On the possibility of implementing the Landau–Hopf scenario of transition to turbulence in the generalized model “multiplier-accelerator”

In this paper, we consider two boundary-value problems for the multiplier-accelerator model taking into account spatial effects. We show that, under an appropriate choice of the control parameter, invariant tori of increasing dimensions arise in both boundary-value problems and the invariant torus of the highest dimension is stable. Our results are based on such methods of the theory of dynamical systems with infinite-dimensional phase spaces as the method of integral manifolds, the Poincaré method of normal forms, and F. Takens' plan for implementing the Landau—Hopf scenario as a cascade of Andronov—Hopf bifurcations. For solutions that belong to invariant tori, we obtain asymptotic formulas.