---
tag: 1W4X6J3948
kind: definition
name: Dynamic equation on time scales
msc: [34N05, "26E70"]
defines: BKXYBS57T9
uses: [C4B1FA1H0J, 30G1M35TK1, 0NT64JV96A, 0FYX44GP92]
sources:
  - url: https://en.wikipedia.org/wiki/Time-scale_calculus
---

## Definition

Let $\mathbb{T}$ be a time scale, that is, a nonempty closed subset of $\R$.
Let $\sigma$ be its forward jump operator, and let $\mathbb{T}^\kappa$ be as in the [[definition of the delta derivative|C4B1FA1H0J]].
For a function $x \colon \mathbb{T} \to \R^n$, write $x^\Delta$ for its delta derivative, taken componentwise.
Let $f \colon \mathbb{T} \times \R^n \to \R^n$ be a function.
The *first-order dynamic equation* on $\mathbb{T}$ with right side $f$ is
$$x^\Delta(t) = f(t, x(t)), \qquad t \in \mathbb{T}^\kappa.$$
A *solution* is a function $x \colon \mathbb{T} \to \R^n$ that is delta differentiable at every $t \in \mathbb{T}^\kappa$ and satisfies the equation there.
Dynamic equations of higher order are defined in the same way with iterated delta derivatives.

## Remarks

For $\mathbb{T} = \R$, the delta derivative is the ordinary derivative and the equation is the [[ordinary differential equation|0NT64JV96A]] $x' = f(t, x)$.
For $\mathbb{T} = \Z$, the equation reads $x(t + 1) = x(t) + f(t, x(t))$, a first-order [[recurrence relation|0FYX44GP92]].
Example: on $\mathbb{T} = h\Z$ with $h > 0$, the function $x(t) = (1 + h p)^{t/h}$ solves $x^\Delta = p x$, since $\frac{x(t + h) - x(t)}{h} = \frac{(1 + hp) - 1}{h} x(t) = p x(t)$.
Non-example: on $\mathbb{T} = \Z$, the function $x(t) = e^t$ does not solve $x^\Delta = x$, because $x(t + 1) - x(t) = (e - 1) e^t \ne e^t$.
At a point $t$ with $\sigma(t) > t$, the equation fixes $x(\sigma(t))$ from $x(t)$, so solutions of initial value problems advance across gaps by an explicit step.