Solution and Stability of a 1-Periodic Differential Equation

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Consider an equation of the form

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,

where is periodic with period 1, that is, . The particular function that we examine is taken from [1]:

.

Because the differential equation is non-autonomous in , the solution trajectory, denoted by , depends on both and .

By selecting the "time trajectory" tab and varying the sliders and , you can explore the solution's dependence on these two parameters. In all cases the solution approaches a 1-periodic solution , that is:

as .

Another interesting property of 1-periodic equations is

.

By selecting "amplitude shifted" and varying the sliders , , and , you can examine how the trajectory is affected by comparing it with . As noted earlier, all trajectories ultimately approach a 1-periodic state.

By selecting "time-amplitude shifted" and taking , you can see that the trajectories and are simply translates of each other in time.

By integrating the 1-periodic differential equation in time, the solution trajectory always approaches a periodic state, if that state is stable. It turns out for this 1-periodic equation that there is another 1-periodic state, but it is not stable and so cannot be found by time integration. One method for investigating the family of periodic states is to examine the Poincaré map drawn from the solution trajectories. By selecting "Poincaré map", you can examine the Poincaré map for the differential equation. You can vary the iterations on the map with the slider The map is constructed by finding the iterates of

.

The intersection of the Poincaré map with the line gives the fixed points for the map. In this case there are two fixed points that are 1-periodic solutions to the differential equation. The intersection near is an unstable fixed point. The stability of a fixed point can be deduced from the slope of the Poincaré map at the intersection point or by computing the Floquet exponents, which is done in this Demonstration. Negative exponents represent stable solutions.

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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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