Difference and functional equations Definition

Nonlinear difference equation

Definition

Let $K$ be a field and let $k \ge 1$ . Let $f \colon \N \times K^k \to K$ be a function, and consider the recurrence relation $u_{n+k} = f(n, u_n, u_{n+1}, \dots, u_{n+k-1}) \qquad (n \in \N).$ It is linear if there are functions $a_0, \dots, a_{k-1}, g \colon \N \to K$ with $f(n, y_0, \dots, y_{k-1}) = a_0(n) y_0 + \cdots + a_{k-1}(n) y_{k-1} + g(n)$ for all $n \in \N$ and all $y_0, \dots, y_{k-1} \in K$ . It is a nonlinear difference equation if it is not linear.

Remarks

Example: the logistic map recurrence $u_{n+1} = r u_n (1 - u_n)$ over $\R$ with $r \ne 0$ is nonlinear. Indeed, for fixed $n$ an affine function $y \mapsto a y + c$ has constant second difference zero, while $f(y) = r y - r y^2$ gives $f(2) - 2 f(1) + f(0) = -2r \ne 0$ . Non-example: the Fibonacci recurrence $u_{n+2} = u_{n+1} + u_n$ is linear, with $a_0 = a_1 = 1$ and $g = 0$ . The same definitions apply with $K$ replaced by $\R^m$ and $f$ by a map to $\R^m$ , which gives nonlinear systems. An autonomous first-order equation $u_{n+1} = f(u_n)$ is the iteration of the map $f$ , so its study belongs to discrete dynamical systems.

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Recurrence relation definition
Nonlinear difference equation definition

Tag	`WC92BCCFHV`
URI	`m2://WC92BCCFHV`
Kind	definition
Writing	W0 stub
MSC	39A05, 39A20
Defines	Nonlinear difference equation
Uses	Recurrence relation

Nonlinear difference equation

Definition

Remarks

Reading path

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