Hi! I have recently started some Maple in chaos in dynamical systems and I am thinking about counting experimental invariant denisty measure (which is in brief  "how often the point visits the given interval") for some discrete mappings (in this case it's logistic mapping 3.7*x*(1-x)).

restart;
with(plots);
x := array(1 .. 10^6 + 2);

x[1] := 0.2;
for i to 10^6 do
    x[i + 1] := 3.7*x[i]*(1 - x[i]);
end do;

counter := 0;
for i to 10^6 do
    if 0 <= x[i] and x[i] < 0.1 then counter := counter + 1; end if;
end do;
counter;
counter := 0;
for i to 0^6 do
    if 0.2 <= x[i] and x[i] < 0.3 then counter := counter + 1; end if;
end do;
counter;
..........

counter := 0;
for i to 10^6 do
    if 0.9 <= x[i] and x[i] < 1 then counter := counter + 1; end if;
end do;
counter;
display(plot([[0, 0], [0.3, 0]]), plot([[0.3, 74089], [0.4, 74089]]), plot([[0.4, 57290], [0.5, 57290]]), plot([[0.5, 86726], [0.6, 86726]]), plot([[0.6, 122087], [0.7, 122087]]), plot([[0.7, 269178], [0.8, 269178]]), plot([[0.8, 185490], [0.9, 185490]]), plot([[0.9, 115405], [1, 115405]]))

I don't know how can I automate this code. I need smaller intervals because I took length 0.1 which is not good enough.

I want to get something like this (it is for logistic map such as above but in below example they plotted this graph for 4*x*(1-x) )

thanks in advance 

Hello everyone,

The "rows and series" chapter is coming to an end. But im not getting this question. Ive got a feeling they are not really specific with this book. But that could just be me.

Any way here is the question:

"In classical physics there is the kinetic energy of a body with the mass m0 and the speed v given by E1=1/2*m0*v^2. According to Einstein the kinetic energy E2=(m*c^2)-(m0*c^2)=((m0*c^2)/sqrt(1-(v/c)^2))-m0*c^2, at which m is the relativistic mass with a speed v, and m0 the mass in rest. Further c is the speed of light. Wright down E2 as a linear function of v^2 and show that E2=E1 when v is small compared to c." 

Now i cant see what they did to get this answer:

A taylor series was probably used, the question before it also used a taylor series. 

If someone knows what they did. What did they do to get the the answer the book gave? 

Thank you!

Greetings,

The Function

But ODEsteps supports simliar ODEs (see attached).
Is the ODESteps command not generic enough to cover the pendulum or have I missed something?

ODESteps.mw

Hi everyone, how can i plot nonlinear phase portraithere k,w, alpha,K, k, gamma, beta are arbitrary constants and i have three equilibrium points:

I hope the resulting graphics are as follows :

How can I plot these phase portraits? Thanks in advance.

hello,
i want to animate a sequence from a nomeric solution i recived, i have a sequene for displacment and another one for time, is there a way to do that ? 

Has somebody experienced something like that before?

I was searching for a bug in a sheet, and was absolute unable to find out why some results in a sheet were different from another sheet.

Found out that it actually was a pure graphics problem. With normal zoom (100%) the minus signum is not visible.

Blowing up the zoom to 125% shows the signum again.

I have an expression of the form

Expr := n0*C[0] + n1*C[1] + ... + nk*C[k] + n = 0,

where the numbers n0,...,nk and n are known to Maple (after it made some calculations), whereas C[0],...,C[k] are undetermined.

I would like to know the values of all of n0,...,nK and n. For n0,...,nk, I found them with 

coeff(Expr, C[m], 1),

with m in 0,...,k. But I don't know how to get the value of the "independent term" n.

Can someone help me with this?

I have a list of lists made from a combinat and was wondering how to make a new list from it that only contains lists that the elements sum up to 0. 

