Versal unfoldings of predator-prey systems with ratio-dependent functional response

In this paper we study the versal unfolding of a predator-prey system with ratio-dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations. We first construct the unfolding and prove its versality and degeneracy of codimension 2. Then we discuss all its possible bifurcations, including transcritical bifurcation, Hopf bifurcation, and heteroclinic bifurcation, give conditions of parameters for the appearance of closed orbits and heteroclinic loops, and describe the bifurcation curves. Phase portraits for all possible cases are presented.     

Ważewski's Principle for retarded functional differential equations

Wa?ewski Principle is an important tool in the study of the asymptotic behavior of solutions of ordinary differential equations. A direct extension of this principle to retarded functional differential equations (RFDEs) can be obtained by noticing that solutions of RFDEs generate processes on C = C([?r, 0], Rⁿ) and by using the general version of Wa?ewski Principle for flows on topological spaces. The resulting method is of little use in applications, due to the infinite-dimensionality of the space C. This paper presents a “Razumikhin-type” extension of Wa?ewski's Principle, which is widely applicable to concrete examples. The main results are Corollaries 3.1 and 3.2. Also, an extension of the method to RFDEs with a merely continuous right-hand side is given, and a few examples illustrate the use of the method. Throughout the paper, a standard notation is used.     

Curvature for curves in semi-Euclidean spaces

We show a formula for curvatures of curves in a semi-Euclidean space (or pseudo-sphere) with respect to Frenet–Serre type frame in terms of volumes. We also investigate versality of height unfolding and distance squared unfolding for a curve.     

Random fuzzy differential equations under generalized Lipschitz condition

We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.     

Topological dynamics of retarded functional differential equations

We prove that a local flow can be constructed for a general class of nonautonomous retarded functional differential equations (RFDE). This is an extension to a result of Artstein (J. Differential Equations 23 (1977) 216) and fits in the classical theory of R. Miller and G. Sell. The main tool in this paper are generalized ordinary differential equations according to Kurzweil (Czech. Math. J. 7 (82) (1957) 418). In obtaining our results, we must prove the space of RFDEs can be embedded in a space of generalized ordinary differential equations. In opposition to the technical hypotheses of Oliva and Vorel (Bol. Soc. Mat. Mexicana 11 (1996) 40), this auxiliary result, as we present, is advantageous in the sense that our assumptions have an explanatory character. Applications based on topological dynamics techniques follow naturally from our results. As an illustration of this fact we show how to achieve in this setting a theorem on continuous dependence on initial data of solutions of RFDEs.     

Suboptimality Estimates for the Semi-Discretized LQR Problem for Parabolic PDEs

The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite-dimensional Hilbert space equivalent of the finite-dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite-dimensional case become operator equations in the infinite-dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite-dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

Toroidal normal forms for bifurcations in retarded functional differential equations I: Multiple Hopf and transcritical/multiple Hopf interaction

For finite-dimensional bifurcation problems, it is well known that it is possible to compute normal forms which possess nice symmetry properties. Oftentimes, these symmetries may allow for a partial decoupling of the normal form into a so-called “radial” part and an “angular” part. Analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings. In this paper, we are interested in the case where such bifurcations occur in retarded functional differential equations, and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay-differential equations. Our analysis allows us to recover and considerably generalize recent results by Faria and Magalhães [T. Faria, L.T. Magalhães, Restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities, J. Dynam. Differential Equations 8 (1996) 35-70] and by Buono and Bélair [P.-L. Buono, J. Bélair, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations 189 (2003) 234-266].     

The Superalgebra spl(p,q) and Differential Operators

Series of finite-dimensional representations of the superalgebrasspl(p,q) can be formulated in terms of linear differentialoperators acting on a suitable space of polynomials. We sketch the generalingredients necessary to construct these representations and presentexamples related to spl(2,1) and spl(2,2). By revisiting the products ofprojectivised representations of sl(2), we are able to construct new sets ofdifferential operators preserving some space of polynomials in two or morevariables. In particular, this allows us to express the representation ofspl(2,1) in terms of matrix differential operators in two variables. Thecorresponding operators provide the building blocks for the construction ofquasi-exactly solvable systems of two and four equations in two variables.We also present a quommutator deformation of spl(2,1) which, by constructionprovides an appropriate basis for analyzing the quasi exactly solvablesystems of finite difference equations.     

GLOBAL EXISTENCE FOR SEMILINEAR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

This paper is concerned with the global existence of solutions for abstract retarded functional differential equations (RFDEs) with infinite delay,via a fixed point approach.Some sufficient conditions are established under which the existence of a globally mild solution are obtained by using Leray-Schauder.     

