Generic Bifurcations of Planar Filippov Systems via Geometric Singular Perturbations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Root system of singular perturbations of the harmonic oscillator type operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues μ k satisfying μ k+1 − μ k ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.

Regularization of discontinuous vector fields on R-3 via singular perturbation

Singular perturbations problems in dimension three which are approximations of discontinuous vector fields are studied in this paper. The main result states that the regularization process developed by Sotomayor and Teixeira produces a singular problem for which the discontinuous set is a center manifold. Moreover, the definition of' sliding vector field coincides with the reduced problem of the corresponding singular problem for a class of vector fields.

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in R-n without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

THREE TIME SCALE SINGULAR PERTURBATION PROBLEMS AND NONSMOOTH DYNAMICAL SYSTEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Geometric singular perturbartion theory for non-smooth dynamical systems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Analysis of dynamic process models for structural insight and model reduction .1. Structural identification measures

An important consideration in the development of mathematical models for dynamic simulation, is the identification of the appropriate mathematical structure. By building models with an efficient structure which is devoid of redundancy, it is possible to create simple, accurate and functional models. This leads not only to efficient simulation, but to a deeper understanding of the important dynamic relationships within the process. In this paper, a method is proposed for systematic model development for startup and shutdown simulation which is based on the identification of the essential process structure. The key tool in this analysis is the method of nonlinear perturbations for structural identification and model reduction. Starting from a detailed mathematical process description both singular and regular structural perturbations are detected. These techniques are then used to give insight into the system structure and where appropriate to eliminate superfluous model equations or reduce them to other forms. This process retains the ability to interpret the reduced order model in terms of the physico-chemical phenomena. Using this model reduction technique it is possible to attribute observable dynamics to particular unit operations within the process. This relationship then highlights the unit operations which must be accurately modelled in order to develop a robust plant model. The technique generates detailed insight into the dynamic structure of the models providing a basis for system re-design and dynamic analysis. The technique is illustrated on the modelling for an evaporator startup. Copyright (C) 1996 Elsevier Science Ltd

Hamiltonian formalism for nonlocal Lagrangians

A Hamiltonian formalism is set up for nonlocal Lagrangian systems. The method is based on obtaining an equivalent singular first order Lagrangian, which is processed according to the standard Legendre transformation and then, the resulting Hamiltonian formalism is pulled back onto the phase space defined by the corresponding constraints. Finally, the standard results for local Lagrangians of any order are obtained as a particular case.

A new approach to evaluate imperfection sensitivity in asymmetric bifurcation buckling analysis

A direct procedure for the evaluation of imperfection sensitivity in bifurcation problems is presented. The problems arise in the context of the general theory of elastic stability for discrete structural systems, in which the energy criterion of stability of structures and the total potential energy formulation are employed. In cases of bifurcation buckling the sensitivity of the critical load with respect to an imperfection parameter e is singular at the state given by epsilon =0, so that, a regular perturbation expansion of the solution is not possible. In this work we describe a direct procedure to obtain the relations between the critical loads, the generalized coordinates at the critical state, the eigenvector, and the amplitude of the imperfection, using singular perturbation analysis. The expansions are assumed in terms of arbitrary powers of the imperfection parameter, so that both exponents and coefficients of the expansion are unknown. The solution of the series exponents is obtained by searching the least degenerate solution. The formulation is here applied to asymmetric bifurcations, for which explicit expressions of the coefficients are obtained. The use of the method is illustrated by a simple example, which allows consideration of the main features of the formulation.

Continuity of the dynamics in a localized large diffusion problem with nonlinear boundary conditions

This paper is concerned with singular perturbations in parabolic problems subjected to nonlinear Neumann boundary conditions. We consider the case for which the diffusion coefficient blows up in a subregion Omega(0) which is interior to the physical domain Omega subset of R(n). We prove, under natural assumptions, that the associated attractors behave continuously as the diffusion coefficient blows up locally uniformly in Omega(0) and converges uniformly to a continuous and positive function in Omega(1) = (Omega) over bar\Omega(0). (C) 2009 Elsevier Inc. All rights reserved.

