Attraction in Nonmonotone Planar Systems and Real-Life Models
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Attraction in Nonmonotone Planar Systems and Real-Life Models Alfonso Ruiz-Herrera1 Dedicated to Prof. Rafael Ortega on the occasion of his 60th birthday Received: 24 March 2020 / Revised: 27 July 2020 / Accepted: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Let h : V ⊂ R2 −→ R2 be an embedding. The aim of this paper is to analyze the dynamical behavior of h depending on the number of fixed points and 2-cycles, their local behaviors and the features of V . Our approach allows us to extend some celebrated results of the theory of monotone flows, namely the order interval trichotomy, for non-monotone maps. Moreover, we discuss several applications in classical models. In the particular case of the Ricker system, we recover some recent results deduced from computer assistance. Keywords Global attraction · Trivial dynamics · Order interval trichotomy · Embeddings · Ricker system with overcompensation Mathematics Subject Classification 37E30 · 92B05 · 37C65 · 37C75
1 Introduction A map h : V ⊂ R2 −→ V which is continuous and injective will be called an embedding. We stress that h is not necessarily onto. From an embedding h : V −→ V , we can define the discrete dynamical system xn+1 = h(xn ) with
x0 ∈ V .
(1.1)
Understanding the dynamical behaviour of (1.1) is a central problem from a theoretical and applied point of view, but this is not an easy task. The main difficulty comes from the reduced number of available tools to study (1.1). Moreover, it is well-known that (1.1) can exhibit complex dynamics [1,17]. The main goal of this paper is to analyze the dynamical behavior of (1.1) depending on the number of fixed points and 2-cycles, their local behaviours and the features of V . We mainly discuss two dynamical aspects: The global attraction to an equilibrium and the presence of
B 1
Alfonso Ruiz-Herrera [email protected] Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain
123
Journal of Dynamics and Differential Equations
trivial dynamics (i.e. the omega limit of any orbit is contained in the fixed point set). As we will see, the presence of a locally asymptotically stable fixed point p and the absence of 2-cycles are enough to guarantee the global attraction of p in many situations (see Theorems 3.2 and 3.3). To prove our results, we propose two different strategies. If there exists a fixed point p that is locally asymptotically stable, we argue as follows: First we construct a topological disk D with p ∈ I nt D and so that all the orbits enter in D in the future. This topological disk will have the additional property of being positively invariant. Next, we define an auxiliary homeomorphism on R2 with the same dynamical behaviour on D as the original embedding. Finally, we apply the theory of prime ends developed in [10,16,18]. The second strategy is based on the theory of translation arcs [3,17]. Under certain conditions, this theory guarantees the presence of additional fixed points when the embedding does not have trivial dynamics. Thus,
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