On polylinear differential realization of the determined dynamic chaos in the class of higher order equations with delay

A. V. Banshchikov ; A. V. Lakeev ; V. A. Rusanov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2023), pp. 3-21.

Résumé

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of $M_2 $-operators and the functional apparatus of the entropy Relay–Ritz operator represent the basis of this theory. It has been shown that — in the case of the finite bundle of the controlled trajectory curves — the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay–Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

Détail

Keywords: differential modeling of chaotic dynamics, higher-order evolution equations with delay, multilinear non-autonomous differential realization, functional Relay–Ritz operator, principle of entropy maximum.

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