This paper clarifies some properties of the Poincar map of the almost periodic oscillation, which is generated by a periodically excited nonlinear system described by a Linard equation. Arguments in this paper are based on the extended Linard theorem already published by the present authors and are focused on the almost periodic oscillations which may occur in the Linard system under a certain constraint on the external force. As the main result, it is shown that the Poincar map of the almost periodic oscillation drawn on the Linard plane forms a simple closed continuous curve, under an explicitly given condition on the external force e (t) = A sin (ωt+θ), for arbitrary value of the amplitude A except for a set of the values ω with zero measure.

This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)f(x)g(x)e(t)where f and g are not necessarily even or odd with respect to x, respectively. First, basic theorems on the existence of limit cycles and properties of singularities are proved in the case where e(t) is equal to a constant bias denoted by e(t)const.A cos ωτ, and effects of f and g on the portrait of trajectories of the systems are clarified. Then, the dynamical behavior of the system, where the external force is periodic, i.e., e(t)A cos ωt, is represented in relation to the singularity which varies periodically in time; an obtained result makes it clear and easy to understand the dynamical behavior. Further, some conditions which are necessary for the system mentioned above to generate a chaotic solution are presented. Finally, the results of the argument above are applied to the periodically forced van der Pol equation, and it is concluded that chaotic solutions hardly exist in this case.

As is well known, Linard's equation +µf (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Linard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.