Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

Barrio, Roberto; Blesa, Fernando; Serrano, Sergio

doi:10.1016/j.physd.2009.03.010

Cited by 104 publications

(77 citation statements)

References 34 publications

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“…Understanding how the template and the spectrum of orbits populating it change across bifurcations is a robust way to characterize how the chaotic dynamics unfolds in parameter space. For example, it was shown recently in [16,17] how a global bifurcation can increase the number of branches in the topological template of dissipative systems like the Rössler model (see also [18,19]). Here, we observe in neuron models that changes in the topological structure along a series of spike-adding bifurcations is more gradual: the number of branches does not change, however the spectrum of periodic orbits is modified in a systematic way, and accordingly how the attractor is visited by chaotic orbits.…”

mentioning

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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mentioning

confidence: 99%

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

Barrio¹,

Lefranc²,

Martínez³

et al. 2015

EPL

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“…. to screw shaped" [10]. This curve passes through a central or focal point of a bifurcation structure that is strikingly similar to that near a T-point of the Lorenz system.…”

Section: Further First Foliation Tangencies In Thementioning

confidence: 86%

“…Published by SIAM under the terms of the Creative Commons 4.0 license Downloaded 05/12/18 to 54.245.55.244. Redistribution subject to CCBY license creation of additional funnel structures [7,10]; a transition from scroll to funnel attractor has also been observed in a Chua circuit experiment in this way [50]. It would be interesting to find and visualize the stable and unstable manifolds in these systems, to determine how their interactions organize the observed topological changes, and to compute loci of first foliation tangencies directly by continuation.…”

Section: Further First Foliation Tangencies In Thementioning

confidence: 90%

“…We expect that this will also be relevant for the study of other systems, especially of other three-dimensional dissipative systems. Examples are the Rössler system and the Rosenzweig-MacArthur models considered by Barrio, Blesa, and Serrano [7,9,10]. In particular, they [10, Figure 1] identify a boundary curve in a twoparameter plane that "determines a change in the topological structure of chaotic attractors [of the Rössler system and a tritrophic food chain] from spiral .…”

Section: Further First Foliation Tangencies In Thementioning

confidence: 99%

See 1 more Smart Citation

Finding First Foliation Tangencies in the Lorenz System

Creaser¹,

Krauskopf²,

Osinga³

2017

SIAM J. Appl. Dyn. Syst.

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Abstract. Classical studies of chaos in the well-known Lorenz system are based on reduction to the onedimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the threedimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits-before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

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“…This system contains three prototype first-order diferential equations with three dynamical variables in defining the phase space and three parameters. This system has been thoroughly studied by many researchers; (see [3,4,5] for more details). Continuing his work, In 1979 Rossler proposed another dynamical system which was made of four first order differential equations as following (see [6]).…”

Section: Introductionmentioning

confidence: 99%

Implementation of multi-step differentialtransformation method for hyperchaotic Rossler system

Biazar

Houlari

Asayesh

2016

IJAMR

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In this work, the multi-step differential transformation method (MSDTM) is applied to approximate a solution of the hyperchaotic Rossler system. MSDTM is adapted from the differential transformation method (DTM). In this method, DTM is implemented in each subinterval.Results are compared with a fourth-order Runge Kutta method and a standard DTM. The results show that the MSDTM is an efficient and powerful technique for solving hyperchaotic Rossler systems and this method is more accurate than DTM.

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Qualitative analysis of the Rössler equations: Bifurcations of limit cycles and chaotic attractors

Cited by 104 publications

References 34 publications

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

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

Finding First Foliation Tangencies in the Lorenz System

Implementation of multi-step differentialtransformation method for hyperchaotic Rossler system

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