@tomleslie 

Thank you very much.

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

@Rouben Rostamian  

Hi,

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  
Tanx!

@Rouben Rostamian  

Hi,

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?

Thanks

@Rouben Rostamian  
Thanks for your help.

@Kitonum 

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?

@acer 
Thank you !

@Carl Love Thanks!

@tomleslie 

First of all, thank you very much for your attention to my problem and for providing a solution to it.

I think there is a misunderstanding. I meant the proposed solution by @Christopher2222  (and not yours),  which was to use *coeff(A,epsilon)*  to extract all of epsilon's power, doesn't work for integer power of epsilon (I uploaded a screen shot of my Worksheet  for him). This fact was accepted by him in his next post.

****************************************************

> A := d*n0/dt+d*epsilon*n1/dt+d*epsilon^(3/2)*n2/dt+d*epsilon^2*n3/dt+epsilon*d*n0*u1/dx+epsilon^(3/2)*d*n0*u2/dx+d*n0*epsilon^2*u3/dx+d*epsilon^2*n1*u1/dx+d*epsilon^(5/2)*n1*u2/dx;

coeff(A, epsilon^2);
%;
Error, unable to compute coeff
> coeff(A, epsilon);
%;
Error, unable to compute coeff

****************************************************

Anyway, unfortunately I haven't had a chance to check your submitted code yet, but as I said before, your idea is definitely great.

Meanwhile, I also use the Mapel 11 version, which is apparently older than your version, so I hope there is no problem using your code.

Thank you and all the friends who tried to solve my problem.

@tomleslie 

Dear friend,

I could pull out the fractional power of epsilon from the expersion with coeff (), but it does not work for integer power of epsilon!!

Please, check the expersion!!

However, as you say, it seems that introducing a dummy variable is a good idea for integer power of epsilon!

Thanks

@Christopher2222 

Thanks Christopher2222 

Yes, d means differential with respect to x/t

Also, n0, n1,n2,u1,.. are functionn of x/t, but epsilon is a constant!

I tried to send the file, Please see the attached file.

Thanks Christopher2222 5030 

Yes, d means differential with respect to x/t

Also, n0, n1,n2,u1,.. are functionn of x/t, but epsilon is a constant!

I tried to send the file, Please see the attached file.

@Kitonum 

Thanks Sir!

But another question:

why can't I extract the coefficints of $epsilon^2$  or $epsilon^1$ with your proposed response?

see:

> A := d*n0/dt+d*epsilon*n1/dt+d*epsilon^(3/2)*n2/dt+d*epsilon^2*n3/dt+epsilon*d*n0*u1/dx+epsilon^(3/2)*d*n0*u2/dx+d*n0*epsilon^2*u3/dx+d*epsilon^2*n1*u1/dx+d*epsilon^(5/2)*n1*u2/dx; coeff(A, epsilon^2);
%;
Error, unable to compute coeff
> coeff(A, epsilon);
%;
Error, unable to compute coeff
 

@ecterrab 

Thanks you for your attention. Really, I have tried to do that by 'dcoeffs', but it did not work well.

I'll appreciate if you help me.

Please, see the attached file.