Waldyr M. Oliva

Dynamics in infinite dimensions

Preface.- Introduction.- Invariant Sets and Attractors.- Functional Differential Equations on Manifolds.- The Dimension of the Attractor.- Stability and Bifurcation.- Stability of Morse-Smale Maps and Semiflows.- One-to-oneness, Persistence and Hyperbolicity.- Realization of Vector Fields and Normal Forms.- Attractor Sets as C1-Manifolds.- Monotonicity.- The Kupka-Smale Theorem.- Appendix A: Conley Index Theory in Noncompact Spaces.- References.- Index.

Jacobi matrices and transversality

The paper deals with smooth nonlinear ODE systems in ℝ n , ẋ = f ( x ), such that the derivative f ′( x ) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f ′( x ( t )) y . Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝ n is then explicitly constructed.

Transversality between Invariant Manifolds of Periodic Orbits for a Class of Monotone Dynamical Systems

We consider a special class of monotone dynamical systems and show that in this special class the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally. The proof is based on the existence of a family of positively invariant nested cones.

A Note on the Conservation of Energy and Volume in the Setting of Nonholonomic Mechanical Systems

This note concerns the analysis of conservation of energy and volume for a series of well known examples of nonholonomic mechanical systems, with linear and non-linear constraints, and aims to make evident some geometric aspects related with them.

Stability of Morse-Smale Maps

We will deal in this section with smooth maps f: B → E, B being a Banach manifold imbedded in a Banach space E. The maps f belong to Cr(B, E), the Banach space of all E-valued Cr-maps defined on B which are bounded together with their derivatives up to the order r ≥ l. Let Cr(B, B) be the subspace of Cr(B, E) of all maps leaving B invariant, that is, f(B) ⊂ B. Denote by A(f) the set

Functional Differential Equations—Generic Theory

\begin{array}{*{20}{c}} {A\left( f \right) = \left\{ {x \in B:\,there\;exists\,a\,sequence} \right.\,\left( {x = {{x}_{1}},{{x}_{2}}, \ldots } \right) \in B,} \\ {\left. {\mathop{{\sup }}\limits_{j} \left\| {{{x}_{j}}} \right\| < \infty \;and\,f\left( {{{x}_{j}}} \right) = {{x}_{{j - 1}}},j = 2,3, \ldots } \right\}.} \\ \end{array}

One-to-oneness for linear retarded functional differential equations

Dopamine (DA) is a neuromodulator in the brainstem involved with the generation and modulation of the autonomic and respiratory activities. Here we evaluated the effect of microinjection of DA intracisterna magna (icm) or into the caudal nucleus tractus solitarii (cNTS) on the baseline cardiovascular and respiratory parameters and on the cardiovascular and respiratory responses to chemoreflex activation in awake rats. Guide cannulas were implanted in cisterna magna or cNTS and femoral artery and vein were catheterized. Respiratory frequency (f(R)) was measured by whole-body plethysmography. Chemoreflex was activated with KCN (iv) before and after microinjection of DA icm or into the cNTS bilaterally while mean arterial pressure (MAP), heart rate (HR) and f(R) were recorded. Microinjection of DA icm (n=13), but not into the cNTS (n=8) produced a significant decrease in baseline MAP (-15+/-1 vs 1+/-1mmHg) and HR (-55+/-11 vs -11+/-17bpm) in relation to control (saline with ascorbic acid, n=9) but no significant changes in baseline f(R). Microinjection of DA icm or into the cNTS produced no significant changes in the pressor, bradycardic and tachypneic responses to chemoreflex activation. These data show that a) DA icm affects baseline cardiovascular regulation, but not baseline f(R) and autonomic and respiratory components of chemoreflex and b) DA into the cNTS does not affect either the autonomic activity to the cardiovascular system or the autonomic and respiratory responses of chemoreflex activation.