4 years, 76 days

## @Kitonum how to find those values algebr...

@Kitonum how to find those values algebraically?

I mean I have: epsilon = sqrt((x-2)^2(3-x))

and then I plug the values x=0, x=2 or 3 and get:

+-2*sqrt(3), 0.

Are those the values I am looking for?

I am not sure understand why?

## @Carl Love thanks. Perhaps you know how...

@Carl Love thanks.

Perhaps you know how to answer my following question from Murdock's text which I asked on stackexchange and didn't get an answer.

"Using the graph of y=(x-2)^2(x-3), sketch the bifurcation diagram of (x-2)^2(x-3)+epsilon^2=0 for all real epsilon.

Notice that this diagram is symmetrical about the x axis. Find the values of epsilon at the two pair pair bifurcation points".

My question is about the last sentence how to find these values?

## @Carl Love Well the equation is epsilon ...

@Carl Love Well the equation is epsilon = -(x-2)^2(x-3), in that case how to use the Bifurcation command here, I've only seen an example for the logisitc equation.

## @Rouben Rostamian  I know there'...

@Rouben Rostamian  I know there's a feature in maple where you can use iterativemaps, I looked at it in the help and also the web when searching bifurcation with maple.

Do you know perhaps know how to use this feature here?

Thanks!

@vv thanks.

## @vv How does it work for every t>0 ? ...

@vv How does it work for every t>0 ?

## @vv Is there a way for maple to show me ...

@vv Is there a way for maple to show me the calculation explicitly?

## @vv I agree, but in my case the argument...

@vv I agree, but in my case the argument depends on other terms that are explicit functions of t, so if we maximize the all whole argument, then if it were to depend on p_0(t) then the max would depend on max p_0(t).

Here are a few examples to what I mean:

\sup_{t\in [0,1/epsilon} |t+p_0(t)| = 1/\epsilon +\sup_{t\in [0,1/\epsilon)}|p_0(t)|

\sup_{t\in [0,1/epsilon) |t-p_0(t)| = 1/\epsilon +\max{-\inf_{t\in [0,1/epsilon)} |p_0(t)|,\sup_{t\in [0,1/epsilon) |p_0(t)|}

Both cases are legitimate instances of what I am searching, can it be done in maple 2017 with my case?

Obviously the dependence of \delta_1(\epsilon) will be on \sup_{t\in [0,1/\epsilon)} or \inf_{t\in [0,1/\epsilon} of |p_0(t)|

but I want to see this dependence explciitly.

## @tomleslie Thanks for that. I tried to ...

@tomleslie Thanks for that.

I tried to compute \delta1(\epsilon) but I get that it doesn't compute the max with respect to t.

I think the x dependence gets cancelled.

I am attaching a file with what I tried to do here.

Perturbation.mw

Can you help me with defining the correct \delta(\epsilon) that it will compute it correctly?

## I think I understand how to define most ...

I think I understand how to define most of the definitions here, but I am not sure how to define that p_0(t) indeed will be taken as a function of t and the max function will regard it as such.

Here are the functions that I've defined so far:

f := proc (x, y1, y2, z) x+y1^2*z end proc

h := proc (x, y1, y2) y1^2 end proc

q_0 := proc (s) exp(-s)*(q0+int(h(x0, cos(r), sin(r))-p0, r = 0 .. s)) end proc

g := proc (x, t) Limit(int(f(x, cos(s), sin(s), p_0(t)+q_0(s)), s = 0 .. T)/T, T = infinity) end proc

Now I need to compute the maximum such that p_0(t) will be regarded as an unkown function of t.

BTW, when I try to compute g(x,t) I get the following answer:

/    1    //           2
lim      |- ------ \\1360 cos(T)  q0 - 2720 cos(T) sin(T) q0
T -> infinity \  6800 T

- 680 cos(2 T) T p0 + 1360 sin(2 T) T p0

- 3400 cos(T) sin(T) exp(T) p_0(t) + 340 cos(2 T) T

+ 408 cos(2 T) p0 - 680 sin(2 T) T + 544 sin(2 T) p0

+ 100 cos(4 T) + 25 sin(4 T) + 136 cos(2 T) - 102 sin(2 T)

- 3400 exp(T) p_0(t) T - 6800 x T exp(T) + 2992 p0 exp(T)

- 4080 q0 exp(T) - 3400 T p0 - 1936 exp(T) + 1700 T - 3400 p0

\        \\
+ 2720 q0 + 1700/ exp(-T)/|
/

Why doesn't it compute the limit?

THanks.

## @Rouben Rostamian  yes you are righ...

@Rouben Rostamian  yes you are right, as I formed it doesn't seem meaningful.

I'll repost my question after I refine exactly what I am looking for.

## @acer can you give me exact instructions...

@acer can you give me exact instructions how to fix this? I mean what exactly did you do to fix this in your system?

Thanks!

## So I think I succesully installed both m...

So I think I succesully installed both maple 2017.3 and the toolbox for matlab.

