A geometic method for the periodic problem in ordinary differential equations

Abstract

In this monograph we present foundations and basic applications of a new method for the periodic problem for ordinary differential equations. The method was introduced in [Sr2,3] and is based on the Lefschetz Fixed Point Theorem and the Wazewski Topological Principle. The paper consists of two parts, each of which is divided into chapters. The chapters are in turn divided into sections. In Part I we introduce necessary definitions and prove the main theorems concerning the geometric method, while in Part II we apply the method to various classes of ordinary differential equations. The geometric method is based on the concept of periodic block (see Definition 2.2.4), which is a modification of the notion of isolating block, introduced by C.Conley. Its intuitive description is presented in Section 1.1, but its rigorous treatment is postponed until Section 2.2. Some motivation for that concept comes also from the Floquet theory, as we shall see in Section 3.3. In Section 3.2 we present a construction of periodic blocks for ordinary differential equations. The construction is further developed in Section 3.4, where Proposition 3.4.1 is stated. That result enables us to determine which equations admit periodic blocks and is used in many applications. A most important role in the method is played by the Lefschetz number, an invariant associated to periodic blocks. It is described in Sections 1.1 and 2.2 (see Definition 2.2.3). Theorems A, B, and C established in Section 1.2 are the main results for the method. They are formulated in a purely topological way, without referring to any differential structure. Section 2.3 contains their proof. In Section 4.2 we modify them, and the resulting Theorems 4.2.1 and 4.2.2 apply directly to ordinary differential equations. They might be compared with other results on the periodic problem, recalled in Section 4.1. At first we apply the geometric method to perturbed linear equations in Section 5.1. Proposition 5.1.3 is the most original result in that section. Its proof, as well as the proofs of Propositions 5.1.1. and 5.1.2, represents a standard way for applying the method. In Section 5.2 we establish a relation between the Conley index and the fixed point index. The most important applications of the method presented in this paper are contained in Chapter 6. Theorems 6.3.1 and 6.3.3 on planar polynomial equations and Theorem 6.7.1 on rational equations are the main results of Chapter 6, which is based on the papers [Sr4] and, [KS]. The theorems concern nonautonomous equations obtained from hyperbolic and elliptic polynomlals (compare Definition 6.2.2), and are motivated by some geometric observations given in Section 6.1. Several examples Illustrating them are presented in Sections 6.3 and 6.7. In Section 7.1 we apply the geometric method to results on second order scalar equations. Section 7.2 is based on [SS] and presents two results on scalar equations of arbitrary order. Unless otherwise is stated, all definitions and theorems concerning the geometric method as well as results obtained by their applications presented in this paper were established in [KS], [Srl-4], and [SS] or appear here for the first time.