On the Possibility of Implementing the Landau–Hopf Scenario of Passage to Turbulence
Two boundary value problems for a nonlinear evolutionary partial differential equation have been considered. The possibility of the well-known Landau–Hopf scenario of passage to turbulence has been shown for these boundary value problems. А plan as the cascade of the Andronov–Hopf bifurcation to realization the scenario proposed later by F. Takens has been implemented. The studied boundary value problems can be interpreted as generalized forms of the well-known mathematical “multiplier–accelerator” model of macroeconomics or a form of the generalized Van der Pol equation with distributed parameters. The justification of the results is based on the application of methods of qualitative theory of partial differential equations such as the method of integral manifolds, the Poincaré–Dulac theory of normal forms, asymptotic methods of analysis of dynamic systems with infinite-dimensional phase space, and perturbation theory for linear differential operators. In particular, asymptotic formulas for the solutions forming invariant tori are obtained by application of the generalized Krylov–Bogoliubov method. DOI: 10.1134/S2304487X18050085