On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property
Received 21 January 2019; accepted 21 March 2019
2019, Vol. 15, no. 1, pp. 97108
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 righthand
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 onetoone 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 righthand sides holds (however, the
righthand 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 lowdimensional 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.
Download File PDF, 280.99 Kb 
References 

[1]  Anikushin, M. M., “On the Liouville Phenomenon in Estimates of Fractal Dimensions of Forced QuasiPeriodic Oscillations”, Vestnik St. Petersb. Univ. Math., 52:3 (2019) (to appear) 
[2]  Anikushin, M. M., Dimensional Aspects of Almost Periodic Dynamics, https://www.researchgate.net/project/DimensionalAspectsofAlmostPeriodicDynamicsAnalyticalandNumericalApproaches, 2019 
[3] 
Burkin, I. M., “Method of “Transition into Space of Derivatives”: 40 Years of Evolution”, Differ. Equ., 51:13 (2015), 
[4] 
Cartwright, M. L., “Almost Periodic Differential Equations and Almost Periodic Flows”, J. Differential Equations, 5 (1969), 
[5] 
Glendinning, P., Jäger, T. H., and Keller, G., “How Chaotic Are Strange NonChaotic Attractors?”, Nonlinearity, 19:9 (2006), 
[6]  Feudel, U., Kuznetsov, S., and Pikovsky, A., Strange Nonchaotic Attractors: Dynamics between Order and Chaos in Quasiperiodically Forced Systems, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 56, World Sci., Hackensack, N.J., 2006 
[7]  Fink, A. M., Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer, Berlin, 1974, 342 pp. 
[8]  Leonov, G. A., Kuznetsov, N. V., and Reitmann, V., Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Springer, Cham, 2019 (to appear) 
[9]  Levitan, B. M. and Zhikov, V. V., Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982 
[10] 
O'Brien, G. C., “The Frequencies of Almost Periodic Solutions of Almost Periodic Differential Equations”, J. Austral. Math. Soc., 17 (1974), 
[11] 
Popov, S. and Reitmann, V., “Frequency Domain Conditions for FiniteDimensional Projectors and Determining Observations for the Set of Amenable Solutions”, Discrete Contin. Dyn. Syst., 34:1 (2014), 
[12] 
Robinson, J. C., “Inertial Manifolds and the Strong Squeezing Property”, Nonlinear Evolution Equations & Dynamical Systems: NEEDS'94 (Los Alamos, N.M.), World Sci., River Edge, N.J., 1995, 
[13] 
Smith, R. A., “Massera's Convergence Theorem for Periodic Nonlinear Differential Equations”, J. Math. Anal. Appl., 120:2 (1986), 
[14] 
Smith, R. A., “Convergence Theorems for Periodic Retarded FunctionalDifferential Equations”, Proc. London Math. Soc. (3), 60:3 (1990), 
[15] 
Suresh, K., Prasad, A., and Thamilmaran, K., “Birth of Strange Nonchaotic Attractors through Formation and Merging of Bubbles in a Quasiperiodically Forced Chua's Oscillator”, Phys. Lett. A, 377:8 (2013), 
[16]  Yakubovich, V. A., Leonov, G. A., and Gelig, A. Kh., Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, Ser. Stab. Vib. Control Syst. Ser. A, 14, World Sci., River Edge, N.J., 2004 
This work is licensed under a Creative Commons AttributionNoDerivs 3.0 Unported License