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Bifurcation |
This gif illustrates the saddle-node bifurcation of the one-parameter family dy/dt = y^2 + a occurring at a=0. The parameter a varies from -4 to 4. Upper left: graphs of the right-hand side y^2 +a. Upper right: slope field with equilibrium solutions plotted. Lower left: bifurcation diagram. Lower right: solution to IVP y(0)=-1. |

Phase plane to x(t), y(t) |
This gif shows how a solution to a 2D system of differential equations (viewed as a parametrized curve in the phase plane) relates to the graphs of its component functions x(t) and y(t) viewed separately on the t-x and t-y planes. The system in question is in fact linear, exhibiting a spiral sink at the origin. |

Bifurcation of 2D system |
This gif illustrates bifurcations of a 2D system of differential equations. The parameter n varies from -4 to 10. Upper left: the path taken through the trace-determinant plane. Upper right: direction field with equilibrium solutions plotted. Lower left: solution to IVP (x(0),y(0))=(-2,2). Lower right: x(t) and y(t) graphs for the solution to this IVP. |

Near-resonance and resonance |
This gif shows graphs of particular solutions to an undamped harmonic oscillator with periodic forcing. The forcing frequency is varied. As the forcing frequency approaches the natural frequency, beats appear. When the forcing frequency exactly equals the natural frequency, resonance occurs. Visually, this looks like one infinitely-sized beat. |