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I'm reading Strogatz book on Nonlinear Dynamics and am a bit confused about the distinction between Conservative Systems and Gradient Systems. On page 160, it is claimed that Conservative Systems typically have saddles or centers. On page 199, it is shown that Gradient Systems cannot have closed orbits. But aren't Gradient Systems conservative too (because a vector field is conservative iff it is the gradient field of a potential function)? If so, shouldn't they be able to have closed orbits, or are Gradient systems just special cases of Conservative systems in which closed orbits cannot occur?

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  • $\begingroup$ Gradient systems are not conservative. $\endgroup$ Aug 31 '13 at 14:49
  • $\begingroup$ I'm not sure why. A vector field $\vec{F(\vec{X})}$ is conservative iff $\vec{F}$ = $\nabla V$ for some continuously differentiable scalar funtion $V(\vec{X})$. A Gradient system meets this definition, so why is it not conservative? $\endgroup$
    – Adam
    Aug 31 '13 at 16:21
  • $\begingroup$ Conservative vector field is different than conservative dynamical system. $\endgroup$ Aug 31 '13 at 16:42
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You are confusing conservative dynamical systems and conservative vector fields.

Conservative dynamical systems preserve phase space volume, i.e. for a continuous time dynamical system defined as $\dot{x}=f(x)$, $x\in R^n$ the jacobian $D_xf(x)$ will have trace 0. In other words, the sum of eigenvalues will be 0. This volume preservation hence disallows the presence of sinks or sources, since near fixed points mean phase space volume is shrinking or growing.

Conservative vector fields (the term mostly used in physics) are force fields that come from a potential. Simplest example is a spring attached to a unit mass. The potential is $(1/2)kx^2$, and force is hence $-kx$. Newton's law states $\ddot{x}=-kx$ Now if you had to make a dynamical system out of this , it would need to be two dimensional. $\dot{x}=v$

$\dot{v}=-kx$

Note that this dynamical system coming out of the conservative vector field is not of the form $\dot{x}=-\nabla V(x)$, and hence is NOT a gradient system.

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  • $\begingroup$ Thank you for the clarification. Your response was very helpful :) $\endgroup$
    – Adam
    Sep 1 '13 at 6:57
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You could define Gradient Systems. Without further explanation, I would think two body gravitation is a Gradient System (it is derived from the gradient of the gravitational potential) and it has closed (elliptic) orbits.

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    $\begingroup$ A Gradient System is a system of the form $\vec{X'}$ = - $\nabla V$ for some continuously differentiable, single-valued scalar function $V(\vec{X})$. $\endgroup$
    – Adam
    Aug 31 '13 at 5:18

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