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Differential equations of the form $$\frac{d}{dt}\vec{x} = - \nabla E(\vec{x})$$ can be analyzed using phase portrait method. In particular, if the function $E$ (we call it energy) has local minimums, then the solution $\vec{x}(t)$ evolve towards it. These special points are called fixed points or stagnation points. If we start from them then the vector would not change in time. Here I want to ask about the similar problem regarding the space of functions, namely consider the equation: $$\frac{d}{dt}\psi(\vec{x},t) = -\frac{\delta E[\psi, \psi^*]}{\delta \psi^*}$$ or $$\frac{d}{dt}(\psi_1, \psi_2, \psi_3) = -\left(\frac{\delta E}{\delta \psi_1^*}, \frac{\delta E}{\delta \psi_2^*}, \frac{\delta E}{\delta \psi_3^*}\right)$$ where $\psi^*$ is the complex conjugate of $\psi$. Can we state that the function $\psi$ evolves towards the minimum of the energy $E[\psi,\psi^*]$? (These kind of equations arise in the study of ground state of the dilute Bose gases and come with a name: imaginary-time evolution).

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    $\begingroup$ There is a huge literature on this, with many flavors. A classical problem where this idea is applied is the question of existence of closed geodesics. A keyword is calculus of variations. $\endgroup$ – Thomas Rot Dec 7 '15 at 22:16
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In connection with your question, there is the theory of "gradient systems", productively defined in terms of Lyapunov functions, which allows being a bit less attached to the particular equation at hands. My best suggestion for a references is the beautiful book by Jack Hale "Asymptotic Behavior of Dissipative Systems", and specifically Section 3.8.

Not surprisingly, Hale concentrates the discussion of possible applications on functional equations (delay equations) and on reaction-diffusion equations. But before that he gives a very clear description of the general theory.

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