An integral curve

If a vector field $f$ is given, then a velocity vector is defined at each point using (8.10). Imagine a point that starts at some $x_0 \in {\mathbb{R}}^n$ at time $t=0$ and then moves according to the velocities expressed in $f$. Where should it travel? Its trajectory starting from $x_0$ can be expressed as a function $\tau : [0,\infty) \rightarrow {\mathbb{R}}^n$, in which the domain is a time interval, $[0,\infty)$. A trajectory represents an integral curve (or solution trajectory) of the differential equations with initial condition $\tau(0) = x_0$ if

$$\frac{d\tau}{dt}(t) = f(\tau(t))$$ (8.14)

for every time $t \in [0,\infty)$. This is sometimes expressed in integral form as

$$\tau(t) = x_0 + \int_0^t f(\tau(s)) ds$$ (8.15)

and is called a solution to the differential equations in the sense of Caratheodory. Intuitively, the integral curve starts at $x_0$ and flows along the directions indicated by the velocity vectors. This can be considered as the continuous-space analog of following the arrows in the discrete case, as depicted in Figure 8.2b.

Example 8..9 (Integral Curve for a Constant Velocity Field)   The simplest case is a constant vector field. Suppose that a constant field $x_1 = 1$ and $x_2 = 2$ is defined on ${\mathbb{R}}^2$. The integral curve from $(0,0)$ is $\tau(t) = (t,2t)$. It can be easily seen that (8.14) holds for all $t \geq 0$. $\blacksquare$

Example 8..10 (Integral Curve for a Linear Velocity Field)   Consider a velocity field on ${\mathbb{R}}^2$. Let ${\dot x}_1 = -2 x_1$ and ${\dot x}_2 = - x_2$. The function $\tau(t) = (e^{-2t},e^{-t})$ represents the integral curve from $(1,1)$. At $t=0$, $\tau(0) = (1,1)$, which is the initial state. If can be verified that for all $t > 0$, (8.14) holds. This is a simple example of a linear velocity field. In general, if each $f_i$ is a linear function of the coordinate variables $x_1$, $\ldots$, $x_n$, then a linear velocity field is obtained. The integral curve is generally found by determining the eigenvalues of the matrix $A$ when the velocity field is expressed as ${\dot x}= A x$. See [192] for numerous examples. $\blacksquare$

A basic result from differential equations is that a unique integral curve exists to ${\dot x}= f(x)$ if $f$ is smooth. An alternative condition is that a unique solution exists if $f$ satisfies a Lipschitz condition. This means that there exists some constant $c \in (0,\infty)$ such that

$$\Vert f(x) - f(x')\Vert \leq c \Vert x - x'\Vert$$ (8.16)

for all $x,x' \in X$, and $\Vert\cdot\Vert$ denotes the Euclidean norm (vector magnitude). The constant $c$ is often called a Lipschitz constant. Note that if $f$ satisfies the Lipschitz condition, then it is continuous. Also, if all partial derivatives of $f$ over all of $X$ can be bounded by a constant, then $f$ is Lipschitz. The expression in (8.16) is preferred, however, because it is more general (it does not even imply that $f$ is differentiable everywhere).