Frist of all, I found post Find the bifurcation points for the following system of ODEs by Antonio Vargas and comment by Pragabhava very useful, nevertheless could anyone be pleased to be more specific about the following text in THAT post :
$$f(x,y) = f(x_0,y_0) + \pmatrix{(1-y_0) x -x_0 y \\ x -2y_0 y} + \ldots$$
I assume this must be Taylor series of a $f(x,y)$ near the point $(x_0,y_0)$, nevertheless I do not get it.
I am also reading paper by John David Crawford about bifurcations and got stuck at some point, if somebody read this article , could you please explain, why when we rewrite our DE as: $$\frac{d}{dt}\pmatrix{x \\ y} = \pmatrix{y\\ -y-x-x^3}$$ And say let : $$\dot x = V(\mu,x)$$ So after this equilibrium points can be found from $V(0,0) = 0$, What does it mean?
3rd question
I have a bit more complicated system of diff. eq. that is why I can not apply(actually do not know how to apply and repeat the article steps. For instance if we do the Taylor expansion of the function: $$\dot x = V(\mu,0) + D(V(\mu,0))+O(x^2)$$ as $V(\mu,0) = 0$ and $O(x^2)$ is not used. then $$\dot x = D(V(\mu,0))$$ next step is to make the something like the following matrix: $$\frac{d}{dt}\pmatrix{x_i \\...\\ x_{i+1}} = \pmatrix{0,0,\lambda\\ ... \\ -\lambda_{i+1},-\lambda_i,- \lambda_i^3} \times \pmatrix{x_i \\...\\ x_{i+1}}$$ (that matrix was done to find the solution easier(page 994)) FINALY, MY question is the following: As I said earlier, I have the system of ODE, but a little bit more complicated than in the example in an article, that is why Taylor expansion and further procedures I do not know how to do. Is it possible to represent the next system of ODE in a matrix form?
Here is the system: $$ [ \ddot{\theta} ]= [ sin(\theta)cos(\theta)(\dot{\phi})^2 + \omega_{af}^2(sin(\theta)^3cos(\theta)cos(\phi)^4+sin(\theta)^3cos(\theta)sin(\phi)^4 - cos(\theta)^3sin(\theta)) ] + [ 2 \gamma_{af} \dot{\theta} - 2 j sin(\theta)cos(\theta)cos(\phi)sin(\phi) ]$$ $$[\ddot{\phi}] = - [cot(\theta)\dot{\phi}\dot{\theta}] + [\omega_{af}^2(-sin(\theta)^2cos(\phi)^3sin(\phi) +sin(\theta)^2sin(\phi)^3cos(\phi)) ] + [2\gamma_{af} \dot{\phi} - j (cos(\phi)^2 - sin(\phi)^2 ) ] $$ So, how can I find bifurcation in this system. My main parameter I can modify is j
My work to the last question: if I represent my system in a bit other way( $L_x = sin(\theta)cos(\phi)$, $L_y=sin(\theta)sin(\phi),L_z = cos(\theta)$), then my system looks like this:
$$ \ddot{L_x} = - \omega_{af}^2 L_x - 2 \gamma_{af} \dot{L_x} - j L_y $$
$$ \ddot{L_y} = - \omega_{af}^2 L_y - 2 \gamma_{af} \dot{L_y} + j L_x $$ then $$ A w^2 - \omega_{af}^2 A - 2 \gamma_{af} A iw - j B = 0 $$
$$ B w^2 - \omega_{af}^2 B - 2 \gamma_{af} B iw + j A = 0 $$ then using De Moivre's formula I get some $w$, however
as the saddle point is when $Re_1 >0, Re_2<0$ I get nonsense.