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If we have a dynamical system for a vector of scalar variables $x_t$: $$\dot x_t=f(x_t)$$

Then the jacobian is the matrix of partial derivatives $$J(x_t)=Df(x_t)$$

Let's say we have a fixed point $x_t^*$: $f(x_t^*)=0$.

Then what is the (intuitive) interpretation of $|J(x_t^*)|$? and of $sign |J(x_t^*)|$?

user56834
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  • @ArnaudMortier, my question is not about "what the jacobian is and how to compute it" (as that question you referred to is about), but about the DETERMINANT of the jacobian or about the sign of that determinant, and their interpretation. – user56834 Feb 01 '18 at 18:29
  • In general the determinant of a matrix tells you by how much this matrix seen as a linear map dilates or contracts volumes. The sign tells you whether the map respects the orientation or reverses it. – Arnaud Mortier Feb 01 '18 at 18:42
  • @ArnaudMortier, I know. I am specifically asking for the interpretation of the determinant of a jacobian of a dynamical system. What makes it harder for me to think about this intuitively is that the jacobian of the dynamical system doesn't actually map points at time $t$ to time $t+1$, but only gives the infinitessimal change. As a result, its hard for me to interpret what it means to "flip the orientation" for a system of ODE's. – user56834 Feb 01 '18 at 19:01
  • You might be surprised, but @ArnaudMortier 's comment is essentially the interpretation of Jacobian for a dynamical system. Intuitive explanation will come if you sit and work around the formula that gives you the change of the volume of a domain under the action of the flow,i.e. $$ V(t) = \int\limits_{\phi^t(G)}, dx = \int\limits_{G} \left \vert \dfrac{D\phi^t}{Dx} \right \vert , dx. $$ (Also, taking derivative of $V(t)$ and using formula ${\rm det}, (I + \varepsilon A) = 1 + \varepsilon \cdot {\rm tr}, A + o(\varepsilon)$ would be useful here) ... – Evgeny Feb 02 '18 at 10:07
  • ... However, for flows, as far as I remember, Jacobian is always posiive. So, no flipping of orientation. This is sometimes can be used for checking whether diffeomorphism can be embedded in a flow or not. – Evgeny Feb 02 '18 at 10:09
  • @Evgeny, why would the jacobian of flows be positive always? Just take any linear model with a matrix that has a negative determinant – user56834 Feb 02 '18 at 11:06
  • Ignore the second comment for a moment. As it is written it is certainly not true, you are right. The fact that I wanted to say is that $\phi^t$ preserves orientation (it is true), but I need to be more careful linking it with properties of vector field. – Evgeny Feb 02 '18 at 11:54
  • Why is it closed? It is actually not duplicate – Carlos Feb 04 '18 at 21:26
  • @Carlos, no it's not. But I've stopped trying to convince people that a post is not a duplicate. They tend not to listen anyway, and usually decide to close a post clearly without reading it. – user56834 Feb 05 '18 at 05:55
  • I see. Though, what i understand of the determinant of the Jacobian of an ode, evaluated at a stationary point, is that it shows when it is zero that ar least one sub-ode of the system has no linear component in a neighbourhood of the stationary point. – Carlos Feb 05 '18 at 07:19
  • @Programmer2134 I voted for reopening it. – Evgeny Feb 09 '18 at 07:45

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