TOPOLOGICAL DEGREE AND STABILITY OF PERIODIC SOLUTIONS FOR CERTAIN DIFFERENTIAL EQUATIONS RAFAEL ORTEGA
1. Introduction This paper analyses the stability of periodic solutions of a class of differential equations by means of degree theory. Associated with a differential equation which is periodic in the independent variable is the Poincare-Andronov operator, also called the translation operator, which reduces the search for periodic solutions to a fixedpoint problem in IRn. A classical method for proving the existence of periodic solutions consists in finding a domain in Un where the degree associated to the fixedpoint problem is not zero. This method is of great interest from the point of view of existence, but no general conclusion on the stability of these periodic solutions can be derived from it. However, if one is restricted to the class of first-order scalar equations, this method provides very accurate stability information. In [6, 7] the author obtained characterizations of asymptotic stability in terms of topological index for certain two-dimensional systems, in particular for second-order equations of Duffing type. These results were based on linearization techniques and restricted to periodic solutions of hyperbolic type. This is understood in the sense that Floquet multipliers of the corresponding variational equation do not lie on the unit circle. In [11], R. A. Smith introduced a class of n-dimensional systems for which Massera's convergence theorem still holds. Such classes of equations must behave similarly to scalar equations and it is natural to expect, in such cases, characterizations of asymptotic stability in terms of degree or index (without imposing conditions of hyperbolic nature). In this paper a result of this nature will be obtained and, following the interests of the author, applied to the forced pendulum equation. The proof of this result will make constant use of techniques developed by Smith in [9,10,11]. In particular, the concept of an amenable solution will allow us to reduce the question of the stability of a periodic solution to the stability of fixed points of certain discrete dynamical systems which are one dimensional. The paper is organized as follows. In Section 2 the main results are stated and compared with other related results. In Section 3, the forced pendulum equation is considered and a necessary and sufficient condition for existence of stable periodic solutions of this equation is obtained. The last section, divided in three parts, is devoted to proofs. Received 9 June 1989. 1980 Mathematics Subject Classification (1985 Revision) 34C25. The author was supported by 'Direction General de Investigation Cientifica y Tecnica, M.E.C. Espana, Proyecto PB86-0458' and 'Junta de Andalucia'. J. London Math. Soc. (2) 42 (1990) 505-516
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RAPAEL ORTEGA
2. Statement of results n
n
Let / : U x U -* U be a continuous function with period T in t and continuous partial derivative fx{t, x). Let us consider the equation x'=f[t,x), (•) and make the following assumptions. (H-l) There exists a constant symmetric n x n matrix P, having exactly one negative eigenvalue and n — 1 positive eigenvalues, and positive constants X, e such that PfJLUx) +fx(t,x)*P+IIP < -2el n for all (t,x)eMx Un, (Here * denotes the transpose matrix, /„ is the identity matrix and the inequality is understood in the usual sense for symmetric matrices.) (H-2) There exist a constant symmetric matrix A > 0, and constants a, b such that x*Af[t,x) ^ a\x\2 + b for all (t,x)eUx Un. An equivalent formulation of (H-l), also suitable for functions without derivative fx, was introduced in [11]. Several criteria for its verification are also included in the same paper. Assumption (H-2) guarantees that each solution of (•) has an interval of existence of the form (0, oo). It is not clear whether a condition of this kind is really needed for our result or is simply a limitation of the method of proof. Later we shall see that this hypothesis can be eliminated for equations of Duffing type. Denote by (t,p) the solution of (•) with (Q,p)=p and consider the Poincare-Andronov operator 0>p = (T,p),
peUn.
It is well known that fixed points of & correspond to initial values of T-periodic solutions. Of course, when the degree of /— 0* is not zero in some region, at least one T-periodic solution exists (/ is the identity in IRn). The next result shows that the stability of this periodic solution can be studied in certain cases. THEOREM
2.1. Assume that (H-l) and (H-2) hold and that there exists a bounded
domain Q in Un such that Then there exist at least d stable T-periodic solutions of(*) with initial values lying on Cl. (Here deg refers to Brouwer degree.) REMARKS. 1 A stable periodic solution of (*) which is isolated is also asymptotically stable when (H-l) holds; see [11]. 2 The existence of a stable T-periodic solution was proved in [11] assuming (H-l) and the existence of a bounded open set Q in IRn such that Q 2 0>(£i). (Actually, the result was more general since (*) was not necessarily defined in the whole space.) When Q is homeomorphic to a ball this condition implies that the degree is one, but in general there is no immediate connection between the two results. 3 The other classical degree approach to the periodic problem is based on Leray—Schauder degree (cf. [8]). Probably it is possible to use the so-called 'duality principle' [1, 2] to obtain an analogue of Theorem 2.1 in that setting.
TOPOLOGICAL DEGREE
507
Given an isolated T-periodic solution of (*), say (p(t), the index of cp is defined as where B is a neighbourhood of ^(0) not containing other fixed points of &. When
0, g is a real entire function, 27r-periodic and with mean value zero. For example, g(y) = asiny. We also assume that max£'0/)
