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 ... .


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