Complexity and stability of soliton management in periodically modulated and random systems

A brief introduction is given to the concept of the soliton management, i.e., stable motion of localized pulses in media with strong periodic (or, sometimes, random) inhomogeneity, or conditions for the survival of solitons in models with strong time‐periodic modulation of linear or nonlinear coefficients. It is demonstrated that a class of systems can be identified, in which solitons remain robust inherently coherent objects in seemingly “hostile” environments. Most physical models belonging to this class originate in nonlinear optics and Bose‐Einstein condensation, although other examples are known too (in particular, in hydrodynamics). In this paper, the complexity of the soliton‐management systems, and the robustness of solitons in them are illustrated using a recently explored fiber‐optic setting combining a periodic concatenation of nonlinear and dispersive segments (the split‐step model) for bimodal optical signals (i.e., ones with two polarizations of light), which includes the polarization mode dispersion, i.e., random linear mixing of the two polarization components at junctions between the fiber segment. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

Exponential stability of uncertain dynamic systems including state delay

In this letter, the exponential stability of uncertain time-delay systems is investigated. Based on the Lyapunov method, a novel stability criterion has been derived in terms of matrix inequalities which can be easily solved using efficient convex optimization algorithms. Two numerical examples are included to show the effectiveness of the proposed method.     

Spectral and dispersion properties of pair soliton states in a quasi-one-dimensional anisotropic antiferromagnet withS=1/2

By solution of the Schrödinger equation in the continuum approximation, it is shown analytically that there exist excited eigenstates of the quasi-one-dimensional Ising antiferromagnet with spinS=1/2 in the form of spatially localized quantum states. Computer modeling of a discrete model of interacting solitons with allowance for the symmetry of the solutions gives eigenvalues of the Sturm sequence that differ from the solutions of the continuum approximation. The spectral and dispersion properties of the nonlinear bound states of lowest energy and the selection rules in resonance transitions in an external magnetic field applied parallel to and perpendicular to the axis of magnetic anisotropy are calculated.L. V. Kirenski Physics Institute, Siberian Branch, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 112–119, April, 1992.     

Modulation instability of soliton trains in fiber communication systems

The linear stability problem for a soliton train described by the nonlinear Schrödinger equation is exactly solved using a linearization of the Zakharov-Shabat dressing procedure. This problem is reduced to finding a compatible solution of two linear equations. This approach allows the growth rate of the soliton lattice instability and the corresponding eigenfunctions to be found explicitly in a purely algebraic way. The growth rate can be expressed in terms of elliptic functions. Analysis of the dispersion relations and eigerfunctions shows that the solution, which has the form of a soliton train, is stable for defocusing media and unstable for focusing media with arbitrary parameters. Possible applications of the stability results to fiber communication systems are discussed.     

Critical cases of stability. Converse implicit function theorem for dynamical systems with cosymmetry

Stability criteria of boundary equilibria for dynamical systems in the three critical cases, (n, k)=(3, 0), (2, 1), and (1, 1), are obtained.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 572–578, April, 1998.The author wishes to thank V. I. Yudovich for useful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01791 and by the International Science Foundation under grant NRQ000.     

Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay

The global exponential stability for a class of switched neutral systems with interval-time-varying state delay and two classes of perturbations is investigated in this paper. LMI-based delay-dependent and delay-independent criteria are proposed to guarantee exponential stability for our considered systems under any switched signal. The Razumikhin-like approach and the Leibniz–Newton formula are used to find the stability conditions. Structured and unstructured uncertainties are studied in this paper. Finally, some numerical examples are illustrated to show the improved results from using this method.     

The computation of stability boundaries in state space for a class of biochemical engineering systems

The stability of a class of biochemical processes defined by a set of m biochemical reactions involving n components is analysed. The processes operate in a continuous mode and possess at least two stable equilibrium states: the normal operating point and a biological wash out state. Using a canonical state representation of the process dynamics the geometric structure of the operating point's stability boundary is characterized. Numerical algorithms are developed to evaluate this boundary and to visualize its extent in state space. The proposed technique is illustrated with a representative engineering example.     

Critical chirality in elliptic systems

We establish the regularity in 2 dimension of $L^2$ solutions to critical elliptic systems in divergence form involving chirality operators of finite $W^{1,2}$-energy.     

