Gradient Systems

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Today, more than three centuries after Isaac Newton wrote down his treatise on the theory of fluxions, and after the rise of the calculus of variations, it has turned out that a large number of evolution models may be expressed in the mathematical language of ordinary and partial differential equations. A system of the form

where \( {\bf x} = \left( x_1 (t), x_2 (t) , \ldots , x_n (t) \right)^{\text T} \in U \subset \mathbb{R}^n , \quad G\,:\, \mathbb{R}^n \to \mathbb{R} \) is a continuously differentiable function defined on an open subset U, and k is a constant, is called a gradient system. (The negative sign in this system is traditional because critical points are the same \( \nabla\cdot G({\bf x}) = - \nabla\cdot G({\bf x}) =0. \) ) Such system is the prototype example of a dissipative evolution equation. Recall that an equilibrium point is called stable if nearby solutions stay nearby for all future time. If additionally, all nearby solutions approach the equilibrium for large time, then the critical point is said to be asymptotically stable.

Suppose that the function G(x) has an isolated local minimum/maximum value at the point x^*. Then this point will be a critical point of the given system of differential equations. Its orbits follow the path of steepest descent/increase of G depending on the sign of k.

Let \( U \subset \mathbb{R}^n \) be an open set and let \( G\,:U \to \mathbb{R}^n \) be a continuous function. A function \( E\,: U \to \mathbb{R} \) is called an energy function for the first order vector differential equation

if for every solution x of this system of equations the composition \( E({\bf x}) = E \circ {\bf x} \) is decreasing. Very often, an energy function is also called Lyapunov function. In some places of the literature, an energy function is also called cost function, mainly because of applications to optimization problems. Differential equations admitting an energy function may be called dissipative system.

The gradient (denoted by nabla: ∇) is an operator that associates a vector field to a scalar field. Both scalar and vector fields may be naturally represented in Mathematica as pure functions. However, there is no built-in Mathematica function that computes the gradient vector field (however, there is a special symbol \[ EmptyDownTriangle ] for nabla). The command Grad gives the gradient of the input function. R requires a package to plot the direction field

library(pracma)
##f <- function(x, y) x^2 + y^2 *x - 3*y
f1=expression(x^2 + y^2 *x - 3*y)

z <- D(f1,'y')
z
f <- function(x,y) 2 * x + y^2
xx <- c(-2, 2); yy <- c(-2, 2)
## Not run:
vectorfield(f, xx, yy, scale = 0.1)
for (xs in seq(-2, 2, by = 0.25)) {
  sol <- rk4(f, -2, 2, xs, 100)
  lines(sol$x, sol$y, col="darkgreen")
}
grid()

Then we apply VectorPlot to defined above potential function. We present some codes using Mathematica with different options. Since not every vector field is a gradient field of any function, we need a procedure that calculate it. Finally, here is a subroutine to plot a gradient field based on a (scalar) potential function. Here is an example of how to use the function. The contour lines of the potential are shown in white, and the streamlines of the gradient field are orange.

 

A system of autonomous differential equations is a gradient system if and only if there is a scalar valued function G so that In one dimensional case, a gradient system coinsides with an autonomous equation In a two-dimensional case, a system is a gradient sytem if and only if there is a function G such that A necessary and sufficent condition for the system to be gradient is the equality of mixed partials, In the general case, the necessary and sufficient condition is again equality of mixed partials expressed as

 

Let V be a C¹ function defined on an open neighborhood U₁ ⊂ U ⊂ ℝⁿ of the equilibria x^* ∈ U₁. Let us define the associated function via differential equation If x(t) is a solution of Eq.\eqref{EqGrad.1}, it follows from the chain rule that The study of gradient systems \eqref{EqGrad.1} is particularly simple due to the formula where x(t) is a solution of Eq.\eqref{EqGrad.1}. To understand a gradient flow geometrically we look at thelevel surfacesof the function G :  ℝⁿ → ℝ. These are the subsets \( G^{-1} (c) \) with c ∈ ℝ. Solutions of the gradient vector field cross the level sets of the function G orthogonally except at critical points. Another consequence of Eq.\eqref{EqGrad.4} is that a gradient system cannot have any periodic solutions. Example: Gradient system There is a related concept of the gradient field, where you look at the gradient (vector derivative) of a function:

 

References