For an autonomous differential equation, why is any non-equilibrium solution entirely contained in exactly one region between the critical points? I know that for a continuous function f(y) there are no zeros in a region between critical points, making the derivative dy/dt either positive or negative everywhere, meaning the solution curve is either increasing everywhere or decreasing everywhere, but I don't understand how this is the case. Due to the Picard–Lindelöf theorem, no non-equilibrium solution y to the ODE f(y) can pass through an equilibrium point. This means their derivative can't pass through zero, and so the solution is always increasing/decreasing. But what does this mean exactly? Why can't there be a zero between critical points?
For an autonomous differential equation, why is any non-equilibrium solution entirely contained in exactly one region between the critical points? I know that for a continuous function f(y) there are no zeros in a region between critical points, making the derivative dy/dt either positive or negative everywhere, meaning the solution curve is either increasing everywhere or decreasing everywhere, but I don't understand how this is the case. Due to the Picard–Lindelöf theorem, no non-equilibrium solution y to the ODE f(y) can pass through an equilibrium point. This means their derivative can't pass through zero, and so the solution is always increasing/decreasing. But what does this mean exactly? Why can't there be a zero between critical points?
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