A Topological Computation Approach to the Interior Crisis Bifurcation

NOLTA, IEICE Paper A topological computation approach to the interior crisis bifurcation Hiroshi Kokubu 1 a) and Hiroe Oka 2 b) 1 Department of Mathematics/JST-CREST, Kyoto University Kyoto 606-8502, Japan 2 Department of Applied Mathematics and Informatics Faculty of Science and Technology, Ryukoku University Seta, Otsu 520-2194, Japan a) [email protected] b) [email protected] Received May 8, 2012; Revised August 27, 2012; Published January 1, 2013 Abstract: We study the interior crisis bifurcation from the viewpoint of the graph-based topological computation developed in [2]. We give a new formulation of the interior crisis bifurcation in terms of a change of the attractor-repeller decompositions of the dynamics, and prove that the attractor before the crisis disappears by creating a chain connecting orbit to the repeller at the moment of the interior crisis. As an illustration, we discuss the interior crisis bifurcation in the Ikeda map. Key Words: dynamical system, bifurcation, interior crisis, Morse decomposition, Conley index, topological computation 1. Introduction A crisis bifurcation refers to a discontinuous change of a chaotic attractor for a family of dynamical systems, when a parameter is varied. There are typically two types of the crisis bifurcation; boundary crisis and interior crisis. The boundary crisis occurs when the chaotic attractor suddenly disappears at the moment of bifurcation, while the interior crisis does when the chaotic attractor suddenly changes its size and/or shape. These crisis bifurcations are studied phenomenologically; the boundary crisis, for instance, is typically explained as the collision of the chaotic attractor with an unstable invariant set, such as an unstable periodic orbit, but there is no single universal mechanism that characterizes the boundary crisis. The situation is similar for the interior crisis. See [1, 4, 9] for more details and related information. Therefore, although there are many examples of dynamical systems known as exhibiting such crisis bifurcations, they are always studied individually, and there has been no attempt, at the authors’ knowledge, to characterize these bifurcations as an abstract global bifurcation. In [2], we, together with our collaborators, have developed a new method for analyzing global dynamics and bifurcations by using a graph-based topological computation. By this method, one 97 Nonlinear Theory and Its Applications, IEICE, vol. 4, no. 1, pp. 97–103 c IEICE 2013 DOI: 10.1587/nolta.4.97 describes the information of global dynamics in terms of the so-called Conley-Morse graph, which is a directed graph representing a Morse decomposition of the dynamical system on a given domain in the phase space, combined with the associated Conley index of each Morse set. The bifurcation of the global dynamics is hence viewed as a change of the Conley-Morse graphs when a parameter is varied. We have tested this method for several examples and have confirmed that one can obtain a reasonably good understanding of a rough structure of the dynamics and bifurcations in these examples. These results naturally lead us to a question: how one can see the crisis bifurcation from the viewpoint of the topological computation in terms of the Conley-Morse graphs? Is it possible to characterize the crisis bifurcation as a change of the Morse decomposition? The purpose of this paper is to answer these questions in the case of the interior crisis in the simplest setting of attractor-repeller decompositions. Namely, we shall propose a mathematical definition of the interior crisis bifurcation in terms of attractor-repeller decompositions, and proves the existence of a chain-connecting orbit from the “attractor” to the “repeller” at the moment of the interior crisis bifurcation. This means that the discontinuous change of the size and shape of the attractor is due to the creation of a connecting orbit in the weakest sense from the attractor at the bifurcation point. The description of the nature of the interior crisis bifurcation is completely general and topological, since our formulation does not make any assumptions about the structure of the attractor and its basin boundary, such as an unstable periodic point on the basin boundary and its stable manifold. Similar results for the boundary crisis and for more general Morse decompositions can be obtained, which will be treated in a separate paper, since we need more technically involved arguments. As an illustration of our formulation and the result, we study the Ikeda map and the Hénon map, which are known as typical examples of dynamical systems that exhibit the interior crisis. We compute the Conley-Morse graphs of the Ikeda map before and after the interior crisis and show how our main result may be applied to this case. We conclude the paper by giving a few remarks concerning the topological computational approach to global bifurcations. 