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For a non-autonomous differential equation $\dot{x}=f(t,x)$, if I change the initial condition from $x(t_1)=x_0$ to $x(t_2)=x_0$, where $t_1 \neq t_2$, will the behavior of the system significantly change?
I find that although translation in the direction of $x$-axis doesn't produce another solution, the change seems not so "significant". For example, consider the equation $$\dot{x}=sin(t)-x, x(t_0)= x_0.$$ Vary the value of $t_0$, we'll get different solutions, which cannot be obtained by translating one another, but look quite the same: they oscillate in some interval, and then explode.
So when can I say that a system significantly change its behavior?

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  • $\begingroup$ You could consider your system as a system with an input, as in $\dot{x}(t) = -x(t) + u(t)$, where $u$ is the input. Then if the input $u$ and initial time/condition are shifted by the same amount, the solution will be correspondingly shifted. $\endgroup$ – copper.hat Oct 9 '14 at 14:55
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Yes. Consider the equation $$ x'=x^2-t,\quad x(t_0)=1. $$ It can be shown that there exists $t^*\sim0.56685$ such that:

  1. if $t_0<t^*$ then the solution blows-up, that is, it is defined on a maximal interval $[t_0,T)$ and $\lim_{t\to T^-}x(y)=\infty$
  2. if $t_0\ge t^*$ then the solution is global.

Thus, a small change of $t_0$ around $t^*$ produces a big difference in behavior.

In three or more dimensions, there are systems, like the Lorenz system, that exhibit chaos, that is, strong sensitivity to initial conditions.

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  • $\begingroup$ Thanks for the great example! This is exactly what I've been looking for! $\endgroup$ – AaronS Oct 11 '14 at 16:00
  • $\begingroup$ But isn't the Lorenz system autonomous? Could you provide some other examples from physics? $\endgroup$ – AaronS Oct 11 '14 at 16:01
  • $\begingroup$ Any system is equivalent to an autonomous system. The equation $x'=f(x,t)$ is equivalent to the two-dimensional system $x'(\tau)=f(x,\tau)$, $t'(\tau)=1$, where $\tau$ is a new variable, and $x$, $t$ the unknowns. $\endgroup$ – Julián Aguirre Oct 11 '14 at 16:18

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