E1ementary Bifurcations of Non—Critical but N0n—Hyperbolic Invariant Tori

Consider the time-periodic peturbations of n-dimensional autonomous systems with non-hyperbolic but non-critical closed orbits in the phase space.The elementary bifurcations,such as the saddle-node,transcritical,pitchfork bifurcation to a non-hyperbolic but non-critical invariant torus of the unperturbed systems in the extended phase space(x,t),are sutdied.Some conditions which depend only on ithe original systems and can be used to determine the bifurcation structures of these problems are obtained.The theory is applied to two concrete examples.     

Coordinatewise estimates for vector outputs of multivariable phase control systems

Two classes of multidimensional phase control systems with differentiable vector periodic functions are considered, the class of continuous systems described by ordinary differential equations and the class of discrete systems described by difference equations. The number of cycle slips for angular coordinates in phase systems with differentiable nonlinearities is studied. The study is based on the direct Lyapunov method and uses periodic Lyapunov functions, extensions of the phase space of the system, and the Yakubovich-Kalman lemma. This lemma provides necessary and sufficient conditions for the existence of Lyapunov functions by using the transfer matrix of the linear part of the system. As a result, for phase systems possessing global asymptotics, frequency criteria making it possible to sharpen estimates for the deviation of angular coordinates from their initial values are obtained. These criteria contain multiparameter frequency inequalities with variable parameters satisfying certain algebraic inequalities.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

Integrable Systems on Phase Spaces with a Nonflat Metric

We study the integrability problem for evolution systems on phase spaces with a nonflat metric. We show that if the phase space is a sphere, the Hamiltonian systems are generated by the action of the Hamiltonian operators on the variations of the phase-space geodesics and the integrability problem for the evolution systems reduces to the integrability problem for the equations of motion for the frames on the phase space. We relate the bi-Hamiltonian representation of the evolution systems to the differential-geometric properties of the phase space.     

The structure of Gibbs states and phase coexistence for non-symmetric continuum Widom Rowlinson models

Summary We prove the existence of phase transitions in non-symmetric r-component continuum Widom-Rowlinson models. Our results are based on an extension of the Pirogov-Sinai theory of phase transitions in general lattice spin systems to continuum systems. This generalizes Ruelle's extension of the Peierls argument for lattices to symmetric continuum Widom-Rowlinson models. The Pirogov-Sinai picture of the low temperature phase diagram for spin systems goes over into a phase-diagram of the Widom-Rowlinson model at large fugacities z=(z₀,..., z _r–1). There is in z-space a point where the system has r-pure phases, lines with r–1 phases, two dimensional surfaces with r–2 phases, etc.Supported in part by the National Science Foundation Grant DMR 81-14726-01     

On the existence,uniqueness and stability of periodic solutions of large scale systems with unbounded delay

In this paper, we consider the periodic solution problems for the systems with unbounded delay, and the existence, uniqueness and stability of the periodic solution are dealt with unitedly. First we establish the suitable delay-differential inequality, then study seperately the problems of periodic solution for the systems with bounded delay, with unbounded delay and the Volterra integral-differential systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré's periodic operator in the dfferent phase spaces. A series of simple criteria for the existence, uniqueness and stability of these systems are obtained.     

The asymptotic behavior of degenerate oscillatory integrals in two dimensions

A theorem of Varchenko gives the order of decay of the leading term of the asymptotic expansion of a degenerate oscillatory integral with real-analytic phase in two dimensions. His theorem expresses this order of decay in a simple geometric way in terms of its Newton polygon once one is in certain coordinate systems called adapted coordinate systems. In this paper, we give explicit formulas that not only provide the order of decay of the leading term, but also the coefficient of this term. There are three rather different formulas corresponding to three different types of Newton polygon. Analogous results for sublevel integrals are proven, as are some analogues for the more general case of smooth phase. The formulas require one to be in certain “superadapted” coordinates. These are a type of adapted coordinate system which we show exists for any smooth phase.     

Symplectic topology of integrable dynamical systems. Rough topological classification of classical cases of integrability in the dynamics of a heavy rigid body

Physical and mechanical systems with four-dimensional phase space are considered. The classification of nondegenerate integral systems is studied. A “physical zone,’ i.e., the systems connected with real physical applications, is determined. Bibliography: 27 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1996, pp. 104–183.     

