Talk:Andronov-Hopf bifurcation

In section "Two-dimensional Case" the sytem of ODEs would imply that $$\dot{x}_1 = \dot{x}_2$$ for an initial value $$x_1(0) = x_2(0)$$. Shouldn't this be two different functions?

Yu.A.: Of course, $$x_1(0) = x_2(0)=a$$ does not imply $$\dot{x}_1 = \dot{x}_2$$, since

$\dot{x}_1=\beta a - a + \sigma a(a^2+a^2)=(\beta-1)a+2\sigma a^3$

but

$\dot{x}_2= a + \beta a + \sigma a(a^2+a^2)=(\beta+1)a+2\sigma a^3$

So, function $$f$$ isn't unique? It is used for both equations in the ODE system.

Yu.A. The "function $$f$$" assigns to each $$(x_1,x_2)$$ two numbers $$(f_1(x_1,x_2),f_2(x_1,x_2))$$. Please, read some basic ODE textbook.

So this isn't unique. I already read ODE books but I am used to identifying indexes etc.. I am sorry for misunderstanding. This article is altogether more than excellent. But I have to admit that in my eyes the dimensionality of "the" function f is not correct. As you wrote there is a tuple $$(f_1(x_1,x_2),f_2(x_1,x_2))$$ which components should be used in these two equations instead of f.

Yu.A. Both. f is one vector-function of the vector-argument.