Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction-diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction-diffusion population model.     

Attractors for random dynamical systems

Summary A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.     

Time-delay coordinates and polynomial mappings

The use of time-delay coordinates to reconstruct mappings is well known and provides an important practical tool in studying real-world problems. In this note we formulate the underlying mathematical analysis in the natural context of polynomial mappings and real analytic systems. This is particularly well adapted to systems defined by simple algebraic equations where, unlike in the general case, we do not require techniques from differential geometry.     

On the cohomology of flows of stochastic and random differential equations

We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise. Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001     

Classifying 3-trip Lorenz knots

The topological types of closed periodic solutions of the Lorenz equations are in one-to-one correspondence with aperiodic positive words on two generators. The number of syllables in such a word is called the trip number of the corresponding knot. Classifications for knots with trip numbers 1 and 2 are known. This paper gives a complete classification for 3-trip Lorenz knots.     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

The theory of infinite-dimensional lie groups and its applications

The purpose of this paper is to survey the theory of regular Fréchet-Lie groups developed in [1–10]. Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of Lie and Cartan [11]. From the group theoretical standpoint, general relativistic mechanics is a more closed system than Newtonian mechanics. Quantized objects of these classical groups are closely related to the group of Fourier integral operators [12]. These can also be managed as regular Fréchet-Lie groups. However, there are many Fréchet-Lie algebras which are not the Lie algebras of regular Fréchet-Lie groups [13]. Thus, the enlargeability of the Poisson algebra is discussed in detail in this paper. Enlargeability is relevant to the global hypoellipticity [14, 15] of second-order differential operators.     

An application of a global inversion theorem to an existence and uniqueness theorem for a class of nonlinear systems of differential equations

In this paper, a class of systems of nonlinear differential equations at resonance is considered. With the use of a global inversion theorem which is an extended form of a non-variational version of a max–min principle, we prove that this class of equations possesses a unique 2π$2\pi$-periodic solution under a rather weaker condition, for existence and uniqueness, than those given in papers [J. Chen, W. Li, Periodic solution for 2k$2k$th boundary value problem with resonance, J. Math. Anal. Appl. 314 (2006) 661–671; F. Cong, Periodic solutions for 2k$2k$th order ordinary differential equations with nonresonance, Nonlinear Anal. 32 (1998) 787–793; F. Cong, Periodic solutions for second order differential equations, Appl. Math. Lett. 18 (2005) 957–961; W. Li, Periodic solutions for 2k$2k$th order ordinary differential equations with resonance, J. Math. Anal. Appl. 259 (2001) 157–167; W. Li, H. Li, A min–max theorem and its applications to nonconservative systems, Int. J. Math. Math. Sci. 17 (2003) 1101–1110; W. Li, Z. Shen, A constructive proof of existence and uniqueness of 2π$2\pi$-periodic solution to Duffing equation, Nonlinear Anal. 42 (2000) 1209–1220]. This result extends the results known so far.     

Primitive links of non-singular Morse-Smale flows on the special Seifert fibered manifolds

In this paper we consider a non-singular Morse-Smale flow Φ_t on an irreducible, simple, closed, orientable 3-manifold M. We define a primitive flow ψ_t from Φ_t, and call the link type of the closed orbits of ψ_t a primitive link of Φ_t. We show that the link types of the primitive links are finite and every non-singular Morse-Smale flow on M is obtained from a primitive flow by exchanging the flow in a regular neighborhood of attracting or repelling closed orbits.     

Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses

We investigate the dynamics of a class of multi-species predator-prey interaction models with Holling type III functional responses based on systems of nonautonomous differential equations with impulsive perturbations. Sufficient conditions for existence of a positive periodic solution are investigated by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the Lyapunov method.     

Some mathematical problems in computer vision

Several interesting mathematical problems arising in computer vision are discussed. Computer vision deals with image understanding at various levels. At the low level, it addresses issues like segmentation, edge detection, planar shape recognition and analysis. Classical results on differential invariants associated to planar curves are relevant to planar object recognition under partial occlusion, and recent results concerning the evolution of closed planar shapes under curvature controlled diffusion have found applications in shape decomposition and analysis. At higher levels, computer vision problems deal with attempts to invert imaging projections and shading processes toward depth recovery, spatial shape recognition and motion analysis. In this context, the recovery of depth from shaded images of objects with smooth, diffuse surfaces require the solution of nonlinear partial differential equations. Here results on differential equations, as well as interesting results from low-dimensional topology and differential geometry are the necessary tools of the trade. We are still far from being able to equip our computers with brains capable to analyze and understand the images that can easily be acquired with camera-eyes; however the research effort in this area often calls for both classical and recent mathematical results.This work was supported in part by NSF grant DMS-8811084, Air Force Office of Scientific Research Grant AFOSR-90-0024, and the Army Research Office DAAL03-91-G-0019, and by the Technion Fund for Promotion of Research.     

On singular points of quasilinear differential and differential-algebraic equations

We show how certain singularities of quasilinear differential and differential-algberaic equations can be resolved by taking the solutions to be integral manifolds of certain distributions rather than curves with specific parametrization.     

Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces

This paper deals with the existence of mild L-quasi-solutions to the initial value problem for a class of semilinear impulsive evolution equations in an ordered Banach space E. Under a new concept of upper and lower solutions, a new monotone iterative technique on the initial value problem of impulsive evolution equations has been established. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. An example is also given.     

Minimizing movements and evolution problems in Euclidean spaces

We study evolution curves of variational type, called minimizing movements, obtainedvia a time discretization and minimization method. We analyze examples in Euclidean spaces, where some classes of minimizing movements are solutions of suitable ordinary differential equations of gradient flow type. Finally, we construct an example to show that in general these evolution curves are not maximal slope curves. Entrata in Redazione il 3 gennaio 1997.     

A method for computing symmetries and conservation laws of integra-differential equations

A method for computing symmetries and conservation laws of integro-differential equations is proposed. It resides in reducing an integro-differential equation to a system of boundary differential equations and in computing symmetries and conservation laws of this system. A geometry of boundary differential equations is constructed like the differential case. Results of the computation for the Smoluchowski's coagulation equation are given.     

GENERALIZED ERMAKOV SYSTEMS IN TERMS OF sl(2,R) INVARIANTS

Abstract

Conventional generalized Ermakov systems are shown to be a subset of the class of second order ordinary differential equations invariant under sl(2,R) symmetry. When the system is two-dimensional, it can be reduced to a one-dimensional time-dependent simple harmonic oscillator by a suitable choice of new time and distance variables.     

Continuity of attractors for a reaction-diffusion problem with nonlinear boundary conditions with respect to variations of the domain

In this work we show, for a class of dissipative semilinear parabolic problems, that the global compact attractor varies continuously with respect to parameters in the equations. Applications to a parabolic problem with nonlinear boundary conditions are also obtained.     

Solitons of curvature

An intristic geometry of surfaces is discussed. In geodesic coordinates the Gauss equation is reduced to the Schrödinger equation where the Gaussian curvature plays the role of a potential. The use of this fact provides an infinite set of explicit expressions for the curvature and metric of a surface. A special case is governed by the KdV equation for the Gaussian curvature. We consider the integrable dynamics of curvature via the KdV equation, higher KdV equations and (2+1)-dimensional integrable equations with breaking solitons.