Find the fixed point of the following equation?

Questions
1

I have the following differential equation

x'[t] == v[t]/?,
v'[t] == G/Q (2*BesselJ[1, v[t - ?]] + (? v[t] - 3/4 ? v[t]^3)) - ?*x[t] - v[t]/Q

How to find the fixed point of the system.

?=2*pi*11.2*10^6, Q=70, G=2.5, ?=1, ?=.06, ?=15
2

Did you consider assigning values to the variables and plot x’(t); the intercepts on the x-axis will be your fixed points.

3

Sorry I forgot to mention about the parameter values. Actually in the equation of v’[t] one x[t] term is involved . I have tried to find the point using the following command

Plot[G/Q (2*BesselJ[1, v] + (\[Alpha] v - 3/4 \[Gamma] v^3)) - \[Omega]*x -   v/Q, {v, -4, 4}]

But I was not getting anything . Can you please tell me what mistake I made here.

4

It should be easy to integrate x’(t) to get x(t). Then we need to assign some variable values.

variablesList = {G -> 1, Q -> 1, \[Alpha] -> 1, 
  x -> 1, \[Omega] -> 1, \[Gamma] -> 1}

Now plot the equation with the assigned values replacing the variables. I see at least one stable fixed point with the simple assignment.

Plot[
 (G/Q (2*BesselJ[1, v] + (\[Alpha] v - 3/4 \[Gamma] v^3)) - \[Omega]*
     x - v/Q) /. variablesList, {v, -4, 4}]

You should not separate the variable assignments with commas. Use semicolons or a list of rules. Hope this helps.

5

Thank you for the reply can I use DSolve to get x(t).

6

I am not familiar with the type of equation(s) above, but if you know v(t) and one initial condition, it is possible to find x(t) with DSolve.

DSolve[x'[t] == v[t] \[Omega] /. v[t] -> Sin[\[Omega] t], x[t], t]