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These are replies submitted by mary120

@Carl Love 

#   How to solve the following set of nonlinear equations and plot variation of u[i01],u[i02] and phi[d0] versus delta[d] for alpha=0.01,delta[d]=1e-4? 

Eq1:=1.359375000*10^36*delta[d]*phi[d0]-delta[e]+delta[i]+1 = 0:

Eq2:=7.217742610*10^14*u[i10]*(1-2*phi[d0]/u[i10]^2)+2.282450621*10^15*delta[i]*u[i20]*(1-2*phi[d0]/u[i20]^2)-2.206601886*10^17*sqrt(2)*delta[e]*(4*alpha*phi[d0]^2/(3*alpha+1)-16*alpha*phi[d0]/(3*alpha+1)+(24*alpha+1)*exp(phi[d0])/(3*alpha+1)) = 0:

Eq3:=5.882341298*10^8*u[i10]^2+1.203367844*10^9*delta[e]/u[i10]-8.823511948*10^7*u[i20]^2-9.633614643*10^7*delta[e]/(delta[i]*u[i20]) = 0:

Eq4:=2.406735687*10^10*delta[e]/u[i10]+9.633614644*10^8*delta[e]/u[i20]-(delta[e]*(4*alpha/(3*alpha+1)-1)+1.510416667*10^37*delta[d]*phi[d0]^2)*(5.882341298*10^8*u[i10]^2+1.203367844*10^9*delta[e]/u[i10])-1.359375000*10^36*delta[d]*phi[d0]*((1.157401021*10^(-25)*Pi)*u[i10]*sqrt(u[i10]^2+8/Pi)*(1-2*phi[d0]/(u[i10]^2+0.4244131815e-1))^1.0+1.395921345*10^(-54)*u[i10]*sqrt(u[i10]^2+8/Pi)*phi[d0]^2*ln((phi[d0]^2/(1000000000000*(u[i10]^2+0.4244131815e-1)^2)+2.005078125*10^14/(1+0.1666666667e-1*delta[e]+.5000000000*delta[i])^1.0)/(phi[d0]^2/(1000000000000*(u[i10]^2+0.4244131815e-1)^2)+(1/1000000000000)*(1-2*phi[d0]/(u[i10]^2+0.4244131815e-1))^1.0))/(u[i10]^2+0.4244131815e-1)^2+(1.157401021*10^(-25)*Pi)*delta[i]*u[i20]*sqrt(u[i20]^2+8/Pi)*(1-2*phi[d0]/(u[i20]^2+0.8488263632e-1))^1.0+1.395921345*10^(-55)*delta[i]*u[i20]*sqrt(u[i20]^2+8/Pi)*phi[d0]^2*ln((phi[d0]^2/(1000000000000*(u[i20]^2+0.8488263632e-1)^2)+2.005078125*10^14/(1+0.1666666667e-1*delta[e]+.5000000000*delta[i])^1.0)/(phi[d0]^2/(1000000000000*(u[i20]^2+0.8488263632e-1)^2)+(1/1000000000000)*(1-2*phi[d0]/(u[i20]^2+0.8488263632e-1))^1.0))/(u[i20]^2+0.8488263632e-1)^2-4.484955822*10^(-6))+6.240139645*10^(-20)*delta[d]*(7.217742610*10^14*u[i10]*(1-2*phi[d0]/u[i10]^2)+2.282450621*10^15*delta[i]*u[i20]*(1-2*phi[d0]/u[i20]^2)-2.206601886*10^17*sqrt(2)*delta[e]*(4*alpha*phi[d0]^2/(3*alpha+1)-16*alpha*phi[d0]/(3*alpha+1)+(24*alpha+1)*exp(phi[d0])/(3*alpha+1))) = 0:

Thanks for your reply. 
Please, see the attached file.

How can I insert a legend like "n_{0p}/n_{0n}=0.1" in above graph? How can I change the legend font size?

Thanks for your help.
But, there was an error when I run the code (see below)!
I want to export (x,U1(x)) as a two-column .txt or .dat file.
About my third question, I look for a command like " if U1(x)>20, then exit" that stop the code for a specific condition!


Yes, I want to export the numeric data as a two-column txt file ( X, U1(x)). I need a command like " write(1,*) x, U1(x)" in fortran.



I have 3 questions and I hope you  help me:

1- How can I extract the numerical data of (x,U1) to use it in another software such as Origin?
2- What is your suggestion to plot the curves of, for example, (x,U1) for different parameters (T,k,beta) in one figure? How can I do that?
I need to do it for comparing the effects of different values of parameters.

3- How can I stop the numerical calculation when U1(x)>20 ?

@Carl Love 
Thank you for your reply. 


Thank you very much.

@Rouben Rostamian  
Thank you. I'll post the worksheet in future questions!

@Rouben Rostamian  


I want to plot the phase portrait and also the direction field of the following system of equations:

diff(x(t), t) = y(t)*(1+x(t)^2+y(t)^2),

diff(y(t), t) = x(t)*(1+x(t)^2+y(t)^2).

Taking the critical point  (0,0) and its eignvalues (-1,+1), it is clear that it is a saddle point.

Now, is there some trick to provide 'nice' initial conditions, such that the phase portrait shows clearly the solution curves?

I considered the following Ics:

[x(0)=-2, y(0)=-2], [x(0)=3, y(0)=0]

But it is not a good choice, because the solution curves are not nicely shown. How does one specify the initial conditions such that this happens? Is it just by trial and error?


@Rouben Rostamian  


I tried to use your code and its ICs for finding the solution of a similar planar dynamical systems with following coefficients

> a = f*((1/2)*r+1/2)+(1/2)*(1-f)*beta*(r+1)-1/M^2,

  b = -(1/8)*f*(r-3)*(r+1)-(1/8)*(1-f)*(r-3)*beta^2*(r+1)-3/(2*M^4),

 c = (1/48)*f*(3*r^2-14*r+15)*(r+1)+(1/48)*(1-f)*(3*r^2-14*r+15)*beta^3*(r+1)-5/(2*M^6),


where r=0.3, beta=0.24, f=0.42, M=1.65868;


 but unfortunately I encountered an error message about the initial conditions: ‘Error, (in DEtools/DEplot/CheckInitial) too many initial conditions:’.

 How should I define ICs for such parameters to plot a similar phase portrait?
 Sometimes the eigenvalues of the Jacobian are complex, how should change the ICs in such cases?


@Rouben Rostamian  
Thanks for your help.


Thanks for your answer. But,

1- Really, I use Maple11 and I couldn't find EXPLORE commend in its help. May you help me?

2- Also, according to the output of above answer, there isn't a specific value of (x_m<>0) that satisfies both condition (3), (6) for alpha=0.2,beta=0.3,gama=17, simultaneously.

Assuming u0>= (mue+mun*gama)^(-1/2), What do I need to do to find the point x_m<>0  where condition (3), (6) are met simultaneously? (Please, check the plot of A for alpha=0.43,beta=0.4,gama=15 and u0=0.635). 

3- How do I get the intersection of two curves A(x)=0 and diff(A(x),x)=0?

Thank you !

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