Symmetric periodic oscillations in weakly dissipative systems with one degree of freedom are cons... more Symmetric periodic oscillations in weakly dissipative systems with one degree of freedom are considered. Results are obtained which are an extension of Zevin's (1980) stability criteria, established for quasi-linear systems with a harmonic force, insignificantly nonlinear systems, and a broad class of perturbing forces. The proposed approach can also be applied to systems with a nonunique equilibrium state.
A qualitative investigation of a pendulum whose suspension point oscillates harmonically in the h... more A qualitative investigation of a pendulum whose suspension point oscillates harmonically in the horizontal direction is presented. It is demonstrated that periodic oscillations exist in the case of which the system coordinate changes its sign twice during a period. The upper bound of these oscillations is determined; the corresponding amplitude-frequency response characteristics are obtained; and the sufficient stability condition, which does not require the calculation of the solution considered, is found. Numerical calculations are presented.
Asymptotically quadratic and subquadratic autonomous Hamiltonian systems are considered. Lower bo... more Asymptotically quadratic and subquadratic autonomous Hamiltonian systems are considered. Lower bounds for the number of periodic solutions with a prescribed minimal period are obtained. These bounds are expressed in terms of the numbers of frequencies corresponding to the critical points of the Hamiltonian. Results are based on a global analysis of families of periodic solutions emanating from these points. * 1991 Mathematics Subject Classifications: 58F14, 58F22.
Finding conditions for absolute stability of a system containing a linear part and a scalar nonli... more Finding conditions for absolute stability of a system containing a linear part and a scalar nonlinear sector restricted function is a classical Lur'e problem. Most of the corresponding results are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach to stability analysis of the problem based on a direct analysis of the corresponding integral Volterra equation about the input of the nonlinear block. The obtained sufficient stability criterion is applicable to non-autonomous systems with arbitrary time-varying delay in the feedback. The approach is extended to general time-varying systems including a linear block and norm bounded vector nonlinear terms with uncertain time-varying delays. The obtained delay-independent stability conditions are formulated in the terms of the transition matrix of the linear part and the norms of the nonlinear terms. The systems are indicated for which the obtained criteria are not only sufficient but also necessary for any delay function. The obtained results are applied to stability analysis of some systems previously studied in the literature; in all cases less conservative stability bounds are found.
Stability criteria of partly uncertain linear systems with variable coefficients play an importan... more Stability criteria of partly uncertain linear systems with variable coefficients play an important part in various fields of robust control. In this paper we consider linear systems with periodic coefficients that may vary within their known lower and upper bounds depending upon time as well. Stability analysis of systems with variable coefficients is often based on the Lyapunov functions method leading in many cases to rigid sufficient stability conditions. A new approach to stability analysis of such systems is described in this paper. Using this method we obtain an upper bound for the real parts of the characteristic exponents of the system solutions yielding sufficient stability criteria and find the cases in which these bounds are reached. This explicitly determines the "worst" possible periodic coefficients yielding the largest value of the maximal real part of the characteristic exponents. Certain results are extended to the case of nonperiodic time-dependent bounded coefficients.
The Myshkis problem of the maximum Lyapunov exponent of a first-order linear differential equatio... more The Myshkis problem of the maximum Lyapunov exponent of a first-order linear differential equation with an arbitrary bounded delay is solved. The result obtained is generalized to a system of equations of arbitrary order, whose matrix has real eigenvalues. A sufficient condition for exponential stability is obtained for a system with complex eigenvalues.
Nonlinear dynamical systems possess some properties which at the first sight look unexpected and ... more Nonlinear dynamical systems possess some properties which at the first sight look unexpected and even surprising (see, e.g., (1,2)). In this paper a collec- tion of new qualitative results relating to stability of such systems is presented. In particular, for periodic oscillations of systems with non-monotonic elastic forces, new stability and instability effects are found. For some systems with uncertain bounded terms, nec- essary and sufficient stability conditions are obtained; the surprising feature is that they are independent upon arbitrary time-varying delays in the uncertain terms. It is shown that the known mathematical model of a swing - a pendulum with a periodic length - is incor- rect. An unknown feature relating to regions of para- metrical resonances is found.
A system of linear differential equations with a Hurwitz matrix A and a variable delay is conside... more A system of linear differential equations with a Hurwitz matrix A and a variable delay is considered. The system is assumed to be stable if it is stable for any delay function τ(t) ≤ h. The necessary and sufficient condition for stability, expressed using the eigenvalues of the matrix A and the quantity h, is found. It is established that the function τ(t), corresponding to the critical value of h, is constant or piecewise-linear depending on to which eigenvalue of matrix A (complex or real respectively) it corresponds. In the first case, the critical values of h in systems with a variable and constant delay are identical and, in the second case, they differ very slightly.
Estimates for the spectrum of the Cauchy operator and logarithms of solutions of non-autonomous d... more Estimates for the spectrum of the Cauchy operator and logarithms of solutions of non-autonomous differential equations in the space, expressed in an arbitrary matrix norm, are found. For equations with periodic coefficients, the lower bound for the periods of "oscillatory" solutions is obtained. Similar results are derived for a nonlinear periodic equation with zero equilibrium position; in the autonomous case, they are valid for any periodic solution. All the estimates are attained at a scalar equation and, thereby, are accurate for any norm. The estimates are extended to equations with derivatives of any order. Using the obtained results, a condition for the uniqueness of an oscillatory solution is found; stability criteria for a linear Hamiltonian system with periodic coefficients and for periodic solutions of a nonlinear system (expressed through the period and any norm of the Hessian) are obtained. Types of matrices and functions, for which the Euclidean norm has a mi...
