Lemma 4.5.1 on page 77 in the book Averaging Methods in Nonlinear Dynamical Systems

[Alone]

I need your advice on understanding a proof of a lemma from a book I am reading.
I asked my question in overflow:
https://mathoverflow.net/questions/...averaging-methods-in-nonlinear-dynamical-syst

Does anyone understand the proof and can answer my questions?

Thanks!

 

[jvicens]

Hi there,

I am happy to help you understand the proof of Lemma 4.5.1 from the book you are reading. I have taken a look at your question on MathOverflow and I see that you have already received some helpful responses. However, I will also provide my own interpretation of the proof and try to answer your questions.

Firstly, let's start with the statement of the lemma itself. It states that for a given dynamical system, if the dynamics are defined by a continuous flow and the system is bounded, then the average of the flow over a finite time interval is equal to the average of the system over the same interval. This is an important result in nonlinear dynamical systems and is often used in the analysis of such systems.

Now, let's move on to the proof. The first step is to define the average of the flow over an interval [t0, t1]. This is done by taking the integral of the flow over the interval and dividing it by the length of the interval (t1 - t0). This is a standard way of defining the average of a function over an interval.

Next, the proof uses the fact that the flow is continuous to show that the average of the flow over [t0, t1] is equal to the average of the flow over a smaller interval [t0, t1 - h] for any h > 0. This is because the flow is continuous, so as h approaches 0, the values of the flow over the smaller interval will approach the values of the flow over the larger interval. This is a key step in the proof and is based on the definition of continuity.

Finally, the proof uses the fact that the system is bounded to show that the average of the system over [t0, t1 - h] is equal to the average of the system over [t0, t1]. This is because the system is bounded, so the values of the system over the smaller interval will also approach the values of the system over the larger interval as h approaches 0.

Overall, the proof uses the definitions of continuity and boundedness to show that the average of the flow is equal to the average of the system. This is a very important result and has many applications in the analysis of nonlinear dynamical systems.

I hope this helps to clarify the proof for you. If you have any further questions, please do not hesitate to ask. Keep up