View source for Gradient dynamical system

A [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] given by the gradient of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446901.png" /> on a smooth manifold. Direct differentiation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446902.png" /> yields a [[Covariant vector|covariant vector]] (e.g. in the finite-dimensional case in a coordinate neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446903.png" /> with local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446904.png" /> this is the vector with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446905.png" />), while the phase velocity vector is a [[Contravariant vector|contravariant vector]]. The passage from the one to the other is realized with the aid of a Riemannian metric, and the definition of a gradient dynamical system depends on the choice of the metric (as well as on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446906.png" />); the phase velocity vector is often taken with the opposite sign. In the given example the gradient dynamical system in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446907.png" /> is described by the system of ordinary differential equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446908.png" /></td> </tr></table>

where the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g0446909.png" /> form a matrix inverse to the matrix of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g04469010.png" /> of the metric tensor; it is understood that in all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044690/g04469011.png" /> equations the right-hand side is taken with the same  "plus"  or  "minus"  sign. A gradient dynamical system is often understood to mean a system of a somewhat more general type [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Smale,   "On gradient dynamical systems"  ''Ann. of Math. (2)'' , '''74''' :  1  (1961)  pp. 199–206</TD></TR></table>