On maximally superintegrable systems

Locally any completely integrable system is maximally superintegrable system since we have the necessary number of the action-angle variables. The main problem is the construction of the single-valued additional integrals of motion on the whole phase space by using these multi-valued action-angle variables. Some constructions of the additional integrals of motion for the Stäckel systems and for the integrable systems related with two different quadratic r-matrix algebras are discussed. Among these system there are the open Heisenberg magnet and the open Toda lattices associated with the different root systems.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

A quadratic polynomial differential systemcan be identified with a single point of ?¹² through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in ?¹² having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.     

Generalized synchronization in discrete maps. New point of view on weak and strong synchronization

In the present Letter we show that the concept of the generalized synchronization regime in discrete maps needs refining in the same way as it has been done for the flow systems Koronovskii et al. [Koronovskii AA, Moskalenko OI, Hramov AE. Nearest neighbors, phase tubes, and generalized synchronization. Phys Rev E 2011;84:037201]. We have shown that, in the general case, when the relationship between state vectors of the interacting chaotic maps are considered, the prehistory must be taken into account. We extend the phase tube approach to the systems with a discrete time coupled both unidirectionally and mutually and analyze the essence of the generalized synchronization by means of this technique. Obtained results show that the division of the generalized synchronization into the weak and the strong ones also must be reconsidered. Unidirectionally coupled logistic maps and Hénon maps coupled mutually are used as sample systems.     

Singularly perturbed stochastic systems

Problems of singular perturbation of reducible invertible operators are classified and their applications to the analysis of stochastic Markov systems represented by random evolutions are considered. The phase merging, averaging, and diffusion approximation schemes are discussed for dynamical systems with rapid Markov switchings. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 1, pp. 25–34, January, 1997.     

Problems of practical stability and stabilization of motion in discrete-time systems

The paper considers some problems of practical stability of motion for systems of difference equations. Stability theorems and criteria are given for cases with various phase constraints. Stabilization of discretetime systems to practical stability levels is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 118–124, 1988.     

Approximating nonstationaryPh(t)/M(t)/s/c queueing systems

Nonstationary phase processes are defined and a surrogate distribution approximation (SDA) method for analyzing transient and nonstationary queueing systems with nonstationary phase arrival processes is presented. Regardless of system capacityc, the SDA method requires the numerical solution of only 6K differential equations, whereK is the number of phases in the arrival process, compared to theK(c+1) Kolmogorov forward equations required for the classical method of solution. Time-dependent approximations of mean and variance of the number of entities in the system and the number of busy servers are obtained. Empirical test results over a wide range of systems indicate the SDA is quite accurate.This research was partially funded by National Science Foundation grant ECS-8404409.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Global Structure of Planar Quadratic Semi-Quasi-Homogeneous Polynomial Systems

This paper study the planar quadratic semi-quasi-homogeneous polynomial systems(short for PQSQHPS). By using the nilpotent singular points theorem, blow-up technique, Poincaré index formula, and Poincaré compaction method, the global phase portraits of such systems in canonical forms are discussed. Furthermore, we show that all the global phase portraits of PQSQHPS can be-classed into six topological equivalence classes.     

Minimum-Phase Second-Order Systems and the Spectrum of the Semigroup Generator

Controller design is much easier for minimum-phase systems than for non-minimum phase systems. We show that a wide class of second-order in.nite-dimensional systems with velocity measurements are minimum-phase. The system does not need to be exponentially stable. We show that in general the spectrum can be quite arbitrary in the closed left half plane. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods

We devise a hybrid approach for solving linear systems arising from interior point methods applied to linear programming problems. These systems are solved by preconditioned conjugate gradient method that works in two phases. During phase I it uses a kind of incomplete Cholesky preconditioner such that fill-in can be controlled in terms of available memory. As the optimal solution of the problem is approached, the linear systems becomes highly ill-conditioned and the method changes to phase II. In this phase a preconditioner based on the LU factorization is found to work better near a solution of the LP problem. The numerical experiments reveal that the iterative hybrid approach works better than Cholesky factorization on some classes of large-scale problems.     

Stability of positive and monotone systems in a partially ordered space

We investigate properties of positive and monotone dynamical systems with respect to given cones in the phase space. Stability conditions for linear and nonlinear differential systems in a partially ordered space are formulated. Conditions for the positivity of dynamical systems with respect to the Minkowski cone are established. By using the comparison method, we solve the problem of the robust stability of a family of systems.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 462–475, April, 2004.