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

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). We thus conclude from the chain- ∗ ∗ ∗ transitivity of R(fγ)thatAγ is chain-connected to Rγ . In fact, any point in Aγ is chain connected ∗ to any point in Rγ because of the chain transitivity of R(fγ). This completes the proof.

3. Example: Ikeda map The Ikeda map, given as follows, is proposed as a simpliﬁed mathematical model of transmitted light fromaringcavitysystem[5]: iK √ f(z)=A + Bz exp iC − (i = −1), 1+|z|2 where z ∈ C,andA, B, C, K are real parameters.

99 (a) (b) Fig. 1. The phase portraits of the Ikeda map at (a) K =7.2 and (b) K =7.3. In (a), the blue set is an attractor and the black set is a repeller, while the entire black set is an attractor in (b), hence the crisis bifurcation is expected between these two parameter values. The pictures are produced by Ken Tsubotani using the software “cmgraph” [2].

The Ikeda map is well-known as exhibiting an interior crisis at some parameter values. For instance, fixing the parameter values as A =0.84,B =0.9,C =0.4, and choosing K =7.2orK =7.3, we obtain the phase portraits as in Fig. 1. These pictures are obtained by applying a rigorous topological computation method introduced in [2] to the Ikeda map. Indeed, actual computation is done by using the software called “cmgraph” (the name comes from the Conley Morse graph), developed based on this method. In order to discuss the topological crisis bifurcation in the Ikeda map, let us briefly recall the process of computation by this software. Given a dynamical system (in the form of an iterated map) and a cubical grid decomposition defined on a domain of interest in the phase space, the software first computes its rigorous combinatorial outer approximation using the interval arithmetic, and represents it in terms of a directed graph whose vertices are the cubical grids and edges are given by the correspondence of a cubical grid element and a rigorous enclosure of its true image by cubical grid elements. Then a partial order relation can be naturally introduced on the collection of the strongly path connected components of the directed graph, which determines a Morse decomposition of the maximal invariant set of the dynamical system on the chosen domain, since each strongly path connected component, called a combinatorial Morse set, is in fact an isolating neighborhood of an isolated invariant set. This Morse decomposition obtained from the directed graph constructed as above, is called a combinatorial Morse decomposition,oraMorse graph when viewed as a directed graph representing the partial order. The partial order in the combinatorial Morse decomposition, namely the graph structure of the Morse graph, gives information on the gradient-like behavior of the dynamical system under consider- ation, since the recurrent behavior is entirely contained in the Morse sets. The Conley index is often used to represent information on the recurrent behavior of Morse sets, and the Conley-Morse graph is a Morse graph associated with the Conley index of each Morse set. See [2] and references therein for more details about the software and definitions of relevant notion. The pictures in Fig. 1 are the combinatorial Morse sets obtained by the cmgraph software applied to the Ikeda map at K =7.2andK =7.3. In this computation, the phase space domain is chosen to 2 ∼ be the square D =[−2.0, 4.0] × [−2.0, 4.0] ⊂ R = C on which we set a uniform cubical grid given by subdividing each edge into 212 small intervals. From the computation using the cmgraph, the resulting combinatorial Morse decomposition consists of two Morse sets, namely an attractor-repeller decomposition, for K =7.2, and a single Morse set for

100 K =7.3. The software also provides us with their Conley index information, and in particular, the attractor in the case of K =7.2 and the unique Morse set in the case of K =7.3 both have 1 as their non-zero eigenvalues of the 0-th homology of the index map. It follows from Proposition 5.8 in [2] that these combinatorial Morse sets are truly attracting in the sense that they have a trapping region, in other words, they do have an index pair with empty exit set. It should be easy to check that the square domain D =[−2.0, 4.0] × [−2.0, 4.0] remains isolated over the parameter interval I =[7.2, 7.3], hence it verifies the assumption (H1) and (H2a) in Section 2 with N = D ×I,andλ = K with α =7.2 and β =7.3. Verification of (H2b) is not at all trivial, as it need to test all possible choice of attractor-repeller decomposition at a given parameter value. The fact that the cmgraph computation results in a single Morse set does not exclude the possibility of other non-trivial attractor-repeller decomposition; for instance, if one takes a finer grid decomposition of the phase space, one might find a non-trivial attractor-repeller decomposition. The best one could say at this level of computation is the finite resolution transitivity due to Luzzatto and Pilarczyk [6], which says that, at a certain size of uniform grid, the dynamics at K =7.3 is finite resolution transitive and hence there is no non-trivial attractor- repeller decomposition up to that grid size. Although this is not a fully mathematically rigorous verification of the assumption (H2b), we consider that it serves as a verification for practical purposes. Having all these assumptions verified, at least practically, we can conclude that the Ikeda map over the parameter interval I =[7.2, 7.3] undergoes the topological interior crisis bifurcation and the dynamics changes as described in Theorem 2.2. The chain orbit referred to in Theorem 2.2 should in fact correspond to a collision orbit in a geometric explanation of the crisis phenomenon in literature.

