@Doug Meade I am not familiar with Tex but yes its really helpful , thank you

@tomleslie thanks for your answer this is what I want.

@tomleslie 

you are so interesting. if you don't want to help me just don't answer me anymore. I can't fix all the error by reading the help of maple that's why I come here to ask for help and what you did is just : read help and fix it yourself. If I can fix it by reading the help why should I come here and write so much thing to you and you said it complete rubbish?

@tomleslie 

This time there is only one error which I have no idea why it happens so could you help me fix it?

This is the code of my cycle count.mw

I don't know why it doesn't work but most parts of the cycle have no error

because in this code count......mw which is exactly the same code to solve the ODE in the cycle works without any error.

@tomleslie 

I think this time the explaination is quite clear because I give you specific details. Sorry about my bad sentences, English is not my first language.

I explain the functions F() S() T() at the bottom of this code. hw2corr4.mw

@tomleslie 

sorry, I did not say it clearly. What I mean is I fixed the two errors you have mentioned but I didn't make this whole code work. Could you help me combine the original code and the cycle? 

@tomleslie its ok now. sorry for my mistake hw2corr3.mw

@tomleslie 

I have read all the 8 illustrations, thanks for your help, your code is exactly what I want. And this time could you help me fix a cycle? My cycle doesn't work at all because I dont know how to combine the cycle with this code. Let me show you everything I have changed based on my currently need and then I will tell you the aim of cycle.

1 add a new variable L at the top

2 local omega:=x-> 1+sin( eps*x ) to local omega:=x-> 1+cos( eps*x )

3 -10..8*evalf(Pi)/eps, to -10..L*evalf(Pi)/eps,    this L can control the number of periods  

4 a cycle writen by myself at the bottom. because it doesn't work so I put it together at the bottom, it's more clear.

The aim of cycle is to :1 count the times of q(t) jumps up and down in the whole interval. 2 wether q(t) jumps up or down at every jump (funtunetly what I need in this part is just the the first and second time) when I change the initial condition p(0) by a small step at the order smaller than the oder of epsilon , for example epsilon = 1/50 then the step should be something like 1/100 or 1/200. I believe this explaination is not clear but when you see the cycle I believe you will understand what I mean, if you don't please let me know.

this is the new code hw2corr3.mw

@tomleslie What should I do if I want to change the variable t into tau=epsilon*t ?

I want to get all the pictures in this code behivor_of_I(pendulum).new.mw respect to tau rather than t. It means that the t axis will be compressed. Could you help me with it?

@tomleslie Actually you lost a sign "-" in the funcion eta so your results are not correct. But I have changed it so it work really good now. Thanks for your code. This is the new code which is exactlly what I want. behivor_of_I_(pendulum).new.mw

And there exists another problem : It supposed to give more results when interval of t grows because all functions in the new code is periodic but it doesn't show more resualts after 3 period.

Could you help me fix it?

@Carl Love How to improve the eifficient of this code? behivor_of_I_(pendulum).amended.mw

@acer How to improve the eifficient of this code? behivor_of_I_(pendulum).amended.mw

@tomleslie I just checked that all functions in your code are as same as the functions in my code but I don't know why your results are not corrects. I am pretty sure that the behaviour of J in behivor_of_I_(pendulum).mw is correct.

@tomleslie 

The reason why I ask you to fix the code is that your code is wrong.

Your pictures of H and Phi are correct but J is wrong, it's my fault didn't find your mistake in the beginning.

Actually the results should be like this.behivor_of_I_(pendulum).mw

J should be jump about 2 time rather than 1/4 times in your code.

@tomleslie 

It's the same question in a different method. behivor_of_I_(pendulum).amended.mw

When I change the epsilon less than 1/20  the code doesn't work but when epsilon larger than 1/20 the code works pretty good. I think it's a problem about the efficiency.Because there is an error when epsilon less than 1/20.

Error, (in unknown) Maple was unable to allocate enough memory to complete this computation. Please see ?alloc

How to fix it?