Taylor expansion for a nonlinear system and Picard Iterations

[Ulver48]

Hello guys
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.

 

[S.G. Janssens]

Does this concern Section 9.5.1 ("Approximation by a flow") in the third edition?

 

[Ulver48]

Yes. It's in this section.

 

[S.G. Janssens]

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

I would read the equation for $x^{(k+1)}$ indeed as a definition - and not more than that - and then proceed to verify its claimed properties, for which I don't think any references to Picard's method are necessary.

As a possible motivation, you could also think about $x^{(k+1)}$ as the solution of the inhomogeneous problem
$$
\dot{x}^{(k+1)} = \Lambda x^{(k+1)} + h^{(k+1)}, \qquad x^{(k+1)}(0) = x,
$$
where $h^{(k+1)}(t) = F^{(2)}(x^{(k)}(t)) + \cdots + F^{(k+1)}(x^{(k)}(t))$ is a known forcing function. (If you solve this for $x^{(k+1)}$ using variation-of-constants, then of course you find the book's definition (9.26) for $x^{(k+1)}$ that you quoted.)

(As an aside, it is my impression that for solving the original nonlinear system using Picard iteration, we would usually take the initial value $x$ in the spot where you write $x_n(0)$.)