QUASIPERIODIC SOLUTIONS OF SYSTEMS

Let / s C(tT ) and let co be some basis. The function F(f) = f((ot) is called a quasi-periodic function with a frequency basis to. The definition of a quasiperiodic function implies that it is defined for all -oo r oo. The set of all these functions forms a linear space which is denoted by C(to). We introduce a norm in C(to) by 

The product (1.2) satisfies all the axioms of scalar product and, consequently, induces the norm 

by completing the space T(9 ) in norm ll-H., we obtain the (separable) Hilbert space if (O In this space, the theorem on compacmess is valid which states that from any infinite sequence bounded in norm of if one can always extract a subsequence convergent in norm of H (tr, where s r. 

By using the Riesz-Fischer theorem, one can easily show that the space iH (cT can be identified (in the sense of isomorphism) with the space L2( r ) of square integrable functions with the scalar product (1.2) (for r = 0), moreover, the Parseval equality 

Then nothing but a subspace of functions from // ( T ,) which have gene- 

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QUASIPERIODIC SOLUTIONS OF SYSTEMS WITH LAG. BUBNOV-GALERKIN S METHOD... 

Construction of Quasiperiodic Solutions of Systems uith Lag by Bubnov-Galerkin s Method... 

Galekrin s method for construction of quasiperiodic solutions of systems with lag. - In IX Intemat. Conf. on Nonlinear Oscillations, 107-108, Nauk. Dumka, Kiev, 1981. 

Given a quasiperiodic solution of the system (1.20), one can construct an invariant set M corresponding to this solution. The set M is defined by... 

The problem of existence of quasiperiodic solutions of the system (1.19) is thus reduced... 

In the uniform metric, any quasiperiodic function can be approximated by trigonometric polynomials as accurately as desired. Therefore, one can try to construct a quasiperiodic solution of the system (1.19) (or a periodic solution of the equation (1.24)) in the form of a sequence of trigonometric polynomials... 

Section 3 Construction of Quasiperiodic Solutions of Perturbed Systems... 127... 

On reducibility and construction of the solutions of systems of linear deferential-functional equations with quasiperiodic coefficients. Dif.Uravn., 15, (1979), 771-783. 

The solutions of these equations (the trajectories) will for long times (i.e., after transient effects associated with switching on the external parameters have decayed) approach so-called limit sets, which may be classified into fixed points (stationary states), limit cycles (periodic oscillations), mixedmode oscillations, quasiperiodic oscillations, and chaotic behavior. Transitions between these states may occur upon variation of the external parameters pk and are called bifurcations. Experimental evidence for these effects with the system CO + 02/Pt(110) will be briefly presented without going further into details of the underlying general theory (see 16, 17). 

Assume that the system (2.1) possesses a sufficiently smooth quasiperiodic solution jc (r) with frequency basis co. Then there exists a function (f) given on the torus such that A (r) = u (o>r) this function is a solution of the equation... 

By using Lemmas 3.3-3.S and Theorem 2.1, we now prove the existence of Bubnov-Gal kin s approximations for the system of differential equations (2.1) furthermore, we prove that these approximations converge to the exact quasiperiodic solution. 

Existence theorems of quasiperiodic solutions to nonlinear differential systems. Funkc. Ekvacioj, 15(1), (1972), 65-100. 

In particular, for a system of third order we have m = 1, i.e. its limit-quasiperiodic solutions have the form... 

If a finite-dimensional system has an almost-periodic solution, that is not quasi-periodic, then the coefficients An are linear compositions of a finite number of basis frequencies. .., ujrn with rational factors. Such solutions are called limit-quasiperiodic. For this case Pontryagin [112] had proven that the dimension m of the minimal set must satisfy the following inequality... 

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