Example

Needs["Graphics`Colors`"]

The choice for the parameter a:

a = 1/2 + 1/16 Exp[-π]

The solution to the linearized system:

\(lp = 
      DSolve[{\(x'\)[t] == \(-a\)\ \ x[t], 
          \(y'\)[t] == \ \((\(-2\)\ a + Sin[Log[t]] + Cos[Log[t]])\) y[t], 
          x[1] == 1, y[1] == 1}, 
	{x[t], y[t]}, t]\)

A solution for the non-linear equation:

\(nlp = 
      DSolve[{\(x'\)[t] == \(-a\)\ \ x[t], 
            \(y'\)[t] == \ 
              \((\(-2\)\ a + Sin[Log[t]] + Cos[Log[t]])\) y[t] + x[t]^2}, 

          	{x[t], y[t]}, t] /. {C[1] -> 0, C[2] -> 1}\)

nlp[[1, 2, 2, 2, 2, 2]] = 1

nlp

Plots for the linearized equation are blue, while plots for the non-linearized equation are red.

Plot[lp[[1, 2, 2]], {t, 1, 30}, PlotRange -> All, 
      PlotStyle -> {{AbsoluteThickness[2], Blue}}];

Plot[nlp[[1, 2, 2]], {t, 1, 30}, PlotRange -> All, 
      PlotPoints -> 200, PlotStyle -> {{AbsoluteThickness[2], Red}}];

Exp[2.5  π]

Perron predicts the first hump at e^2.5π, and there it is, no kidding!

Plot[nlp[[1, 2, 2]], {t, 2000, 3000}, PlotRange -> All, 
      PlotPoints -> 200, PlotStyle -> {{AbsoluteThickness[2], Red}}];

Etc.