# Solution and Stability of a 1-Periodic Differential Equation

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Consider an equation of the form

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Contributed by: Brian G. Higgins and Housam Binous (January 2013)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The example used in this Demonstration is adapted from [1].

The idea of a Poincaré map is to take an arbitrary initial value at and map it to the value of the solution at , which is . We represent the map as

.

Then the iterate of the map is given by

.

The intersection of with defines the fixed points of the Poincaré map. The fixed points (periodic solutions) of the Poincaré map are solutions to

.

The stability of a fixed point is found by determining the Floquet exponents (using Floquet theory):

.

Reference

[1] J. H. Hubbard and B. H. West, *Differential Equations: A Dynamical Systems Approach*, New York: Springer, 1991.

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