Bifurcations in a harmonic oscillator equation

[smithnya]

Hello everyone,

I've been trying to figure out how to determine bifurcation values in a harmonic oscillator when either the spring constant α or damping coefficient β act as undefined parameters. I understand bifurcations in first-order DEs, but I can't figure them out in a second-order equation such as a harmonic oscillator. Could anyone give me an explanation or tip on how to achieve this?

Thanks in advance

 

[HallsofIvy]

The general second order, constant coefficients, homogeneous linear differential equation can be written
$$\frac{d^2y}{dx^2}+ B\frac{dy}{dx}+ Cy= 0$$

Yes, this can be interpreted as the motion of a spring where 'B' gives the damping and 'C' the spring force. The characteristic equation for this would be $r^2+ Br+ C= 0$ which can be solve by the quadratic equation:
[tex]r= \frac{-B\pm\sqrt{B^2- 4C}}{2}[/quote]

That equation has either (a) two real roots, (b) a single real root, (c) two complex roots (which gives oscilatory motion) depending upon the discriminant, [itex]\sqrt{B^2- 4C}[/tex]. The solution "bifurcates" when that is 0.

 

[smithnya]

Thanks so much. So what happens when you end up with complex roots or with two distinct roots?