The Hénon map h(x, y)=(a−x2 +by, x) is another well-known example of dynamical systems that exhibits the interior crisis bifurcation. Between the parameter values (a, b)=(2.01, −0.3) and (a, b)= (2.03, −0.3), the cmgraph computation detects a change of attractor-repeller pair decompositions of the maximal invariant set of (the second iterate of) the Hénon map, to which our definition and the theorem of the topological interior crisis bifurcation can be applied, just in the same way as to the Ikeda map discussed above. See Fig. 2. We believe that our formulation is so simple that it will be applicable to (almost) all examples of the interior crisis bifurcation in literature.

(a) (b) Fig. 2. The phase portraits of the second iterate of the H´enon map at (a) a =2.01 and (b) a =2.03, for b = −0.3 ﬁxed. In (a), the red and green sets are combinatorial Morse sets corresponding to true attractors, and a Morse set (colored blue) sits in-between, which looks like a saddle point. On the other hand, the blue set in (b) is a single Morse set which correspond to an attractor. There is a neighborhood that isolates these combinatorial Morse sets (red, green, and blue sets), and hence the crisis bifurcation is expected between these two parameter values (a) and (b). Note that there is another Morse set colored black in both (a) and (b), looking like a horseshoe invariant set, which is isolated away from the observed change of the attractor-repeller decomposition, and thus is irelevant to the interior crisis bifurcation. The pictures are also produced by Ken Tsubotani using the software “cmgraph”

101 4. Discussion In this paper, we have discussed the interior crisis bifurcation from a topological viewpoint, and in the simplest situation of attractor-repeller decompositions. There are many examples of dynamical systems that exhibit interior crisis bifurcations. In the study of these examples, the collision of the attractor with the stable manifold of an unstable object such as an unstable periodic orbit or a horseshoe is verified by a detailed analysis of the system with an aid of careful numerical computation, which explains, as a result, a sudden change of the size and shape of an attractor to which the name “interior crisis” is given. Our approach is in some sense opposite: we assume there is a qualitative change of an attractor, in our case in terms of the change of attractor-repeller decompositions, and study what we can say about the moment of the change in a most general manner. Naively speaking, our definition of the interior crisis bifurcation is very natural, as we consider the interior crisis as a “bifurcation” that happens to an attractor, and try to describe it as a change of attractor-repeller decomposition. The main point of our definition is that it does not say anything about the cause of the change of the attractor, and thus it may potentially include quite large va- riety of phenomena occurring to an attractor. The value of the definition is therefore measured by the conclusions that can be drawn from it, and what the main theorem tells about is one of such conclusions. Our theorem might be seen quite limited, as it only shows the existence of a chain-connecting orbit from the attractor to the repeller. However, it says it occurs in any systems that satisfy our assumptions of the topological interior crisis bifurcation. As said above, our formulation of the interior crisis is quite weak as it merely refers to just a change of an attractor. The existence of a chain-connecting orbit, a weakest kind of connecting orbits, from the limit set of the attractor to the limit set of repeller in the limit of the parameter variation, is thus quite general. Notice that the direction of the chain-connecting orbit is opposite to the natural direction of connecting orbits in the attractor-repeller decomposition. One can consider that this chain-connecting orbit is the trigger of the breakdown of the attractor before the crisis, and of the creation of a new attractor after the crisis. If one can somehow capture the chain-connecting orbit, which is certainly beyond the capability of our coarse topological-computation method, one may be able to locate finer structure of the mechanism of the crisis, and may be able to classify the crisis bifurcation according to possible types of the occurrence of the chain-connecting orbits in various examples. We also consider that our approach may provide an important view to the study of global bifurcations of dynamics. In practical situation such as experiments and observations in experimental sciences and engineering, one sees the change of dynamics by comparing the observed behaviors of the system at different parameter values. If the change is simple enough, such as generation of a periodic behavior, one may guess that it may be caused by a Hopf bifurcation. There are however other possibility of changes in a global structure of dynamics that can generate periodic behaviors, and hence it is not straightforward to identify what kind of changes of dynamics may correspond to some known bifurcations or not. Therefore it must be important to be able to compare the change of global dynamics in a general and coarser manner, without referring to some detailed geometric structures such as unstable periodic orbits and their stable/unstable manifolds, because one may not have sufficiently fine enough mathematical models when one sees such changes of global dynamics. Our topological approach may be important and useful for such purposes, and we consider that the result in this paper is one of the simplest types of the results in this approach. We hope to be able to extend this approach to a broader class of global bifurcations. For instance, we believe that our result on the topological interior crisis bifurcation as a change of attractor-repeller decompositions can be extended to a more general Morse decompositions. We also consider that the boundary crisis bifurcation can be treated similarly. These extensions, however, need a little more technically involved arguments, and hence they shall be treated separately and will appear elsewhere.

102 Acknowledgments The authors thank Marcio Gameiro, Ken Tsubotani, and Ippei Obayashi for their help in carrying out the computation for the Ikeda map and Hénon map using the cmgraph software. HK is partially supported by Grant-in-Aid for Scientific Research (No. 21340035), Ministry of Education, Science, Technology, Culture and Sports, Japan. HO is partially supported by Grant-in- Aid for Scientific Research (No. 21540231), Ministry of Education, Science, Technology, Culture and Sports, Japan.

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