Carmen Chicone

Evolution Semigroups in Dynamical Systems and Differential Equations

Introduction Semigroups on Banach spaces and evolution semigroups Evolution families and Howland semigroups Characterizations of dichotomy for evolution families Two applications of evolution semigroups Linear skew-product flows and Mather evolution semigroups Characterizations of dichotomy for linear skew-product flows Evolution operators and exact Lyapunov exponents Bibliography List of notations Index.

Bifurcation of limit cycles from quadratic isochrones

Abstract For a one parameter family of plane quadratic vector fields X (.,e) depending analytically on a small real parameter e, we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous (or linearizable) center. Techniques are developed for treating the bifurcations of all orders, and these are applied to prove the following results. For the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three. For one class of nonlinear isochrones, at most one continuous family of limit cycles can emerge, whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge. Moreover, for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center.

The monotonicity of the period function for planar Hamiltonian vector fields

If the Hamiltonian system with Hamiltonian H(x, y) = 12y2 + V(x) has a center at the origin it is shown that the period function for the family of periodic trajectories surrounding the center is monotone when the function V(V′)2 is convex. This theorem is applied to determine existence and uniqueness of solutions for the Neumann boundary value problem X″ = a(x′)2 + bx + c, x′(0) = x′(1) = 0.

Bifurcations of nonlinear oscillations and frequency entrainment near resonance

A unified approach to the Poincare–Andronov global center bifurcation and the subharmonic Melnikov bifurcation theory is developed using S. P. Diliberto’s integration of the variational equations of a two-dimensional system of autonomous ordinary differential equations and a Lyapunov–Schmidt reduction to the implicit function theorem. In addition, the subharmonic Melnikov function is generalized to the case of subharmonic bifurcation from an unperturbed system whose free oscillation is a limit cycle. Thus, results on frequency entrainment are obtained when an external periodic excitation is in resonance with the frequency of the limit cycle. The theory is applied to the subharmonic bifurcations of two coupled van der Pol oscillators running in resonance.

Finiteness for critical periods of planar analytic vector fields

LIMBURGS UNIV CENTRUM,DEPT MATH,B-3610 DIEPENBEEK,BELGIUM.CHICONE, C, UNIV MISSOURI,DEPT MATH,COLUMBIA,MO 65211.

Inertial and slow manifolds for delay equations with small delays

Abstract Yu.A. Ryabov and R.D. Driver proved that delay equations with small delays have Lipschitz inertial manifolds. We prove that these manifolds are smooth. In addition, we show that expansion in the small delay can be used to obtain the dynamical system on the inertial manifold. This justifies “post-Newtonian” approximation for delay equations.

The generalized Jacobi equation

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed 2−1/2c ≈ 0.7c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored.

On General Properties of Quadratic Systems

where the polynomials Pn and Qn are relatively prime and at least one of them has degree n. In the special case n = 2, which is most important for this exposition, the system is called quadratic. Of course, the basic problem is to find the qualitative structure of the integral curves in the plane of the general system of form (l)* The study of polynomial systems has a long history in mathematics and in the physical sciences. Of the many possible physical examples we mention the van der Pol oscillator

Line Integration of Ricci Curvature and Conjugate Points in Lorentzian and Riemannian Manifolds.

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.

Separatrix and limit cycles of quadratic systems and Dulac’s theorem

Separatrix cycles for a planar quadratic vector field are studied. The results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles.