- #1

Ulver48

- 12

- 2

I struggle since yesterday with the following problem

I am reading the book "Elements of applied bifurcation theory" by Kuznetsov . At one point he has the following Taylor expansion of a nonlinear system with respect to x=0 where ##x\in \mathbb(R)^n##

$$\dot{x} = f(x) = \Lambda x + F^{2}(x) + F^3(x)+\ldots $$

where ##F^k ## is a smooth polynomial vector-valued function of order k,

$$F_i^k(x)=\sum_{j_1+j_2+\ldots+j_n} b^k_{i,j_1,j_2,\ldots,j_n}x_1^{j_1}x_2^{j_2}\dots x_n^{j_n}$$

Afterwards he tries to approximate the solution of this non-linear system using Picard Iterations. He sets $$x^1(t)=e^{\Lambda t}x $$, which is the solution of the linear approximation for initial data x and defines

$$x^{k+1}=e^{\Lambda t}x + \int_0^t e^{\Lambda(t-\tau)}(F^2(x^k(\tau))+\ldots+F^{k+1}(x^k(\tau)))d\tau $$

I don't understand how he ends up with the last equation by using the Picard Method

$$ x_{n+1}(t) = x_n(0) + \int_0^t f(x_n(\tau)) d\tau $$

Thank you very much for your time.