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Decoupling normalizing transformations and local stabilization of nonlinear systems

S. Nikitin (1996)

Mathematica Bohemica

The existence of the normalizing transformation completely decoupling the stable dynamic from the center manifold dynamic is proved. A numerical procedure for the calculation of the asymptotic series for the decoupling normalizing transformation is proposed. The developed method is especially important for the perturbation theory of center manifold and, in particular, for the local stabilization theory. In the paper some sufficient conditions for local stabilization are given.

Degenerate evolution problems and Beta-type operators

Antonio Attalienti, Michele Campiti (2000)

Studia Mathematica

The present paper is concerned with the study of the differential operator Au(x):=α(x)u”(x)+β(x)u’(x) in the space C([0,1)] and of its adjoint Bv(x):=((αv)’(x)-β(x)v(x))’ in the space L 1 ( 0 , 1 ) , where α(x):=x(1-x)/2 (0≤x≤1). A careful analysis of their main properties is carried out in view of some generation results available in [6, 12, 20] and [25]. In addition, we introduce and study two different kinds of Beta-type operators as a generalization of similar operators defined in [18]. Among the corresponding...

Degenerate Hopf bifurcations and the formation mechanism of chaos in the Qi 3-D four-wing chaotic system

Hongtao Liang, Yanxia Tang, Li Li, Zhouchao Wei, Zhen Wang (2013)

Kybernetika

In order to further understand a complex 3-D dynamical system proposed by Qi et al, showing four-wing chaotic attractors with very complicated topological structures over a large range of parameters, we study degenerate Hopf bifurcations in the system. It exhibits the result of a period-doubling cascade to chaos from a Hopf bifurcation point. The theoretical analysis and simulations demonstrate the rich dynamics of the system.

Degenerating Cahn-Hilliard systems coupled with mechanical effects and complete damage processes

Christian Heinemann, Christiane Kraus (2014)

Mathematica Bohemica

This paper addresses analytical investigations of degenerating PDE systems for phase separation and damage processes considered on nonsmooth time-dependent domains with mixed boundary conditions for the displacement field. The evolution of the system is described by a degenerating Cahn-Hilliard equation for the concentration, a doubly nonlinear differential inclusion for the damage variable and a quasi-static balance equation for the displacement field. The analysis is performed on a time-dependent...

Derivative of the norm of a linear mapping and its application to differential equations

František Tumajer (1992)

Applications of Mathematics

In this paper the notion of the derivative of the norm of a linear mapping in a normed vector space is introduced. The fundamental properties of the derivative of the norm are established. Using these properties, linear differential equations in a Banach space are studied and lower and upper estimates of the norms of their solutions are derived.

Derived cones to reachable sets of a nonlinear differential inclusion

Aurelian Cernea (2014)

Mathematica Bohemica

We consider a nonlinear differential inclusion defined by a set-valued map with nonconvex values and we prove that the reachable set of a certain variational inclusion is a derived cone in the sense of Hestenes to the reachable set of the initial differential inclusion. In order to obtain the continuity property in the definition of a derived cone we use a continuous version of Filippov's theorem for solutions of our differential inclusion. As an application, in finite dimensional spaces, we obtain...

Design of unknown input fractional-order observers for fractional-order systems

Ibrahima N'Doye, Mohamed Darouach, Holger Voos, Michel Zasadzinski (2013)

International Journal of Applied Mathematics and Computer Science

This paper considers a method of designing fractional-order observers for continuous-time linear fractional-order systems with unknown inputs. Conditions for the existence of these observers are given. Sufficient conditions for the asymptotical stability of fractional-order observer errors with the fractional order α satisfying 0 < α < 2 are derived in terms of linear matrix inequalities. Two numerical examples are given to demonstrate the applicability of the proposed approach, where the...

Determination of the diffusion operator on an interval

Ibrahim M. Nabiev (2014)

Colloquium Mathematicae

The inverse problem of spectral analysis for the diffusion operator with quasiperiodic boundary conditions is considered. A uniqueness theorem is proved, a solution algorithm is presented, and sufficient conditions for the solvability of the inverse problem are obtained.

Determining the domain of attraction of hybrid non–linear systems using maximal Lyapunov functions

Szabolcs Rozgonyi, Katalin M. Hangos, Gábor Szederkényi (2010)

Kybernetika

In this article a method is presented to find systematically the domain of attraction (DOA) of hybrid non-linear systems. It has already been shown that there exists a sequence of special kind of Lyapunov functions V n in a rational functional form approximating a maximal Lyapunov function V M that can be used to find an estimation for the DOA. Based on this idea, an improved method has been developed and implemented in a Mathematica-package to find such Lyapunov functions V n for a class of hybrid (piecewise...

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