Taylor expansion for a nonlinear system and Picard Iterations

In summary, the conversation discusses the book "Elements of applied bifurcation theory" by Kuznetsov and his Taylor expansion of a nonlinear system. The speaker also mentions Picard Iterations and how the author uses it to approximate the solution of the system. They discuss a specific equation and how it relates to Picard's method.
  • #1
Ulver48
12
2
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.
 
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  • #2
Does this concern Section 9.5.1 ("Approximation by a flow") in the third edition?
 
  • #3
Yes. It's in this section.
 
  • #4
Ulver48 said:
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)##.)
 

1. What is the purpose of using Taylor expansion for a nonlinear system?

The purpose of using Taylor expansion for a nonlinear system is to approximate the behavior of the system around a given point. This is useful in situations where it is difficult or impossible to solve the system analytically, or when obtaining an exact solution is not necessary.

2. How is Taylor expansion for a nonlinear system different from linear systems?

In linear systems, the equations can be solved using simple algebraic methods. However, in nonlinear systems, the equations are more complex and require numerical methods like Taylor expansion to approximate the solution.

3. What is the process of performing a Taylor expansion for a nonlinear system?

The process of performing a Taylor expansion for a nonlinear system involves expanding the system's equations into a series of terms, with each term representing a higher order of approximation. This series is then used to iteratively solve for the system's solution using numerical methods like Picard Iterations.

4. What are the limitations of using Taylor expansion for a nonlinear system?

One limitation of using Taylor expansion for a nonlinear system is that it can only provide an approximation of the solution. The accuracy of the approximation depends on the order of the expansion and the size of the step used in the iteration process. Additionally, Taylor expansion may not converge if the system is highly nonlinear or if the initial guess is too far from the actual solution.

5. How can Picard Iterations improve the accuracy of the Taylor expansion for a nonlinear system?

Picard Iterations can improve the accuracy of the Taylor expansion by using the previous iteration's solution as the initial guess for the next iteration. This allows for a more refined approximation of the solution and can lead to faster convergence. Additionally, Picard Iterations can be combined with other numerical methods, such as Newton's method, to further improve the accuracy of the solution.

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