Method of “transition into space of derivatives”: 40 years of evolution

Burkin, I. M.

doi:10.1134/s0012266115130054

Cited by 7 publications

(8 citation statements)

References 55 publications

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“…This condition is well-known in control theory ( [3,8,16]). For the proof of this fact we refer to [16].…”

Section: The Proofsmentioning

confidence: 84%

“…In [15] it was shown that for certain parameters the appearance of strange nonchaotic attractors in such a system is possible. Conditions (A3) (the squeezing property 3 ) and (A4) can be effectively verified for a certain class of control systems by using frequency-domain methods (see [13] and [3]). For various applications of frequency-domain methods in the dimension theory of dynamical systems (including, in particular, the study of dimensional-like properties of almost periodic oscillations) we refer to [1,2,8].…”

Section: Introductionmentioning

confidence: 99%

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On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property

Anikushin¹

2019

Nelin. Dinam.

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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 Cartwright'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 condition is well-known in control theory ( [3,8,16]). For the proof of this fact we refer to [16].…”

Section: The Proofsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property

Anikushin¹

2019

Nelin. Dinam.

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“…In particular, we are interested in such properties as the global stability, convergence and the Poincaré-Bendixson trichotomy. Many concrete examples are given by R. A. Smith (see [60,58] for convergence in periodic systems and [59,57,56,55] for the Poincaré-Bendixson theory; for the latter see also the papers of the author [6] and I. M. Burkin [10]) and some are given by the present author [1,3,5]. Since there are hundreds of papers and monographs studying these problems, we mention only few, which go in the spirit of (or extended by) our theory.…”

Section: Introductionmentioning

confidence: 97%

Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces

Anikushin¹

2020

Preprint

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We present a geometric theory of inertial manifolds for compact cocycles (non-autonomous dynamical systems), which satisfy a certain squeezing property with respect to a family of quadratic Lyapunov functionals in a Banach space. Under general assumptions we show that these manifolds posses classical properties such as exponential tracking, differentiability and normal hyperbolicity. Our theory includes and largely extends classical studies for semilinear parabolic equations by C. Foias, G. R. Sell and R. Temam (based on the Spectral Gap Condition) and by G. R. Sell and J. Mallet-Paret (based on the Spatial Averaging) and their further developments. Besides semilinear parabolic equations our theory can be applied also to ODEs, ODEs with delay (extending the inertial manifold theories of Yu. A. Ryabov, R. D. Driver and C. Chicone for delay equations with small delays), parabolic equations with delay and parabolic equations with boundary controls (nonlinear boundary conditions). In applications, the squeezing property can be verified with the aid of various versions of the Frequency Theorem, which provides optimal (in some sense) and flexible conditions. This flexibility gives a possibility in applications to obtain conditions for low-dimensional dynamics, including, in particular, developments of the Poincaré-Bendixson theory and convergence theorems from a series of papers by R. A. Smith.

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“…A review of several works extending the Poincaré-Bendixson theory for certain high-dimensional ODEs is contained in the paper of B. Li [21]. Another review for ODEs is done by I. Burkin [8], who especially treats the works of R. A. Smith and their connection with the frequency-domain methods (see also [20]). In [29,27,28] a comparison of the presented results with many others in the field of delay and parabolic equations is given by R. A. Smith.…”

mentioning

confidence: 99%

The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces

Anikushin

2020

Preprint

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We study semi-flows satisfying a certain squeezing condition with respect to the quadratic form of a bounded self-adjoint operator acting in a Hilbert space. Under certain compactness assumptions from our previous results it follows that there exists an invariant topological manifold that attracts all compact trajectories. In the case of a two-dimensional manifold we obtain an analog of the Poincaré-Bendixson theorem on the trichotomy of ω-limit sets. Moreover, we obtain the conditions of existence of an orbitally stable periodic trajectory. We present applications of our results to study certain nonlinear systems of delay equations and reaction-diffusion systems. For these the required operator is obtained as a solution to certain operator inequalities with the use of the Yakubovich-Likhtarnikov frequency theorem for C 0 -semigroups and its properties are established from the Lyapunov inequality and dichotomy of the linear part.

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Method of “transition into space of derivatives”: 40 years of evolution

Cited by 7 publications

References 55 publications

On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property

On the Smith Reduction Theorem for Almost Periodic ODEs Satisfying the Squeezing Property

Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces

The Poincaré-Bendixson theory for certain compact semi-flows in Banach spaces

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