Mobility limits of the Lyapunov exponents of linear systems under small average perturbations

Mobility limits of the Lyapunov and central exponents of linear systems of differential equations, under arbitrarily small average linear perturbations are found with the aid of V. M. Millionshchikov's method of rotations. One obtains stability criteria for these exponents with respect to the mentioned perturbations, as well as criteria for the stabilizability or destabilizability of the zero solution.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 32–73, 1986.     

The Lyapunov exponents of regular systems under small on the average perturbations

Precise bounds of mobility of Lyapunov exponents are described in the class of regular systems under arbitrarily small on the average linear perturbations, and some estimates for these bounds are proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 13, pp. 166–176, 1988.     

Exact boundaries of upper mobility of the Lyapunov exponents of linear differential systems under exponentially decaying perturbations of the coefficient matrices

For a linear differential system, we obtain formulas for the computation of the exact boundaries of upper mobility for Lyapunov exponents under exponentially decaying perturbations of its coefficient matrix on the basis of the Cauchy matrix.     

Simultaneous attainability of central exponents of a linear Hamiltonian system under Hamiltonian perturbations

We show that, for any linear Hamiltonian system, there exists an arbitrarily close (in the uniform metric on the half-line) linear Hamiltonian system whose upper and lower Lyapunov exponents coincide with the upper and lower upper-limit central Vinograd–Millionshchikov exponents, respectively, of the original system and whose upper and lower Perron exponents coincide with the respective lower-limit exponents of the original system.     

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.     

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε ^?1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).     

On the properties of Lyapunov exponents of regular linear systems

We establish the sharp Baire class of the Lyapunov exponents on the space of Lyapunov regular linear systems with continuous bounded coefficients equipped with the topology of uniform or compact convergence of the coefficients on the half-line.     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

Lyapunov inequalities and stability for linear Hamiltonian systems

In this paper, we will establish several Lyapunov inequalities for linear Hamiltonian systems, which unite and generalize the most known ones. For planar linear Hamiltonian systems, the connection between Lyapunov inequalities and estimates of eigenvalues of stationary Dirac operators will be given, and some optimal stability criterion will be proved.     

Simultaneous attainability of the central exponents of small-dimensional linear Hamiltonian systems

We claim that the upper and lower central exponents of linear Hamiltonian systems of second and fourth orders are simultaneously attainable under uniformly small and infinitesimal Hamiltonian perturbations.     

On inequalities of Lyapunov for linear Hamiltonian systems on time scales

In this paper, we establish several new Lyapunov type inequalities for linear Hamiltonian systems on an arbitrary time scale T when the end-points are not necessarily usual zeroes, but rather, generalized zeroes, which generalize and improve all related existing ones including the continuous and discrete cases.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

The efficiency of perturbations in the class of linear Hamiltonian systems

It is proved that the set of all limiting values of solutions’ arbitrary indicators under uniformly small perturbations of coefficients of a linear Hamiltonian system is the same as the similar set obtained by uniformly small Hamiltonian perturbations.     

Nondiagonable systems under small perturbations

By a nondiagonable system, we mean a system whose state matrix is nondiagonable, i.e. having nonlinear elementary divisors. In this paper, closed form formulae are given for the shifts occuring in the eigenvalues and eigenvectors of these types of systems due to small variations in the system elements.

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Zusammenfassung Ein System heißt nicht-diagonalisierbar, wenn seine Zustandsmatrix nichtlineare Elementarteiler hat In dieser Arbeit werden geschlossene Formeln für die Änderungen der Eigenwerte und der Eigenvektoren eines nicht-diagonalisierbaren Systems angegeben, welche durch kleine Änderungen in den Systemelementen verursacht werden.

     

On the control of Lyapunov exponents of two-dimensional linear systems with locally integrable coefficients

We show that if a two-dimensional linear nonstationary control system with locally integrable and integrally bounded coefficients is uniformly completely controllable, then the complete spectrum of Lyapunov exponents of the corresponding closed system whose feedback is linear in the phase variables is globally controllable.