A Tour of Second-Order Ordinary Differential Equations

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This Demonstration is a tour of autonomous second-order ordinary differential equations (ODEs). The systems chosen represent most of the possible important qualitative behaviors. The general form of a second-order ODE is:

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Some of the systems are most naturally described in polar coordinates:

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The polar coordinates are then transformed to rectangular coordinates.

Phase portraits can be selected from a number of systems. Stable fixed points are indicated by solid disks, while unstable points are shown as open circles. Each system has a parameter that you can control using its slider bar. Drag the locator to highlight a single trajectory starting from any initial state. The dynamics of the selected trajectory can then be visualized using the slider bar for . To focus on a single trajectory only, set the density of the stream points to "none", select an initial state, and move the slider for .

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Contributed by: Brian Vick (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Some of the characteristics of the systems of ODEs selected for this tour are described. These insights suggest interesting explorations.

For the linear system, produces an unstable spiral, is a center, and produces a stable spiral.

For the van der Pol equation, reduces to a linear center. As increases, a limit cycle forms.

The critical value of the parameter for all the bifurcation cases occurs at , where the number and stability of the fixed points change, causing a qualitatively different dynamical picture.

The saddle node has no fixed points for and fixed points at for .

The transcritical fixed points at and switch stability at .

The supercritical pitchfork has one stable fixed point at for . When , becomes unstable and a pair of symmetric stable points forms at .

The subcritical pitchfork has a stable fixed point at and a symmetric pair of unstable points at for . When , becomes unstable and all trajectories diverge.

For both the supercritical and subcritical Hopf cases, the origin is a stable spiral when and bifurcates to an unstable spiral when . When , the trajectories approach limit cycles.

Reference:

S. H. Strogatz, Nonlinear Dynamics and Chaos, Jackson, TN: Perseus Books Publishing, 1994.



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