On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property
2019, Vol. 15, no. 1, pp. 97-108
Author(s): Anikushin M. M.
This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License
Author(s): Anikushin M. M.
We give a supplement to the Smith reduction theorem for nonautonomous ordinary differential
equations (ODEs) that satisfy the squeezing property in the case when the right-hand
side is almost periodic in time. The reduction theorem states that some set of nice solutions
(including the bounded ones) of a given nonautonomous ODE satisfying the squeezing property
with respect to some quadratic form can be mapped one-to-one onto the set of solutions of
a certain system in the space of lower dimensions (the dimensions depend on the spectrum of
the quadratic form). Thus, some properties of bounded solutions to the original equation can
be studied through this projected equation. The main result of the present paper is that the
projected system is almost periodic provided that the original differential equation is almost
periodic and the inclusion for frequency modules of their right-hand sides holds (however, the
right-hand sides must be of a special type). From such an improvement we derive an extension
of Cartwrightâ€™s result on the frequency spectrum of almost periodic solutions and obtain
some theorems on the existence of almost periodic solutions based on low-dimensional analogs
in dimensions 2 and 3. The latter results require an additional hypothesis about the positive
uniformly Lyapunov stability and, since we are interested in nonlinear phenomena, our existence
theorems cannot be directly applied. On the other hand, our results may be applicable
to study the question of sensitive dependence on initial conditions in an almost periodic system
with a strange nonchaotic attractor. We discuss how to apply this kind of results to the Chua
system with an almost periodic perturbation. In such a system the appearance of regular almost
periodic oscillations as well as strange nonchaotic and chaotic attractors is possible.
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This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License