On delay-dependent stability for a class of nonlinear stochastic systems with multiple state delays

Global asymptotic stability conditions for nonlinear stochastic systems with multiple state delays are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.     

Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations

By considering an isospectral eigenvalue problem, a hierarchy of soliton equations are derived. Two types of extensions are presented by enlarging the associated spectral problem. With the aid of generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures for the integrable extensions are established.     

Critical thresholds in hyperbolic relaxation systems

Critical threshold phenomena in one-dimensional 2×2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.     

Critical systems thinking and practice

Critical systems thinking and the methodologies associated with it were developed precisely to allow analysis of complex societal problems and intervention to resolve such problems. Early approaches employing systems ideas, such as operational research, systems analysis and systems engineering, were suitable for tackling certain well-defined problems, but were found to have limitations when faced with complex problems involving people with a variety of viewpoints and frequently at odds with one another. Systems thinkers responded with approaches such as system dynamics and organisational cybernetics to tackle complexity; soft systems methodology (SSM) and interactive planning to handle subjectivity; and critical systems heuristics to help the disadvantaged in situations involving conflict. There was a corresponding enlargement of the range of problem contexts in which they felt competent to intervene. It has been critical systems thinking, however, which has supplied the bigger picture, has allowed systems thinking to mature as a discipline and has set out how the variety of methodologies now available can be used together in a coherent manner to promote successful intervention in complex societal problem situations. This paper outlines, at the request of the editor of this special issue, my involvement in developing critical systems thinking and practice, describes its origins, nature and use, and sets out a programme for future research in the area.     

Critical damping in nonviscously damped linear systems

In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are named critical damping surfaces (or critical manifolds, since several parameters can be involved). In this paper the theoretical foundations to determine critical damping surfaces in nonviscously damped systems are established. In addition, a numerical method to obtain critical curves is developed. The approach is based on the transformation of the algebraic equations, which define implicitly the critical curves, into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.     

Critical values of autonomous Lagrangian systems

Let M be a closed manifold and a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Ma?é in [5]. Let c _u (L) denote the critical value of the lift of L to the universal covering of M and let c _a (L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, . Let c ₀ (L) denote the strict critical value of L defined as the smallest critical value of where ranges among all possible closed 1-forms. We show that c _a (L) = c ₀ (L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all , then . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy , thus answering in the negative a question raised by Ma?é. This example also shows that the inequality could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures. Finally, we describe a large class of Lagrangians for which c _u (L) is strictly bigger than the maximum of the energy restricted to the zero section of TM. Received: October 2, 1996     

Critical heuristics of social systems design

Health care planning, city and regional planning, energy and transportation planning, environmental design and other areas of social systems design are becoming issues of increasing concern to policy makers. Faced with complex steering problems that may have far-reaching societal implications, they except help from applied disciplines such as Operational Research, systems science, technology assessment, program evaluation, cost-effectiveness analysis, and similar approaches to scientifically informed planning. But these approaches offer little help in critically reflecting on the normative implications of the problem definitions and solution proposals they inspire.Critical Heuristics seeks to provide both the involved planners and affected citizens with a conceptual framework for identifying and discussing the normative implications of problem definitions, systems designs, program evaluations, etc. The paper introduces some key concepts of Critical Heuristics and briefly discusses their significance to ‘rational’ planning.     

Critical exponents of Ising-like systems in general dimensions

Critical exponents of Ising-like systems are calculated for the case where the dimension of space is non-integer. Calculations are performed within the framework of the fixed-dimension field theoretical approach. Renormalization group functions in the Callan-Symanzik scheme are considered directly in non-integer dimensions. Perturbation theory expansions are resummed with the use of the Pade-Borel transformation.In memory of Prof. D.N.ZubarevInstitute of Physics of Condensed Systems, Ukrainian Academy of Sciences. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 3, pp. 482–495, September, 1993.     

Delay-dependent stability for 2D systems with time-varying delay subject to state saturation in the Roesser model

This paper addresses the problem of delay-dependent stability of 2D systems with time-varying delay subject to state saturation in the Roesser model. By introducing diagonally dominant matrices, new delay-dependent conditions are obtained in terms of linear matrix inequalities (LMIs) where the lower and upper delay bounds along horizontal and vertical directions, respectively, are known. numerical examples are provided to demonstrate the proposed results.