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Decomposing a 4th order linear differential equation as a symmetric product

Mark van Hoeij (2002)

Banach Center Publications

Let L(y) = 0 be a linear differential equation with rational functions as coefficients. To solve L(y) = 0 it is very helpful if the problem could be reduced to solving linear differential equations of lower order. One way is to compute a factorization of L, if L is reducible. Another way is to see if an operator L of order greater than 2 is a symmetric power of a second order operator. Maple contains implementations for both of these. The next step would be to see if L is a symmetric product of...

Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs

Mehmet Emir Koksal (2016)

Open Mathematics

Necessary and sufficiently conditions are derived for the decomposition of a second order linear time- varying system into two cascade connected commutative first order linear time-varying subsystems. The explicit formulas describing these subsystems are presented. It is shown that a very small class of systems satisfies the stated conditions. The results are well verified by simulations. It is also shown that its cascade synthesis is less sensitive to numerical errors than the direct simulation...

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.

Déformations isomonodromiques, forme de Liouville, fonction τ

Bernard Malgrange (2004)

Annales de l’institut Fourier

Cet article améliore des résultats antérieurs de Miwa et de l’auteur sur la “fonction τ ” de l’équation de Schlesinger. On relie cette fonction à la forme de Liouville d’un groupe de lacets associé naturellement à cette équation

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...

Delay differential systems with time-varying delay: new directions for stability theory

James Louisell (2001)

Kybernetika

In this paper we give an example of Markus–Yamabe instability in a constant coefficient delay differential equation with time-varying delay. For all values of the range of the delay function, the characteristic function of the associated autonomous delay equation is exponentially stable. Still, the fundamental solution of the time-varying system is unbounded. We also present a modified example having absolutely continuous delay function, easily calculating the average variation of the delay function,...

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