Polylinear Differential Realization of Deterministic Dynamic Chaos in the Class of Higher Order Equations with Delay

Abstract

A characteristic criterion (and its modifications) of the solvability of differential realization of the bundle of controlled trajectory curves of deterministic chaotic dynamic processes in the class of higher order bilinear nonautonomous ordinary differential equations (with and without delay) in the separable Hilbert space has been found. This formulation refers to inverse problems for the additive combination of higher order nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. This theory is based on constructs of tensor products of Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M₂ operators, and the functional apparatus of the Rayleigh–Ritz nonlinear entropy operator. It has been shown that, in the case of a finite bundle of controlled trajectory curves, the property of sublinearity of the given operator allows one to obtain sufficient conditions for the existence of such realizations. The results obtained in this study are partly of a review nature and can become the basis for the development (in terms of Fock spaces) of a qualitative theory of inverse problems of higher order polylinear evolution equations with generalized delay operators describing, for example, the modeling of nonlinear oscillators of the Van der Pol type or Lorentz strange attractors.