Gradient dynamical system

A flow (continuous-time dynamical system) given by the gradient of a smooth function $f$ on a smooth manifold. Direct differentiation of $f$ yields a covariant vector (e.g. in the finite-dimensional case in a coordinate neighbourhood $U$ with local coordinates $x^1,\dots ,x^n$ this is the vector with components $\partial f/\partial x^1,\dots ,\partial f/\partial x^n$), while the phase velocity vector is a 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 $f$); the phase velocity vector is often taken with the opposite sign. In the given example the gradient dynamical system in the domain $U$ is described by the system of ordinary differential equations

$$\frac{d{x}^i}{dt}=\pm \sum _j{g}^{ij}\frac{\partial f}{\partial x^j},i=1,\dots ,n,$$

where the coefficients $g^{ij}$ form a matrix inverse to the matrix of coefficients $\Vert g_{ij}\Vert$ of the metric tensor; it is understood that in all $n$ 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 [1].

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