Robert J. Sacker

A Spectral Theory for Linear Differential Systems

Abstract This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.

Existence of dichotomies and invariant splittings for linear differential systems, II☆

Abstract This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.

On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations

This article consists of Chapter 2 of the authors 1954 Dissertation bearing the same title and published as Courant Institute report IMM-NYU 333, available in full on the authors personal web site. Chapter 2 consists of the first complete proof of what has come to be known as the Neimark-Sacker bifurcation theorem. It includes the reduction to normal form using weighted monomials which precludes using the Center Manifold theorem which was not known to the author and was published the same year as the dissertation and called “a Reduction Principle” by V. Pliss and later named Center Manifold theorem by A. Kelley (see previous article for citations). After reduction to normal form, the resulting functional equations for the bifurcating invariant curve is solved in detail. Stand-alone decimals provide solutions of linear functional equations, a-priori estimates and interpolation inequalities between derivatives in the sup norm.

Nonautonomous Beverton-Holt Equations and the Cushing-Henson Conjectures

In [3] Jim Cushing and Shandelle Henson published two conjectures (see Section 3) related to the Beverton-Holt difference equation (with growth parameter exceeding one) which said that the B-H equation with periodically varying coefficients (a) will have a globally asymptotically stable periodic solution and (b) the average of the state variable along the periodic orbit will be strictly less than the average of the carrying capacities of the individual maps. They had previously [3] proved both statements for period 2. In [4] the authors solved the first conjecture in the affirmative for arbitrary period and in a metric state space. In addition they showed that the period of the periodic “geometric cycle”, i.e. the projection of the periodic orbit onto the state space, must be a divisor of the period of the underlying system. In [5] the authors solved the second conjecture. Independently Ryusuke Kon [8], [9] discovered a solution to the second conjecture and in fact proved the result for a wider class of difference equations including the Beverton-Holt equation. Also Kocic [7] has given a solution to the second conjecture. In this paper we consider the B-H equation with periodic growth parameter as well as periodic carrying capacity. We first give an estimate relating the averages of the state variable and the carrying capacities. This is done by a modification of the proof of Kocic [7]. We then refine the estimate and actually obtain an equality relating the averages, (in the case of period p = 2) thus laying to rest once and for all the p = 2 case. The general case will be treated elsewhere.

Population models with Allee effect: a new model

In this paper, we develop several population models with Allee effects. We start by defining the Allee effect as a phenomenon in which individual fitness increases with increasing density. Based on this biological assumption, we develop several fitness functions that produce corresponding models with Allee effects. In particular, a rational fitness function yields a new mathematical model, which is the focus of our study. Then we study the dynamics of 2-periodic systems with Allee effects and show the existence of an asymptotically stable 2-periodic carrying capacity.

The spectrum of an invariant submanifold

This paper is concerned with vector fields on smooth compact manifolds. The exponential growth of solutions of the linearized equations is described by the already well-known Spectral Theorem applied to the induced linear flow on the tangent bundle. The spectrum of the tangent bundle flow is compared to the two secondary spectra obtained by first taking the spectrum of the bundle of tangent spaces to an invariant submanifold and second, the spectrum of an induced flow on an arbitrary complementary bundle to the latter. The relationship among the three spectra is studied and it is shown that whenever these secondary spectra are disjoint then an invariant complementary bundle can be found. The results have implications in the theory of perturbation of invariant manifolds. The problem is studied in the setting of skew-product dynamical systems and the results are applicable to block triangular systems of ordinary differential equations.

The splitting index for linear differential systems

Consider the linear differential operator associated with an n-dimensional first order linear system of time varying ordinary differential equations. Conditions are given on the system for time near plus and minus infinity which guarantee that the operator is Fredholm. The splitting index is introduced and it is shown to be the negative of the ordinary index of a Fredholm operator. The splitting index is shown to be invariant under appropriate perturbations and is computable in terms of the asymptotic properties of the coefficient matrix for a wide class of systems. The asymptotic conditions on the system are discussed in various function space topologies and a new concept of admissibility of a pair of Banach spaces is introduced whereby a pair is admissible with respect to all operators whose coefficients lie in a given function space.

Basin of Attraction of Periodic Orbits of Maps on the Real Line

We prove a conjecture by Elaydi and Yakubu which states that the basin of attraction of an attracting 2 k -cycle of the Rickers map is where E is the set of all eventually 2 r -periodic points, The result is then extended to a more general class of continuous maps on the real line.

Finite extensions of minimal transformation groups

In this paper we shall study homomorphisms p: W Y on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-l) extension of Y if and only if W is an Nfold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.

A note on periodic Ricker maps

The proof proceeds as follows. Let f 1⁄4 Rpk21 +Rpk22+· · ·+Rp1 + Rp0 : ð1:3Þ We first establish an interval I 1⁄4 [a, b], a . 0, which is invariant under application of the composite map and into which all points of Rþ are mapped in finitely many applications of f. Clearly, b 1⁄4 exp ðmax pi 2 1Þ and a 1⁄4 min {min pi; b exp ðmin pi 2 bÞ} suffice. Differentiating equation (1.3), and letting x0 1⁄4 x,