Delay differential equation: Difference between revisions

Latest revision as of 02:16, 9 July 2012

Definition

The notion of delay differential equation (abbreviated DDE) is a variant of the notion of differential equation (in other words, delay differential equations are not (ordinary) differential equations).

First-order first-degree case

If we denote the dependent variable by $x$ and the independent variable by $t$ (Which we think of as time), the first-order first-degree case is:

$\frac{d x (t)}{d t} = f (t, x (t), the entire trajectory of x prior to time t)$

General case

The general case of a delay differential equation is of the form:

$F (t, x (t), derivatives of the function x (t) at the point t, the entire trajectory of x prior to time t) = 0$

Note on autonomous case

The delay differential equations that we study are typically autonomous delay differential equations: an equation in the general form above is autonomous if, for any $τ \in R$ , the function $F (t, x (t), derivatives of the function x (t) at the point t, the entire trajectory of x prior to time t)$ is invariant under replacing $x (t)$ by the function $t \mapsto x (t - τ)$ . Intuitively, what this means is that $t$ does not appear explicitly in $F$ , and all the behavior at previous points is specified in terms of how much earlier they were than $t$ .

@@ Line 1: / Line 1: @@
 ==Definition==
-The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations). If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math>, the first-order first-degree case is:
+The notion of '''delay differential equation''' (abbreviated '''DDE''') is a variant of the notion of [[differential equation]] (in other words, delay differential equations are not (ordinary) differential equations).
+===First-order first-degree case===
+If we denote the dependent variable by <math>x</math> and the independent variable by <math>t</math> (Which we think of as time), the first-order first-degree case is:
 <math>\frac{dx(t)}{dt} = f(t,x(t),\mbox{the entire trajectory of } x \mbox{ prior to time } t)</math>
+===General case===
+The general case of a delay differential equation is of the form:
+<math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t) = 0</math>
+===Note on autonomous case===
+The delay differential equations that we study are typically [[autonomous delay differential equation]]s: an equation in the general form above is autonomous if, for any <math>\tau \in \R</math>, the function <math>F(t,x(t), \mbox{derivatives of the function } x(t) \mbox{ at the point } t, \mbox{the entire trajectory of } x \mbox{ prior to time } t)</math> is invariant under replacing <math>x(t)</math> by the function <math>t \mapsto x(t - \tau)</math>. Intuitively, what this means is that <math>t</math> does not appear explicitly in <math>F</math>, and all the behavior at previous points is specified in terms of how much earlier they were than <math>t</math>.