2. Main result Let X be a locally compact complete metric space and fλ : X → X (λ ∈ I =[α, β]) be a one-parameter family of continuous maps, depending continuously on the parameter λ. We define its parametrized map over the parameter interval I as fI : X × I → X × I, (x, λ) → (fλ(x),λ). We give a formulation of the interior crisis bifurcation in terms of attractor-repeller decompositions in the Conley index theory. See [3] and [7] for terminology and related information. Definition 2.1 We say that the family fλ undergoes a topological interior crisis bifurcation over the parameter interval I =[α, β], if the following three conditions are satisfied: (H1) There exists an index pair (N,L)forfI such that L = ∅. This means that, for all λ ∈ I,the maximal invariant set Inv(Nλ) is truly attracting in the sense that it has a trapping region. (H2a) At λ = α, there exists an attractor-repeller decomposition {A0,R0} of Inv(Nλ=α)withA0 being truly attracting. (H2b) At λ = β, there is no attractor-repeller decomposition of Inv(Nλ=β) of the type (H2a). In the next sections we shall give examples of interior crisis bifurcations to which our formulation can be applied, as well as a short discussion for its meaning. Now we state our main theorem, which roughly asserts that, under very mild assumptions at the two ends of a parameter interval in which the type of attractor-repeller decomposition changes, there must exist a chain connecting orbit from the attractor to the repeller (not from the repeller to the 98 attractor) at the moment of crisis bifurcation. In other words, the nature of crisis may be characterized purely topologically in a very general manner, and not necessarily in terms of saddles and their stable manifolds. Theorem 2.2 If a family fλ undergoes the topological interior crisis bifurcation over I =[α, β], then there exists γ ∈ (α, β] such that the following three statements hold: (1) If λ ≥ γ, fλ does not admit an attractor-repeller decomposition of Nλ of the type (H2a). {λ } γ n f (2) There exists an increasing sequence n n=1,2,3,... conversing to such that, for each , λn A ,R N does admit an attractor-repeller decomposition ( λn λn )of λn of the type (H2a). (3) For any sequence λn as in (2), the limit set A∗ A A γ = lim sup λn := λk n→∞ n≥1 k≥n R∗ R∗ A∗ {R } is chain-connected to γ , where γ is similarly defined as γ for λn . Here, a set A∗ is called chain-connected to another set R∗ by a dynamical system f, if, for any ε>0, there exists an ε-chain orbit from a point in A∗ to a point in R∗ under f. Proof: The first two statements are obvious from the definition of the topological crisis bifurcation, once we define γ as the supremum of the parameter values λ at which the corresponding map fλ admits an attractor-repeller decomposition of the type (H2a). Observe that there is no attractor- repeller decomposition of the type (H2a) at the supremum value λ = γ due to the robustness of the attractor-repeller decomposition, namely, an attractor-repeller decomposition (more generally a Morse decomposition) persists under small enough perturbation. See [3, 7] for details. To prove the statement (3), we use the Fundamental Theorem of Dynamical Systems due to C. Con- t ley [3], which says that any flow ϕ = {ϕ }t∈R on a compact metric space admits a continuous function Θ which is strictly decreasing along orbits off of its chain recurrent set R(ϕ) and has the image Θ(R(ϕ)) being nowhere dense in R. Moreover, for each c ∈R(ϕ), the pre-image Θ−1(c)ischain- transitive. This theorem also holds true for invertible discrete time dynamical systems (i.e. an iterated home- omorphism) [10], as well as for non-invertible discrete time dynamical systems (i.e. an iterated endo- morphism) [8]. From the definition of the topological interior crisis, we have that any attractor-repeller decomposition of the type (H2a) does not continue beyond λ = γ. In this case, we claim that Θ(R(fγ)) consists of a single point. Otherwise, there must exist a value c ∈ R \ Θ(R(fγ)) which would separate the set Θ(R(fγ)) into two non-empty subsets, and hence the corresponding pre-images in R(fγ ) would define two non-empty invariant subsets A˜ and R˜, which would clearly give a non-trivial attractor-repeller decomposition of the type (H2a) at λ = γ, a contradiction. Therefore Θ(R(fγ)) is a single point set and the whole chain-recurrent set R(fγ) must be chain-transitive. ∗ ∗ Obviously, the sets Aγ and Rγ are invariant subsets in R(fγ).

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