DAMPED WAVE EQUATIONS WITH FAST GROWING DISSIPATIVE NONLINEARITIES

Let a > 0, Omega subset of R(N) be a bounded smooth domain and - A denotes the Laplace operator with Dirichlet boundary condition in L(2)(Omega). We study the damped wave problem {u(tt) + au(t) + Au - f(u), t > 0, u(0) = u(0) is an element of H(0)(1)(Omega), u(t)(0) = v(0) is an element of L(2)(Omega), where f : R -> R is a continuously differentiable function satisfying the growth condition vertical bar f(s) - f (t)vertical bar <= C vertical bar s - t vertical bar(1 + vertical bar s vertical bar(rho-1) + vertical bar t vertical bar(rho-1)), 1 < rho < (N - 2)/(N + 2), (N >= 3), and the dissipativeness condition limsup(vertical bar s vertical bar ->infinity) s/f(s) < lambda(1) with lambda(1) being the first eigenvalue of A. We construct the global weak solutions of this problem as the limits as eta -> 0(+) of the solutions of wave equations involving the strong damping term 2 eta A(1/2)u with eta > 0. We define a subclass LS subset of C ([0, infinity), L(2)(Omega) x H(-1)(Omega)) boolean AND L(infinity)([0, infinity), H(0)(1)(Omega) x L(2)(Omega)) of the `limit` solutions such that through each initial condition from H(0)(1)(Omega) x L(2)(Omega) passes at least one solution of the class LS. We show that the class LS has bounded dissipativeness property in H(0)(1)(Omega) x L(2)(Omega) and we construct a closed bounded invariant subset A of H(0)(1)(Omega) x L(2)(Omega), which is weakly compact in H(0)(1)(Omega) x L(2)(Omega) and compact in H({I})(s)(Omega) x H(s-1)(Omega), s is an element of [0, 1). Furthermore A attracts bounded subsets of H(0)(1)(Omega) x L(2)(Omega) in H({I})(s)(Omega) x H(s-1)(Omega), for each s is an element of [0, 1). For N = 3, 4, 5 we also prove a local uniqueness result for the case of smooth initial data.

Regularity of solutions on the global attractor for a semilinear damped wave equation

We consider attractors A(eta), eta epsilon [0, 1], corresponding to a singularly perturbed damped wave equation u(tt) + 2 eta A(1/2)u(t) + au(t) + Au = f (u) in H-0(1)(Omega) x L-2 (Omega), where Omega is a bounded smooth domain in R-3. For dissipative nonlinearity f epsilon C-2(R, R) satisfying vertical bar f ``(s)vertical bar <= c(1 + vertical bar s vertical bar) with some c > 0, we prove that the family of attractors {A(eta), eta >= 0} is upper semicontinuous at eta = 0 in H1+s (Omega) x H-s (Omega) for any s epsilon (0, 1). For dissipative f epsilon C-3 (R, R) satisfying lim(vertical bar s vertical bar) (->) (infinity) f ``(s)/s = 0 we prove that the attractor A(0) for the damped wave equation u(tt) + au(t) + Au = f (u) (case eta = 0) is bounded in H-4(Omega) x H-3(Omega) and thus is compact in the Holder spaces C2+mu ((Omega) over bar) x C1+mu((Omega) over bar) for every mu epsilon (0, 1/2). As a consequence of the uniform bounds we obtain that the family of attractors {A(eta), eta epsilon [0, 1]} is upper and lower semicontinuous in C2+mu ((Omega) over bar) x C1+mu ((Omega) over bar) for every mu epsilon (0, 1/2). (c) 2007 Elsevier Inc. All rights reserved.

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system U(A+B)(t, s)0 <= s <= t <= T generated by the sum -(A(t) + B(t)) of two linear, time-dependent, and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express U(A+B)(t, s)0 <= s <= t <= T as the strong limit in C(8) of a product of the holomorphic contraction semigroups generated by -A (t) and - B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t) + B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND(t is an element of)[0,T] D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct U(A)(t, s)0 <=(s)<=(t)<=(T) very simply in terms of the semigroup generated by -A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrodinger type in quantum mechanics.

A Trotter-Kato product formula for a class of non-autonomous evolution equations

In this article dedicated to Professor V. Lakshmikantham on the occasion of the celebration of his 84th birthday, we announce new results concerning the existence and various properties of an evolution system UA+B(t, s)(0 <= s <= t <= T) generated by the sum -(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing G(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)(0 <= s <= t <= T) as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by -A(t) and -B(t), thereby getting a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D subset of boolean AND D-t epsilon[0,D-T](A(t)+B(t)) everywhere dense in B. We then mention several possible applications of our product formula to various classes of non-autonomous parabolic initial-boundary value problems, as well as to evolution problems of Schrodinger type related to the theory of time-dependent singular perturbations of self-adjoint operators in quantum mechanics. We defer all the proofs and all the details of the applications to a separate publication. (C) 2008 Elsevier Ltd. All rights reserved.