# Help! One problem of nonlinear dynamics

1. Dec 5, 2012

### xibeisiber

Show that all vector fields on the line are gradient systems.

This is exercise 7.2.4 in the book "Nonlinear Dynamics and Chaos" by Steven H.Strogatz

Thanks very much!

2. Dec 5, 2012

### pasmith

What's a gradient system on the line?

What's a vector field on the line?

If you can answer those questions, then the proof should be straightforward.

3. Dec 5, 2012

### xibeisiber

I tried to prove that ∂f/∂y=∂g/∂x,as shown in the attachment,but failed...

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4. Dec 5, 2012

### pasmith

You're barking up the wrong tree, I'm afraid.

"The line" means the real numbers. A http://planetmath.org/GradientSystem.html [Broken] on the line is a system in which
$$\dot x = -\frac{dV}{dx}$$
for $x \in \mathbb{R}$ and some V(x). A vector field on the line is essentially just a function $f: \mathbb{R} \to \mathbb{R}$ which is at least continuous.

So, given any continuous function $f: \mathbb{R} \to \mathbb{R}$, can we always find V(x) such that
$$\frac{dV}{dx} = -f(x)?$$

Last edited by a moderator: May 6, 2017
5. Dec 6, 2012

### xibeisiber

But the system I consider is 2D,the V(x,y) is not always there for all f(x,y) and g(x,y).So there must be anything else I haven't realised.

6. Dec 6, 2012

### xibeisiber

em,I feel like I understand that "on the line" means one dimension.:surprised
I wasted too much time on such a simple question!!!!!!!