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?
The statement that any non-equilibrium solution of an autonomous differential equation is entirely contained in exactly one region between the critical points is a consequence of the uniqueness of solutions to ordinary differential equations (ODEs) and the behavior of the solutions in the vicinity of equilibrium points.
Let's break down the reasons behind this statement:
Uniqueness of Solutions: The Picard-Lindelöf theorem, also known as the existence and uniqueness theorem for ODEs, ensures that for a given initial value problem, there exists a unique solution. This means that once you specify an initial condition, there is only one solution curve that passes through that point.
Behavior Near Equilibrium Points: In the context of autonomous ODEs, equilibrium points are where the right-hand side of the ODE (the function f(y)) is equal to zero. At these points, the derivative dy/dt is zero, meaning the solutions remain constant. In other words, equilibrium points are where the solutions "stall."
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