Modelling and studying fuzzy dynamical systems

The goal of this paper is to further develop the Yu. P. Pyt'ev fuzzy set theory. Based on this, the author suggests a new class of differential equations for describing fuzzy dynamics. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Controllability of nonlinear fractional dynamical systems

In this paper we establish a set of sufficient conditions for the controllability of nonlinear fractional dynamical systems. The results are obtained by using the recently derived formula for solution representation of systems of fractional differential equations and the application of the Schauder fixed point theorem. Examples are provided to illustrate the results.     

Lie-point symmetries and nonlinear dynamical systems

Nonlinear symmetries of finite dimensional dynamical systems are related to nonlinear normal forms and center manifolds in the neighbourhood of a singular point. Certain abstract results can be used algorithmically to construct the normal forms and/or the center manifold up to a given order in the perturbation expansion. We also argue that for this task, approximate symmetries are as useful as exact ones.     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.     

Positional robust inversion in nonlinear dynamical systems

The article studies robust inversion of nonlinear dynamical systems using a known phase vector. Inversion algorithms are proposed for the case when the system dynamics is exactly known. These algorithms solve the inversion problem with any prespecified accuracy. Algorithms solving the inversion problem with perturbed system dynamics are also considered. Accuracy bounds are obtained for the various algorithms. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 5–18, 2003.     

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

On-line state estimation for nonlinear dynamical systems

A problem of estimation of states of a dynamical object on the base of incomplete and inexact measurements is studied. The focus is on a set-based estimation of states required for construction of optimal guaranteeing feedbacks. In particular, the aim is to construct at every current time instant polyhedral approximations of sets of all states consistent with the obtained measurements. A problem of constructing estimates, associated with these approximations, is called an optimal observation problem. This note presents a numerical method for solving optimal observation problems for nonlinear dynamical systems. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear matrix differential dynamical systems with fuzzy matrices

In this paper, we investigate linear first-order fuzzy matrix differential dynamical systems where the coefficients matrix is described by a fuzzy matrix. We show some properties of the matrix differential dynamical systems, and their phase portraits are described by means of examples.     

Metastability for nonlinear random perturbations of dynamical systems

In this paper we describe the long time behavior of solutions to quasi-linear parabolic equations with a small parameter at the second order term and the long time behavior of the corresponding diffusion processes.     

Integrability of nonlinear dynamical systems and differential geometry structures

Some aspects of the application of differential geometry methods to the study of the integrability of non-linear dynamical systems given on infinite-dimensional functional manifolds are considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 419–427, March, 1993.     

Compatibly bi-Hamiltonian superconformal analogs of Lax-integrable nonlinear dynamical systems

Compatibly bi-Hamiltonian superanalogs of the known Lax-integrable nonlinear dynamical systems are obtained by using a relation for the Casimir functionals of central extensions of the Lie algebra of superconformal even vector fields and its adjoint semidirect sum. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 7, pp. 887–900, July, 2006.     

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.     

Optimal disturbance rejection control for nonlinear impulsive dynamical systems

In this paper, we develop an optimality-based framework for addressing the problem of nonlinear–nonquadratic hybrid control for disturbance rejection of nonlinear impulsive dynamical systems with bounded exogenous disturbances. Specifically, we transform a given nonlinear–nonquadratic hybrid performance criterion to account for system disturbances. As a consequence, the disturbance rejection problem is translated into an optimal hybrid control problem. Furthermore, the resulting optimal hybrid control law is shown to render the closed-loop nonlinear input–output map dissipative with respect to general supply rates. In addition, the Lyapunov function guaranteeing closed-loop stability is shown to be a solution to a steady-state hybrid Hamilton–Jacobi–Isaacs equation and thus guaranteeing optimality.     

Bounded invariant verification for time-delayed nonlinear networked dynamical systems

We present a technique for bounded invariant verification of nonlinear networked dynamical systems with delayed interconnections. The underlying problem in precise bounded-time verification lies with computing bounds on the sensitivity of trajectories (or solutions) to changes in initial states and inputs of the system. For large networks, computing this sensitivity with precision guarantees is challenging. We introduce the notion of input-to-state (IS) discrepancy of each module or subsystem in a larger nonlinear networked dynamical system. The IS discrepancy bounds the distance between two solutions or trajectories of a module in terms of their initial states and their inputs. Given the IS discrepancy functions of the modules, we show that it is possible to effectively construct a reduced (low dimensional) time-delayed dynamical system, such that the trajectory of this reduced model precisely bounds the distance between the trajectories of the complete network with changed initial states. Using the above results we develop a sound and relatively complete algorithm for bounded invariant verification of networked dynamical systems consisting of nonlinear modules interacting through possibly delayed signals. Finally, we introduce a local version of IS discrepancy and show that it is possible to compute them using only the Lipschitz constant and the Jacobian of the dynamic function of the modules.     

Integrable nonlinear evolution equations and dynamical systems in multidimensions

The investigation of nonlinear evolution equations and dynamical systems integrable in multidimensions constitutes at present our main research interest. Here we survey findings obtained recently as well as over time: solvable equations (both PDEs and ODEs) are reported, philosophical motivations and methodological approaches are outlined. For more detailed treatments, including the display and analysis of solutions, the interested reader is referred to the original papers.On leave while serving as Secretary General, Pugwash Conferences on Science and World Affairs, Geneva, London, Rome.     

A nonrandom lyapunov spectrum for nonlinear stochastic dynamical systems

We present a nonrandom version of the Multiplicative Ergodic (Oseledec) Theorem for a nonlinear stochastic dynamical system on a smooth compact Riemannian Manifold M. This theorem characterises the a.s. asymptotic behaviour of the derivative system. Our approach (based on work of Furstenberg and Kifer, who deal with a linear system) is to consider an associated system on the projective bundle over M and to relate the behaviour of the theorem to the ergodic behaviour of this system. When the system has no random element, our work reduces to an alternative approach to the Multiplicative Ergodic Theorem for a diffeomorphism of M.