How do you call this system? Is there a

Eddie Clarke

Eddie Clarke

Answered question

2022-04-18

How do you call this system?
Is there a specific name for a dynamical system that depends on the relative indexation i±k for some k? For example, consider the following dynamical system defined on a ring of cells by
u˙i=ui+2(vi1+vi+1)v˙i=uivi
for each cell i, where the derivative is with respect to time t.
The main reason I ask this is because I won't to compare this kind of systems with systems involving spatial coordinates, u(x,t), as in reaction-diffusion equations.

Answer & Explanation

glajusv8iv

glajusv8iv

Beginner2022-04-19Added 10 answers

Explanation:
The ODE system is translationally invariant, if iZ or the indices have periodic boundary conditions (identifying i=N+1 with i=1), meaning that no unit i is distinguished. Translation invariance implies that the system specified in matrix form contains a (block) circulant matrix. In addition, the circulant is banded, as stated by Ian.

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