Differential Equations

  • Stan Wagon


A damped pendulum can be described by the differential equation x″ = −x ′−10sinx, where x represents the angular displacement and −x′ is the damping term. The image shows 24 solutions in the phase plane; the solutions almost always converge to one of the equilibrium points, the exception being the separatrix curves. The image also shows the equilibrium points in yellow, the nullcline curves (where x or x ′ vanish) in black, and the regions defined by the nullcline curves (where the direction of motion is either northeast, southeast, northwest, or southwest) in pastel colors. All of these features can be computed for any autonomous two-dimensional system.


Equilibrium Point Phase Plane Duffing Equation Wheel Center Symbolic Solution 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Stan Wagon
    • 1
  1. 1.Department of Mathematics and Computer ScienceMacalester CollegeSt. PaulUSA

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