For example:

L1 := [[1,1,1,1], [1,2,0,0], [0,0,0,0], [1,1,-2,0]];

and I want the result to be 

L2:= [[0,0,0,0], [1,1,-2,0]]

Thank you in advance! I'm new to maple so would appreciate the guidance!

I am studying something about complex numbers. 

What commands are specific to find in Maple for complex numbers ?

• complexplot()
• conformal()
• conformal3d()

It seems that there are a lot of standard calculus statements can be used by adding the word : complex 
Iam using here a package downloaded from Maple website : complex analysis for mathematics and engineering

Got the impression that some modern plot commands for complex numbers are not yet in this book present ..and how about other commands?

an_introduction_to_complex_numbers.mws

Is an analytic solution possible for this problem: find an expression for maximum of power subject to constraints f=0, g=0, h=0? The feasible range is: T1, T2, R, Rthhot, Rthcold, and Z>0.

I know how to solve it using either FindMaximum or Langrage multiplier if I assign numerical values to T1, T2, R, Rthhot, Rthcold, and Z. I'd like to know if it's possible to find an analytic solution if I leave T1, T2, R, Rthhot, Rthcold and Z as parameters

I tried this but it did not work:

f := (X, Y, Rload, Rth, S) -> -(T1 - X)/Rthhot + (X - Y)/Rth + S^2*X*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

g := (X, Y, Rload, Rth, S) -> (Y - T2)/Rthcold - (X - Y)/Rth - S^2*Y*(X - Y)/(R + Rload) + (-1)*0.5*S^2*Rload*(X - Y)^2/(R + Rload)^2;

h := (X, Y, Rload, Rth, S) -> S^2*Rth/R - Z;

power := (X, Y, Rload, Rth, S) -> S^2*(X - Y)^2*Rload/(R + Rload)^2;

LagrangeMultipliers(power(X, Y, Rload, Rth, S), [f(X, Y, Rload, Rth, S), g(X, Y, Rload, Rth, S), h(X, Y, Rload, Rth, S)], [X, Y, Rload, Rth, S]);

Hello everybody,

I managed to get to the point that i could start part 2 of the book series of applied Dutch math. 

This paragraph is about Taylor series.

Now i am being asked to find a solution for the taylorseries with a certain x value. That is all fine, Maple can spit it out. But to put in into something that is there with a sum sign in front of it is something else. I have to convert the solution into something that is written with "x to the k-ed, something something". 

Now i did find to solution to a not so complex one. I will add that one on the maple sheet. But there is this one that is really complex, and i cant get my head around how to get to the final solution that makes the sum go and work well. It does have some series to it. But i dont know how to find that one out. Is there some trick to make it work? 

Without further ado:

Thank you!

Greetings,

The Function

Download Mapleprimes_Book_2_Question_1.mw

Is Maple Flow available as individual licence similar to Maple? 

I want to compute the series expansion of i3_r wrt (x, y, z) at point (x=y=z=0):

i2   := (x,y) -> -(1/2)*I*(exp(I*x)*(sin(x)/x)-exp(I*y)*(sin(y)/y))/(x-y):
i3_r := -(1/2)*I*(i2(y,z)-i2(y,x))/(z-x);

My first attempt was to compute this mulltiple series expansion this way:

ordre := 3:
sx := convert( series(i3_r, x, ordre), polynom);
sy := convert( series(sx  , y, ordre), polynom);
sz := convert( series(sy  , z, ordre), polynom);

But this gives me sy=sz=0 whatever the expansion order.

I then do this:

sx :=              convert(series(i3_r , x, ordre), polynom):
sy := add(map(u -> convert(series(u    , y, ordre), polynom), [op(expand(sx))])):
sz := add(map(u -> convert(series(u    , z, ordre), polynom), [op(expand(sy))]));

and obtained non zero results for both sy and sz (but are they are correct ?).

Could you explain me what happens and tell me how to find the series expansion of i3_r wrt (x, y, z) ?

TIA

Download ODEprobz.mw

H.M.mw

Download H.M.mw