A Nonhorseshoe Template in a Chaotic Laser Model

Topological methods have recently been developed for the analysis of dissipative dynamical systems that operate in the chaotic regime. They were originally developed for three-dimensional dissipative dynamical systems, but they are applicable to all ''low-dimensional'' dynamical systems. These are systems for which the flow rapidly relaxes to a three-dimensional subspace of phase space. Equivalently, the associated attractor has Lyapunov dimension d L Ͻ3. Topological methods supplement methods previously developed to determine the values of metric and dynamical invariants. However, topological methods possess three additional features: they describe how to model the dynamics; they allow validation of the models so developed; and the topological invariants are robust under changes in control-parameter values. The topological-analysis procedure depends on identifying the stretching and squeezing mechanisms that act to create a strange attractor and organize all the unstable periodic orbits in this attractor in a unique way. The stretching and squeezing mechanisms are represented by a caricature, a branched manifold, which is also called a template or a knot holder. This turns out to be a version of the dynamical system in the limit of infinite dissipation. This topological structure is identified by a set of integer invariants. One of the truly remarkable results of the topological-analysis procedure is that these integer invariants can be extracted from a chaotic time series. Furthermore, self-consistency checks can be used to confirm the integer values. These integers can be used to determine whether or not two dynamical systems are equivalent; in particular, they can determine whether a model developed from time-series data is an accurate representation of a physical system. Conversely, these integers can be used to provide a model for the dynamical mechanisms that generate chaotic data. In fact, the author has constructed a doubly discrete classification of strange attractors. The underlying branched manifold provides one discrete classification. Each branched manifold has an ''unfolding'' or perturbation in which some subset of orbits is removed. The remaining orbits are determined by a basis set of orbits that forces the presence of all remaining orbits. Branched manifolds and basis sets of orbits provide this doubly discrete classification of strange attractors. In this review the author describes the steps that have been developed to implement the topological-analysis procedure. In addition, the author illustrates how to apply this procedure by carrying out the analysis of several experimental data sets. The results obtained for several other experimental time series that exhibit chaotic behavior are also described.

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“…(v) Flows over three (or more) adjacent branches should also be possible. These have been observed in simulations (Boulant et al, 1997c).…”

Section: E ''Invariant'' Versus ''Robust''supporting

confidence: 77%

Topological analysis of chaotic dynamical systems

1998

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“…5, a topological template is a branched surface which combines orientable and non-orientable, twisted, stripes (like a Möbius band) [12,14]. The procedure followed is similar to that described in previous works [12,14,23,24] so that we only sketch it briefly here.…”

mentioning

confidence: 99%

“…The linking number of a pair of orbits indicates how many turns one orbit winds around the other. In this work, we determine its value using a recently proposed algorithm [25] for computing the Gauss linking integral [24,26], which is well suited to systems of differential equations (an alternative and more common method is to compute the sum of signed crossings between the projections of the two orbits onto a plane [23]). The next step is to determine the simplest template which carries a set of periodic orbits with the same linking numbers as the detected periodic orbits.…”

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confidence: 99%

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Barrio¹,

Lefranc²,

Martínez³

et al. 2015

EPL

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PACS 05.45.Ac -Low-dimensional chaos PACS 05.45.Pq -Numerical simulations of chaotic systems PACS 87.19.ll -Models of single neurons and networks Abstract -We characterize the systematic changes in the topological structure of chaotic attractors that occur as spike-adding and homoclinic bifurcations are encountered in the slow-fast dynamics of neuron models. This phenomenon is detailed in the simple Hindmarsh-Rose neuron model, where we show that the unstable periodic orbits appearing after each spike-adding bifurcation are associated with specific symbolic sequences in the canonical symbolic encoding of the dynamics of the system. This allows us to understand how these bifurcations modify the internal structure of the chaotic attractors.The wide-range assessment of brain and behaviors is one of the pivotal challenges of this century. To understand how an incredibly sophisticated system such as the brain per se functions dynamically, it is imperative to study the dynamics of its constitutive elements -neurons. Therefore, the design of mathematical models for neurons has arisen as a trending topic in science for a few decades, since Hodgkin and Huxley developed the first model of action potentials in the neuron membrane [1]. Starting from that seminal mathematical model, a lot of variant models describing different kinds of neuron cells in numerous animals have been proposed in the literature. For instance, a reduced model [2-4] of the bursting of leech heart neuron is specified by the following three equations derived through the Hodgkin-Huxley gated variable formalism: As control parameters of the system are varied, its dynamics undergoes bifurcations such as classified by dynamical systems theory that match qualitative metamorphoses in terms specific to neuroscience. A broad range of non-stationary activity types can be observed, which includes regular and irregular tonic spiking, bursting and mixed-mode oscillations and combinations of them, as well as oscillatory transients toward quiescent states. In terms of dynamical system theory, these behaviors correspond to stable periodic and deterministically chaotic orbits in the phase space of the model. Figure 1 represents a twoparameter sweep of the leech model in the (τ K2 , I app )-plane, where the number of spikes per period, measured with the spike-counting method (SPC) [5], is color-coded. Banded structures correlated with zones of chaotic behavior appear clearly in this diagram. They are associated with spike-adding bifurcations, where the number of spikes is incremented by one, as indicated in the figure.Spike-adding bifurcations are special bifurcations that are common in fast-slow systems. They lead to the appearance of extra spikes (turns) in the fast manifold region and are quite important in that they progressively p-1

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“…Another visualization was given by Boulant et al (Fig. 6 of [7]). Such a 3D visualization would allow to be even closer visually to the nature of a chaotic attractor, and thus could provide more intuitive insights.…”

Section: Discussionmentioning

confidence: 98%

Visualizing the Template of a Chaotic Attractor

Olszewski¹,

Meder²,

Kieffer³

et al. 2018

Lecture Notes in Computer Science

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0000−0001−9926−058X] , Jeff Meder 1[0000−0002−2360−8487] , Emmanuel Kieffer 2[0000−0002−5530−8577] , Raphaël Bleuse 1[0000−0002−6728−2132] , Martin Rosalie 2[0000−0003−3676−120X] , Grégoire Danoy 1[0000−0001−9419−4210] , and Pascal Bouvry 1,2[0000−0001−9338−2834]Abstract. Chaotic attractors are solutions of deterministic processes, of which the topology can be described by templates. We consider templates of chaotic attractors bounded by a genus-1 torus described by a linking matrix. This article introduces a novel and unique tool to validate a linking matrix, to optimize the compactness of the corresponding template and to draw this template. The article provides a detailed description of the different validation steps and the extraction of an order of crossings from the linking matrix leading to a template of minimal height. Finally, the drawing process of the template corresponding to the matrix is saved in a Scalable Vector Graphics (SVG) file.

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A Nonhorseshoe Template in a Chaotic Laser Model

Cited by 21 publications

References 23 publications

Topological analysis of chaotic dynamical systems

Topological analysis of chaotic dynamical systems

Symbolic dynamical unfolding of spike-adding bifurcations in chaotic neuron models

Visualizing the Template of a Chaotic Attractor

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