Al86

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These are questions asked by Al86

I have this polynomial equation: (x-2)^2*(x-3)+epsilon =0, I want to draw a bifurcation diagram in the (epsilon , x) plane.

 

How to implement this in maple 2018?

 

Thanks!

 

I have the following ODE perturbation problem which I want maple to solve for me:

q'(\tau)=f(p(eps*\tau)+eps*q(\tau),r(eps*\tau)+s(\tau))-f(p(eps*\tau,r(eps*\tau)+s(\tau))-f(p(eps*\tau),r(eps*\tau))

 

where q(\tau)=q_0(\tau)+eps*q_1(\tau)

p(eps*\tau)=p_0(eps*\tau)+eps*p_1(eps*\tau)

s(\tau)=s_0(\tau)+eps*s_1(\tau)

r(eps*\tau)=r_0(eps*\tau)+eps*r_1(eps*\tau)

I want maple to expand every function that depends on eps in its arguments by a Taylor series around eps=0, i.e h(eps)=h(0)+eps*h'(0)

and also expand the difference above the fs with an eps-expansion around eps=0.

I did all this manually now I want to check if my calculations are correct, eventaully I want to equate same powers of eps of the RHS and LHS of the first ODE I wrote above.

 

Then how to use maple for this?

Thnaks.

 

I want to compute a limit via maple and that it will show me the way how to compute the limit.

 

The limit is:

\lim_{epsilon ->0, t\in [0,1]} 1/(exp((-1+(1-4*epsilon)^(0.5))/(2*epsilon))-exp((-1-(1-4*epsilon)^(0.5))/(2*epsilon)))*[exp((-1+(1-4*epsilon)^(0.5))/(2*epsilon)*t)-exp((-1-(1-4*epsilon)^(0.5))/(2*epsilon)*t)]/(exp(1-t)-exp(1-t/(epsilon)))

 

According to my book it should converge to 1.

I tried manually but got stuck.

 

I am a little bit clueless here, about how to use maple to calculate Christoffel symbol and Ricci tensor and scalar?

 

I read the help, but it got me confused; I have the metric ds^2 = du*dv+F(y,z)du^2+dy^2+dz^2

It's not Minkowski nor Cartesian, so how to use maple to calculate these symbols?

 

I am sure it's easy, but I didn't quite follow maple's instructions.

 

I want to compute the following supremum of the function.

I'll write in Latex code.

I have the following term which I want to estimate:

\delta_1(\epsilon):= \sup_{|x|<100}\sup_{ 0<= t<1/\epsilon} \epsilon \cdot |\int_0^t [ f(x,\cos s , \sin s , p_0(t)+q_0(s))-g(x,t)]ds|

where p_0(t) is an unknown function that depends on t alone.

f(x,y_1,y_2,z):=x+(y_1^2)*z

g(x,t):= \lim_{T\to \infty} 1/T \int_0^T f(x,\cos s , \sin s , p_0(t)+q_0(s))ds

q_0(s):= \exp(-s)(q_0(0)+\int_0^s (h(x_0(0),\cos(r), \sin(r))-p_0(0))dr)

where h(x,y_1,y_2):=y_1^2.

Can someone lend me a hand?

Thanks!

 

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