The problem is while running maple through root, i.e running the root in the terminal, I still get the following error:

Exception in thread "Request id 1" java.lang.InternalError: XXX0 profile[1]: GL3bc -> profileImpl GL4bc !!! not mapped
at com.jogamp.opengl.GLProfile.computeProfileMap(GLProfile.java:2071)
at com.jogamp.opengl.GLProfile.initProfilesForDeviceCritical(GLProfile.java:1924)
at com.jogamp.opengl.GLProfile.initProfilesForDevice(GLProfile.java:1875)
at com.jogamp.opengl.GLProfile.initProfilesForDefaultDevices(GLProfile.java:1843)
at com.jogamp.opengl.GLProfile.access$000(GLProfile.java:80) at com.jogamp.opengl.GLProfile$1.run(GLProfile.java:230)
at java.security.AccessController.doPrivileged(Native Method)
at com.jogamp.opengl.GLProfile.initSingleton(GLProfile.java:216)
at com.jogamp.opengl.GLProfile.getProfileMap(GLProfile.java:2297)
at com.jogamp.opengl.GLProfile.get(GLProfile.java:988)
at com.jogamp.opengl.GLProfile.get(GLProfile.java:1004)
at com.maplesoft.mathdoc.view.plot.Plot3DCanvasView.<clinit>(Unknown Source)
at com.maplesoft.worksheet.view.WmiWorksheetView.wmiWorksheetView_init(Unknown Source)
at com.maplesoft.worksheet.view.WmiWorksheetView.<init>(Unknown Source)
at com.maplesoft.worksheet.view.WmiWorksheetView.<init>(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetWindowManager.createWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetWindowManager.createWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetWindowManager.createWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetWindowManager.createWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetStartupStrategy.createFirstWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetStartupStrategy.openFirstWorksheet(Unknown Source)
at com.maplesoft.worksheet.application.WmiGenericStartupStrategy.doStartup(Unknown Source)
at com.maplesoft.worksheet.application.WmiWorksheetStartupStrategy.doStartup(Unknown Source)
at com.maplesoft.application.Maple.doStartup(Unknown Source)
at com.maplesoft.application.Application.startup(Unknown Source)
at com.maplesoft.application.ServerProtocol$StartApplicationHandler.processCommand(Unknown Source) at com.maplesoft.application.ServerProtocol.executeCommand(Unknown Source) at com.maplesoft.application.ServerProtocol.processNextStep(Unknown Source) at com.maplesoft.application.ExchangeProtocol.executeProtocol(Unknown Source) at com.maplesoft.application.ApplicationManager$Listener.run(Unknown Source)
And maple initial app window gets stuck, I searched google for the first line error:

Exception in thread "Request id 1" java.lang.InternalError: XXX0 profile[1]: GL3bc -> profileImpl GL4bc !!! not mapped

But I am not sure that's relevant for my problem in maple.

Can anyone help me with this?

## @Carl Love  Here's the correct...

Here's the corrected code:

restart:  with(Statistics):  Seed:= randomize(): # set up random numbers   N:= 1000: # number of samples   deltax:= 0.1: # max displacement nm   nm:= 10^(-9): # nanometres   kB:= 1.381*10^(-23)/nm^2: # Boltzmann const J nm-2 K-1   T:= 300.0: # temperature K   kBT:= kB*T: # initial thermal energy    V:= x-> 0.01*kBT*x^2 /2: # pot'l energy kg nm2 s&ndash;2   Etot:= 0.0: # initial energy <E>   E2tot:= 0.0: # initial <E2>  Xtot:=0.0:#initial <X>  X2tot:=0.0:#initial<X2>   x1:= 0.0: # first guess of x in nm   E1:= V(x1): # first guess of energy   for i from 1 to N do # start loop step (2)   x2(i):= x1 + (rand()/1e12 - 0.5)*deltax: # new x position   E2:= V(x2(i)): # new PE   DeltaE:= E2 - E1: # energy difference  # next line is Metropolis sampling   if DeltaE <= 0.0 or exp(-DeltaE/kBT) > (rand()/(1e12)) then   x1:= x2(i): # save new configuration   E1:= E2: # save new energy   end if:   Etot:= Etot + E1: # always add to total <E>  E2tot:= E2tot + E1^2: # add to total <E2>  Xtot:= Xtot + x1: # add to total <X>   X2tot:= X2tot + x1^2: # add to total <X2>   end do: # end loop step (3)   # convert result back to SI units  Rx:=AutoCorrelation(<seq(x2(i),i=1..N)>,1000): #compute autocorrelation of x(t)  RE:=AutoCorrelation(<seq(E2(i),i=1..N)>,1000):#compute autocorrelation of E(t)  ss1:=seq(x2(i),i=1..N):  ss2:=seq(E2(i),i=1..N):  ColumnGraph(Rx,'color'="Black",'style'='polygon');  ColumnGraph(RE,'color'="Red",'style'='polygon');  plot(<ss1>,style=point,symbol=point, color=blue);  plot(<ss2>,style=point,symbol=point,color=green);   Eav:= (Etot/N)*nm^2; # <E> average step (4)    E2av:= (E2tot/N)*nm^4; # <E2>  Xav:=(Xtot/(N))*nm^(2);#<X> average  X2av:=(X2tot/(N))*nm^(4);#<X^(2)> average   CV:= (E2av-Eav^2)/(kB*T^2)/nm^2;

Just one thing that I am not sure of is the conditional, does it satisfy my official problem in my OP?

Here it is:

"perfroming individual steps by selecting an attmept to move x=+-a ,with equal probability, and choose whether to make the actual step according to energy change expected in that step."

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