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.
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)