Postinstability Models in Dynamics

This paper is devoted to the concept of instability in dynamical systems with the main emphasis on orbital, Hadamard, and Reynolds instabilities. It demonstrates that the requirement about differentiability in dynamics in some cases is not consistent with the physical nature of motions, and may lead to unrealistic solutions. Special attention is paid to the fact that instability is not an invariant of motion: it depends upon frames of reference, the metric of configuration space, and classes of functions selected for mathematical models of physical phenomena. This leads to the possibility of elimination of certain types of instabilities (in particular, those which lead to chaos and turbulence) by enlarging the class of functions using the Reynolds-type transformation in combination with the stabilization principle: the additional terms (the so-called Reynolds stresses) are found from the conditions that they suppress the original instability. Based upon these ideas, a new approach to chaos and turbulence as well as a new mathematical formalism for nonlinear dynamics are discussed.