On the structure of a semigroup of operators with finite-dimensional ranges

In the present paper, we describe the structure of a strongly continuous operator semigroup T(t) (where T: ?₊ → End X and X is a complex Banach space) for which ImT(t) is a finite-dimensional space for all t > 0. It is proved that such a semigroup is always the direct sum of a zero semigroup and a semigroup acting in a finite-dimensional space. As examples of applications, we discuss differential equations containing linear relations, orbits of a special form, and the possibility of embedding an operator in a C ₀-semigroup.     

Global continuation of forced oscillations of retarded motion equations on manifolds

We investigate T-periodic parametrized retarded functional motion equations on (possibly) noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations we prove a global continuation result for T-periodic solutions. The approach is topological and is based on the degree theory for tangent vector fields as well as on the fixed point index theory.Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continuation principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions.     

A level set reduced basis approach to parameter estimation

We introduce an efficient level set framework to parameter estimation problems governed by parametrized partial differential equations. The main ingredients are: (i) an “admissible region” approach to parameter estimation; (ii) the certified reduced basis method for efficient and reliable solution of parametrized partial differential equations; and (iii) a parameter-space level set method for construction of the admissible region. The method can handle nonconvex and multiply connected regions. Numerical results for two examples in design and inverse problems illustrate the versatility of the approach.     

Finite-dimensional global attractors in Banach spaces

We provide bounds on the upper box-counting dimension of negatively invariant subsets of Banach spaces, a problem that is easily reduced to covering the image of the unit ball under a linear map by a collection of balls of smaller radius. As an application of the abstract theory we show that the global attractors of a very broad class of parabolic partial differential equations (semilinear equations in Banach spaces) are finite-dimensional.     

THEORY AND APPLICATIONS OF MONOTONE SEMIFLOWS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

The theory of monotone semiflows has been widely applied to functional differential equations (FDEs). The studies on the theory and applications of monotone semiflows for FDEs are very important and interesting. A brief des-cription of our recent works are as follows.By using general monotone semiflow theory, several results of positively invariant sets, monotone solutions and contracting rectangles of retarded functional differential equations(RFDEs) with infinite delay are gained under the assumption of quasimonotonicity; sufficient conditions for the existence, un-iqueness and global attractivity of periodic solutions are also established by combining the theory of monotone semiflows for neutral functional differential equations(NFDEs) and Krasnoselskii's fixed point theorem.     

Bogdanov-Takens bifurcation in a delayed Michaelis-Menten type ratio-dependent predator-prey system with prey harvesting

In this paper, we study a delayed Michaelis-Menten Type ratio-dependent predator-prey model with prey harvesting. By considering the characteristic equation associated with the nonhyperbolic equilibrium, the critical value of the parameters for the Bogdanov-Takens bifurcation is obtained. The conditions for the characteristic equation having negative real parts are discussed. Using the normal form theory of Bogdanov-Takens bifurcation for retarded functional differential equations, the corresponding normal form restricted to the associated two-dimensional center manifold is calculated and the versal unfolding is considered. The parameter conditions for saddle-node bifurcation, Hopf bifurcation and homoclinic bifurcation are obtained. Numerical simulations are given to support the analytical results.     

Versal braid monodromy

We extend the method of Zariski to determine the braid monodromy group of the discriminant of a versal unfolding of a hypersurface singularity from low-dimensional generic subunfoldings to highly non-generic ones. At the expense of an induction over adjacent singularities, it is thus possible to neglect genericity issues and perturb by very simple polynomials only. To cite this article: M. Lönne, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Systems of matrix rational differential equations arising in connection with linear stochastic systems with Markovian jumping

In this paper, a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled Riccati differential equations arising in connection with control of some linear stochastic systems is considered.The system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite-dimensional Hilbert space adequately choosen.This allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of Lyapunov type.Our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions.Conditions providing the existence of a maximal solution (minimal solution respectively) with respect to some classes of global solutions are presented. It is shown that if the coefficients of the equations are periodic functions all these special solutions (stabilizing, maximal, minimal) are periodic functions, too.Whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves.     

Finite-Dimensional Sturm–Liouville Vessels and Their Tau Functions

We introduce a theory of a class of finite-dimensional vessels, a concept originating from the pioneering work of Livšic (Soobshch Akad Nauk Gruzin SSSR 91(2):281–284, 1978). Our work may be considered as a first step toward analyzing and constructing Lax Phillips scattering theory for Sturm–Liouville differentiable equations on the half axis (0,∞) with singularity at 0. We also develop a rich and interesting theory of vessels with deep connections to the notion of the τ function, arising in non linear differential equations (LDE), and to the Galois differential theory for LDEs.