how do i find the bifurcation for this:
d^x/dt^2 = -.3 x - (x - 1)/((x - 1)^2 + a^2)^(3/2) - (x + 1)/((x + 1)^2 + a^2)^(3/2)
we've tried using mathematica but it just runs forever...
Well by inspection,
dx/dt=-.3/2 x^2 - 1/2((x-1)^2+ a^2)^.5+1/2 (((x+1)^2+a^2)^.5
and then
x= -1/20x^3- 1/3 ((x-1)^2+ a^2)^3/2 + you finish it.
Did I miss something?
Finding the bifurcation points of a dynamical system can be challenging, especially for complex equations like the one you provided. There may not always be an analytical solution available, and numerical methods are often required.
However, I can suggest a general approach to help you find the bifurcation points for your equation:
1. Simplify the equation: To make the equation more manageable, you can try simplifying it by expanding the denominators and factoring out common terms.
2. Analyze the equilibrium points: Set the right-hand side of the equation to zero and solve for x. These values of x are the equilibrium points of the system. Analyze the stability of each equilibrium point by linearizing the system around those points.
3. Plot the phase portrait: Plot the graph of x versus its derivative, dx/dt, by treating dx/dt as an independent variable. This phase portrait will provide visual insights into the behavior of the system and help identify potential bifurcation points.
4. Numerical analysis: Utilize numerical methods to explore the behavior of the system. Algorithms like continuation methods (e.g., parameter continuation) or dynamical system software packages (e.g., AUTO) can help you track bifurcation points or study the behavior as a parameter varies.
5. Limiting values: As a parameter (in this case, 'a') changes, investigate the limiting values of x that might imply bifurcation points or changes in the system's behavior.
It's worth noting that finding bifurcation points can be a complex task and may require expert knowledge in dynamical systems analysis. If the equation still proves to be extremely challenging for Mathematica, you may want to seek assistance from experts in the field of dynamical systems or consider simplifying the equation further.
Remember to consult relevant literature, textbooks, or research papers that deal with similar equations, as they might provide additional insights or techniques for analyzing the bifurcation behavior of your system.