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JohnDuck
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Non-linear ODE, "blow-up" phenomenon
On page 40 of Steven H. Strogatz's Nonlinear Dynamics and Chaos, question 2.5.2 says:
I've not been able to show this to my satisfaction. The best I've been able to do is to find an implicit solution for x of the form f(x) - f(x0) = t, where x(0) = x0. Using graphical methods, I've determined that the limit of f(x) - f(x0) as x approaches infinity is a real positive number L for all real x0 (I believe I could show this analytically, however it would be very tedious as f(x) is not at all nice). This leaves me with the equation L = lim t as x approaches infinity. I've interpreted this to mean that the the limit of x as t approaches L diverges to infinity--this seems intuitively correct to me, but I can't seem to justify it using definitions of limits. Could someone tell me whether this interpretation is justified or unjustified (and why)?
Furthermore, this is obviously not the solution Strogatz had in mind. In the book, he finds an exact solution to the ODE x' = 1 + x2, namely x(t) = tan (t + arctan(x0)), and from this it is clear that x "blows up" for some finite t. How might one use this result to solve the above problem?
On page 40 of Steven H. Strogatz's Nonlinear Dynamics and Chaos, question 2.5.2 says:
Show that the solution to x' = 1 + x10 escapes to positive infinity in a finite time, starting from any initial condition. (Hint: Don't try to find an exact solution; instead, compare the solutions to those of x' = 1 + x2.)
I've not been able to show this to my satisfaction. The best I've been able to do is to find an implicit solution for x of the form f(x) - f(x0) = t, where x(0) = x0. Using graphical methods, I've determined that the limit of f(x) - f(x0) as x approaches infinity is a real positive number L for all real x0 (I believe I could show this analytically, however it would be very tedious as f(x) is not at all nice). This leaves me with the equation L = lim t as x approaches infinity. I've interpreted this to mean that the the limit of x as t approaches L diverges to infinity--this seems intuitively correct to me, but I can't seem to justify it using definitions of limits. Could someone tell me whether this interpretation is justified or unjustified (and why)?
Furthermore, this is obviously not the solution Strogatz had in mind. In the book, he finds an exact solution to the ODE x' = 1 + x2, namely x(t) = tan (t + arctan(x0)), and from this it is clear that x "blows up" for some finite t. How might one use this result to solve the above problem?