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Bifurcations in a harmonic oscillator equation

  1. Dec 2, 2011 #1
    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
  2. jcsd
  3. Dec 3, 2011 #2


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

    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 [itex]r^2+ Br+ C= 0[/itex] 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.
  4. Dec 3, 2011 #3
    Thanks so much. So what happens when you end up with complex roots or with two distinct roots?
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