# Fixed Points of ODE

1. Jul 21, 2009

### Apteronotus

In a book on synchronization it is stated that given the ODE

$$\frac{d\psi}{dt}=-\nu+\epsilon q(\psi)$$

there is at least one pair of fixed points if

$$\epsilon q_{min}<\nu<\epsilon q_{max}$$

were $$q_{min}, q_{max}$$ are the min and max values of $$q(\psi)$$ respectively.

While this could be true under particular circumstances (ie. when $$q_{min}<0, q_{max}>0$$), I dont see how it could hold in general; such as the case when $$q(\psi)>0$$.

Can anyone shed some light on this?

Thanks in advance.

2. Jul 21, 2009

### trambolin

Just think about this...

$$0 = -\nu+\epsilon q(\psi)$$

3. Jul 21, 2009

### HallsofIvy

Assuming that q is continuous, the "intermediate value property" gives the answer.

4. Jul 21, 2009

### Apteronotus

Yes, of course! Thank you both.

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