Symmetric periodic oscillations in weakly dissipative systems with one degree of freedom are cons... more Symmetric periodic oscillations in weakly dissipative systems with one degree of freedom are considered. Results are obtained which are an extension of Zevin's (1980) stability criteria, established for quasi-linear systems with a harmonic force, insignificantly nonlinear systems, and a broad class of perturbing forces. The proposed approach can also be applied to systems with a nonunique equilibrium state.
A qualitative investigation of a pendulum whose suspension point oscillates harmonically in the h... more A qualitative investigation of a pendulum whose suspension point oscillates harmonically in the horizontal direction is presented. It is demonstrated that periodic oscillations exist in the case of which the system coordinate changes its sign twice during a period. The upper bound of these oscillations is determined; the corresponding amplitude-frequency response characteristics are obtained; and the sufficient stability condition, which does not require the calculation of the solution considered, is found. Numerical calculations are presented.
Asymptotically quadratic and subquadratic autonomous Hamiltonian systems are considered. Lower bo... more Asymptotically quadratic and subquadratic autonomous Hamiltonian systems are considered. Lower bounds for the number of periodic solutions with a prescribed minimal period are obtained. These bounds are expressed in terms of the numbers of frequencies corresponding to the critical points of the Hamiltonian. Results are based on a global analysis of families of periodic solutions emanating from these points. * 1991 Mathematics Subject Classifications: 58F14, 58F22.
Finding conditions for absolute stability of a system containing a linear part and a scalar nonli... more Finding conditions for absolute stability of a system containing a linear part and a scalar nonlinear sector restricted function is a classical Lur'e problem. Most of the corresponding results are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach to stability analysis of the problem based on a direct analysis of the corresponding integral Volterra equation about the input of the nonlinear block. The obtained sufficient stability criterion is applicable to non-autonomous systems with arbitrary time-varying delay in the feedback. The approach is extended to general time-varying systems including a linear block and norm bounded vector nonlinear terms with uncertain time-varying delays. The obtained delay-independent stability conditions are formulated in the terms of the transition matrix of the linear part and the norms of the nonlinear terms. The systems are indicated for which the obtained criteria are not only sufficient but also necessary for any delay function. The obtained results are applied to stability analysis of some systems previously studied in the literature; in all cases less conservative stability bounds are found.
Stability criteria of partly uncertain linear systems with variable coefficients play an importan... more Stability criteria of partly uncertain linear systems with variable coefficients play an important part in various fields of robust control. In this paper we consider linear systems with periodic coefficients that may vary within their known lower and upper bounds depending upon time as well. Stability analysis of systems with variable coefficients is often based on the Lyapunov functions method leading in many cases to rigid sufficient stability conditions. A new approach to stability analysis of such systems is described in this paper. Using this method we obtain an upper bound for the real parts of the characteristic exponents of the system solutions yielding sufficient stability criteria and find the cases in which these bounds are reached. This explicitly determines the "worst" possible periodic coefficients yielding the largest value of the maximal real part of the characteristic exponents. Certain results are extended to the case of nonperiodic time-dependent bounded coefficients.
The Myshkis problem of the maximum Lyapunov exponent of a first-order linear differential equatio... more The Myshkis problem of the maximum Lyapunov exponent of a first-order linear differential equation with an arbitrary bounded delay is solved. The result obtained is generalized to a system of equations of arbitrary order, whose matrix has real eigenvalues. A sufficient condition for exponential stability is obtained for a system with complex eigenvalues.
Nonlinear dynamical systems possess some properties which at the first sight look unexpected and ... more Nonlinear dynamical systems possess some properties which at the first sight look unexpected and even surprising (see, e.g., (1,2)). In this paper a collec- tion of new qualitative results relating to stability of such systems is presented. In particular, for periodic oscillations of systems with non-monotonic elastic forces, new stability and instability effects are found. For some systems with uncertain bounded terms, nec- essary and sufficient stability conditions are obtained; the surprising feature is that they are independent upon arbitrary time-varying delays in the uncertain terms. It is shown that the known mathematical model of a swing - a pendulum with a periodic length - is incor- rect. An unknown feature relating to regions of para- metrical resonances is found.
A system of linear differential equations with a Hurwitz matrix A and a variable delay is conside... more A system of linear differential equations with a Hurwitz matrix A and a variable delay is considered. The system is assumed to be stable if it is stable for any delay function τ(t) ≤ h. The necessary and sufficient condition for stability, expressed using the eigenvalues of the matrix A and the quantity h, is found. It is established that the function τ(t), corresponding to the critical value of h, is constant or piecewise-linear depending on to which eigenvalue of matrix A (complex or real respectively) it corresponds. In the first case, the critical values of h in systems with a variable and constant delay are identical and, in the second case, they differ very slightly.
Estimates for the spectrum of the Cauchy operator and logarithms of solutions of non-autonomous d... more Estimates for the spectrum of the Cauchy operator and logarithms of solutions of non-autonomous differential equations in the space, expressed in an arbitrary matrix norm, are found. For equations with periodic coefficients, the lower bound for the periods of "oscillatory" solutions is obtained. Similar results are derived for a nonlinear periodic equation with zero equilibrium position; in the autonomous case, they are valid for any periodic solution. All the estimates are attained at a scalar equation and, thereby, are accurate for any norm. The estimates are extended to equations with derivatives of any order. Using the obtained results, a condition for the uniqueness of an oscillatory solution is found; stability criteria for a linear Hamiltonian system with periodic coefficients and for periodic solutions of a nonlinear system (expressed through the period and any norm of the Hessian) are obtained. Types of matrices and functions, for which the Euclidean norm has a mi...
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Papers by Alexandr Zevin