Isochronous systems and their quantization

We review recent results about classical isochronous systems characterized by the presence of an open (hence fully dimensional) region in their phase space in which all their solutions are completely periodic (i.e., periodic in all degrees of freedom) with the same fixed period (independent of the initial data provided they are inside the isochronicity region). We report a technique for generating such systems, whose wide applicability justifies the statement that isochronous systems are not rare. We also present an analogous technique applicable to a vast class of Hamiltonian systems and generating isochronous Hamiltonian systems. We also report some results concerning the quantized versions of such systems.     

Global stability and oscillations of dynamical systems describing synchronous electrical machines

A new mathematical model for a synchronous machine with strong excitation control is proposed. The limit load problem for synchronous machines is considered. The limit admissible load is estimated by the nonlocal reduction method. Criteria for the existence of circular solutions and limit cycles of the second kind for the model of a synchronous machine are obtained.     

On the stochastic stability of some two-dimensional dynamical systems

We study the stochastic stability of a two-dimensional diffusion process described by a system of two nonlinear Itô stochastic differential equations. The nonlinear drift operator is assumed to be additively separated with respect to the variables, and the diffusion is assumed to be degenerate with respect to only one of the two components of the process. We consider cases in which, in the expression for the nonlinear drift vector, two of the four functions of one variable are linear. We obtain sufficient conditions for the stochastic stability of such systems. As an example, we consider a stochastic system whose deterministic part is equivalent to the classical Lienard equation.     

Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems

General two-dimensional autonomous dynamical systems and their standard numerical discretizations are considered. Nonstandard stability-preserving finite-difference schemes based on the explicit and implicit Euler and the second-order Runge–Kutta methods are designed and analyzed. Their elementary stability is established theoretically and is also supported by a numerical example.     

Collapsed indecomposable continua in area preserving two-dimensional dynamical systems

While invariant indecomposable continua can occur in two dimensional area preserving dynamical systems, it is often the case that processes that would normally produce these continua instead produce a collapsed version of the continua because of the area preserving constraints. The collapsed continuum and the dynamics on it have a strong relationship to an indecomposable continuum in a related dynamical system. We also prove that the presence of a homoclinic point of a saddle point in such a system has a branch of its unstable manifold that is inaccessible from the complement of the closure of .

     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

Efficient methods in the search for periodic oscillations in dynamical systems

Efficient methods in the search for the periodic oscillations of dynamical systems are described. Their application to the sixteenth Hilbert problem for quadratic systems and the Aizerman problem is considered. A synthesis of the method of harmonic linearization with the applied bifurcation theory and numerical methods for calculting periodic oscillations is described.     

A stability criterion for periodic solutions of two-dimensional discontinuous dynamical systems

We establish stability conditions for periodic solutions of two-dimensional systems of ordinary differential equations with pulse influence. We study the properties of the jump operator for such systems. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1262–1266, September, 1999.     

A direct method for calculating Lyapunov quantities of two-dimensional dynamical systems

We discuss cosmological models involving homogeneous and isotropic Yang-Mills (YM) fields. Such models were proposed recently as an alternative to scalar models of cosmic acceleration. There exists a unique SU(2) YM configuration (generalizable to larger gauge groups) whose energy-momentum tensor is homogeneous and isotropic in space. It is parameterized by a single scalar field with a quartic potential. In the case of the closed universe the coupled YM-doublet Higgs system admits homogeneous and isotropic configurations as well. While pure Einstein-Yang-Mills (EYM) cosmology with the standard conformally invariant YM action gives rise to the hot universe, Einstein-Yang-Mills-Higgs (EYMH) cosmology has a variety of regimes which include inflationary stages and bounces, and exhibits global cycling behavior reminiscent of the multiverse developed in time. We also discuss other mechanisms of conformal symmetry breaking such as the string-inspired Born-Infeld (BI) modification of the YM action or field-theoretical quantum corrections.     

Boundary jets of holomorphic maps between strongly pseudoconvex domains

In this paper we consider jets taken at a fixed boundary point of germs of holomorphic diffeomorphisms which send one strongly pseudoconvex domain into another. We completely describe possible first and second jets and conditions of extremality in terms of the Chern-Moser normal forms of the domains.     

Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective

We conjecture that the well-known oscillations (3- to 5-yr and 10-yr cycles) of northern mammals are examples of subharmonic resonance which obtains when ecological oscillators (predator-prey interactions) are subject to periodic forcing by the annual march of the seasons. The implications of this hypothesis are examined through analysis of a bare-bones, Hamiltonian model which, despite its simplicity, nonetheless exhibits the principal dynamical features of more realistic schemes. Specifically, we describe the genesis and destruction of resonant oscillations in response to variation in the intrinsic time scales of predator and prey. Our analysis suggests that cycle period should scale allometrically with body size, a fact first commented upon in the empirical literature some years ago. Our calculations further suggest that the dynamics of cyclic species should be phase coherent, i.e., that the intervals between successive maxima in the corresponding time series should be more nearly constant than their amplitude—a prediction which is also consistent with observation. We conclude by observing that complex dynamics in more realistic models can often be continued back to Hamiltonian limits of the sort here considered.     

Center and isochronous center problems for quasi analytic systems

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved. This research is partially supported by the National Natural Science Foundation of China (10671179 and 10771196) and the Natural Science Foundation of Yunnan Province (2005A0092M)