Nonlinear differential equations don’t necessarily have globally defined solutions, even if the equation is autonomous (no time is involved explicitly). The simplest example is with initial condition
: the solution is
, which ceases to exist at
, escaping to infinity.
This kind of behavior can often be demonstrated without solving the ODE. Consider with
,
. I’ve no idea what the explicit solution is, but it’s clear that the quantity
remains constant: indeed,
. (Here,
is analogous to the sum of kinetic and potential energy in physics.)
The solution is increasing and convex. Let
be such that
, e.g.,
. The invariance of
yields
. By the mean value theorem,
for
. Since the series on the right converges, the limit
is finite; this is the blow-up time.
But no blow-up occurs for the equation , where the nonlinear term pushes back toward equilibrium. Indeed, the invariant energy is now
, which implies that
and
stay bounded. In the phase space
the solution with initial values
,
traces out this curve:

Accordingly, the solution is a periodic function of (although this is not a trigonometric function):

Everything so far has been pretty straightforward. But here is a stranger animal: . As in the previous example, nonlinear term pushes toward equilibrium. Using initial conditions
,
, I get this numerical solution up to time
:

As in the previous example, oscillates. But the speed and amplitude of oscillation are increasing.


Rapidly increasing:


In the plane the solution spirals out:

The plots make it clear that the solution ceases to exist in finite time, but I don’t have a proof. The issue is that the function does not escape to infinity in just one direction. Indeed, if
, then regardless of the values of
and
at
, the negative third derivative will eventually make the function
decrease, and so
will attain the value
at some
. After that, the third derivative is positive, guaranteeing the existence of time
when
returns to
again. I haven’t succeeded in proving that the limit
is finite, giving the time when oscillatory explosion occurs.
The plots were made in SageMathCloud:
var('t y y1 y2')
P = desolve_system_rk4([y1,y2,-y^3],[y,y1,y2], ics=[0,1,0,0],ivar=t,end_points=5,step=0.01)
Q=[ [i,j] for i,j,k,l in P]
list_plot(Q, plotjoined=True)