Alexander M. Krasnosel'skii

Nonlinear resonance in systems with hysteresis

A general approach is suggested for the analysis of nonlinear resonance in control systems including an integral linear term and a nonlinear term with hysteresis. This approach uses a special property of nonlinear terms which is valid for many important classes of hysteresis operators. For systems with such hysteresis nonlinearities, we introduce a method to derive conditions for the existence of forced oscillations and (for systems with a parameter) of nonlinear resonance at infinity.

Continua of cycles of higher-order equations

where μj ≥ 0 and μ0 + · · · + μ −1 > 0. This estimate is a natural analog of the ordinary two-sided sector estimate |f(x)| ≤ q|x|. By virtue of (3), zero is an equilibrium of Eq. (1) for each λ. Our main theorems guarantee the existence of a continuum of cycles issuing from zero and going to infinity. (A rigorous definition is given in the following section.) The theorems contain assumptions about the polynomial (2) that guarantee the existence of such a continuum of cycles for Eq. (1) with an arbitrary continuous nonlinearity satisfying the estimate (3) for q < q0, where q0 is determined by the polynomial (2) and the numbers μj. We suggest a constructive method for the computation of q0, which can readily be implemented on a computer. In the examples given in the paper, MAPLE has been used to perform the computations. We point out that the theorems on global continua of cycles apply simultaneously to the entire class of equations (1) with common linear part and common coefficients q and μj in the estimate (3) of the continuous nonlinearity. No information on the differentiability of the nonlinearity at any point is required. Our method can also be used in the analysis of local continua of cycles. The main classical example of the existence of such continua is the theorems on Andronov–Hopf bifurcations (e.g., see [1] and the bibliography therein). The known theorems are based on the linearization of the equation around the equilibrium; information on the linearization permits one to prove the existence of a continuum of cycles in a small neighborhood of this point. Theorems based on the differentiability of the nonlinearity only at the equilibrium were stated for the first time in [2]. Similar theorems are valid for bifurcations at infinity [3]. In the present paper, we neither assume the differentiability nor use linearizations. The main results of [2, 3] for higher-order equations readily follow from the theorems given here. In conclusion, we present results that can be treated as conditions for the existence of cycles in situations where the nonlinearity is known only approximately. Here we prove the existence of a continuum of cycles in the domain lying between two concentric spheres; the equilibrium, whose exact position is not known, is localized in the ball bounded by the inner sphere. Although the stability of cycles is important, related issues are not discussed in the present paper. Their analysis requires additional information about the nonlinearities.

On a Nonlocal Condition for Existence of Cycles of Hysteresis Systems

For the higher-order autonomous equations with hysteresis nonlinearities, nonlocal criteria for existence of cycles were proposed. The basic theorems guarantee existence of at least one cycle and present simple additional general conditions for existence of a one-parametric continuum of different cycles in the equation.

Incomplete corrections in nonlinear problems

A general setting of incomplete iterations: u n+1 (x) = u n (x) + n (x))F u n (x) ? u n (x)] is considered. Using methods from the theory of positive operators general results about convergence of this are given. These results relate the convergence of the iterations to the series P n (x) and its divergence. As an application we consider asynchronous iteration with parallel processors.

Cycle Stability for Hopf Bifurcation Generated by Sublinear Terms

The paper is concerned with the study of small stable cycles of autonomous quasilinear systems depending on a parameter. Sufficient conditions are presented for the existence of such cycles for control theory equations with scalar nonlinearities. The principal distinction of the case considered from usual results on Hopf bifurcations is that the linear part of the problem is degenerate for all the parameter values (not only at a bifurcation point). Small sublinear nonlinearities play the main role in our results. The proofs are based on the theory of monotone operators.

Hopf bifurcations in resonance 2:1

Abstract In this paper Hopf bifurcations for equations of control theory are studied in strong resonance case 2:1. The harmonic linearization approach and topological methods allow to give the sharp analysis of the problem for the case of quadratic principal nonlinearity.

Existence of Continua of Cycles in Hamiltonian Control Systems

Simple conditions for the existence of global continua of cycles in autonomous Hamiltonian single-loop control systems are derived from ordinary linear two-sided nonlinearity estimates. Limiting bounds of sector estimates are determined by the properties of the linear part. The designed methods are useful in determining two-sided estimates for the period of a cycle in a sector.

A degenerate case of Andronov-Hopf bifurcation at infinity

New conditions are suggested for the origin of the Andronov-Hopf bifurcation at infinity in quasilinear control systems, i.e., the conditions of origin of the cycles as large as desired at parameter values that are close to the critical value.

Criteria of resonance origin in a single-circuit control system with saturation

Studies are made of forced periodic oscillations in a single-circuit control system with a parameter, the dynamics of which is described by resonance equations in the linear approximation. Criteria are suggested of the origin of resonance that is understood as an increase to infinity of the amplitude of forced oscillations in the approximation of the parameter to certain critical values. The principal results relate to the case when answers are defined by bounded nonlinearities of the order of a constant; the summands that decrease to zero are of no importance.

Strong Resonances at Hopf Bifurcations in Control Systems

The 0:1 and 1:1 resonances at Hopf bifurcations in control systems with a parameter are investigated. Conditions for the generation of cycles in the neighborhood of the equilibrium position and at infinity are formulated. Nonlinearities with a principle quadratic part and with a principle homogeneous part of the general (nonpolynomial) type in the neighborhood of the equilibrium position are separately studied. The main case of bounded saturation nonlinearities at infinity is also studied.