In this paper, we propose a model of the water wheel under dry friction (the modi ed Lorenz-Malkus system). Dynamical features of this system are investigated and analyzed depending on the dry friction parameter. Particularly, we present the analysis of stationary points, Lyapunov exponents, bifurcation diagrams, and stochastic properties of the considered system. Based on the results of numerical simulations we showed that the dry friction may serve an effective mechanism to control the chaotic dynamics.
In this paper, we analyse basic facts of infinite matrix theory. We construct a similarity transform which allows one to represent matrices in a certain class of 5-th diagonal matrices of a difference operator in a diagonal or block-diagonal form. For such matrices, asymptotic estimates of eigenvalues and eigenvectors are obtained. Such matrices are considered in game theory. They are also used in the study fourth-order difference operators with growing potential. The problem of invariant subspaces is also considered.
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