MSC: 34C05 34C07 37G15Keywords: Hopf bifurcation Non-smooth dynamical system Limit cycle Piecewise linear system As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the HyersUlam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.

Abstract. Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135-163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov-Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov-Takens bifurcations of limit cycles and homoclinic loops of order 2. In most epidemic models (see Anderson and May [3]), the incidence rate (the number of new cases per unit time) takes the mass-action form with bilinear interactions, namely, κS(t)I(t), where S(t) and I(t) are the numbers of susceptible and infectious individuals at time t, respectively, and the constant κ is the probability of transmission per contact. Epidemic models with such bilinear incidence rates usually have at most one endemic equilibrium and do not exhibit periodicity; the disease will be eradicated if the basic reproduction number is less than one and will persist otherwise (Anderson and May [3], Hethcote [11]). There are many reasons for using nonlinear incidence rates, and various forms of nonlinear incidence rates have

Abstract. The concept of characteristic interval for piecewise monotone functions is introduced and used in the study of their iterative roots on a closed interval.

Using the fixed point theorems of Banach and Schauder we discuss the existence, uniqueness and stability of continuous solutions of a polynomial-like iterative equation with variable coefficients. I. Introduction. Let I = [a, b] be a given closed bounded interval. Given a continuous F : I → I such that F (a) = a and F (b) = b, and given continuous functions λ 1 ,. .. , λ n : I → [0, 1] such that n i=1 λ i (x) = 1 for all x ∈ I, we wish to find continuous functions f : I → I such that (1) λ 1 (x)f (x) + λ 2 (x)f 2 (x) +. .. + λ n (x)f n (x) = F (x) for all x ∈ I. Here f i denotes the ith iterate of f (i.e., f 0 (x) = x and f i+1 (x) = f (f i (x)) for all x ∈ I and all i = 0, 1,. . .). We suppose that n ≥ 2. The case in which the λ i 's are constant was considered in [4]-[7] and [9]-[11] for special choices of F and/or n. Similar equations are discussed on pages 237-240 of [5]. Such problems are related both to problems concerning iterative roots (see [1], [3] and [8]), e.g. finding a function f such that f n (x) = F (x), ∀x ∈ I, and to the theory of invariant curves for mappings (see Chapter XI of [5]). Note that we may assume without loss of generality that a = 0 and b = 1.

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