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:



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


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 .


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



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.


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

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