Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

Pereira, Rodrigo Frehse; Camargo, Sabrina; Pinto, Sandro Ely de Souza; Lopes, S. R.; Viana, Ricardo L.

doi:10.1103/physreve.78.056214

Cited by 10 publications

(20 citation statements)

References 41 publications

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“…In general, networks are useful models for studying systems that have a spatial extension. For instance, insect populations whose interaction between them produces the extinction of one of them 8 , the interaction between proteins 9 and the interaction between gears 10 . These networks can be represented by a multiplex network of coupled complex subnetworks 11 12 13 14 15 16 17 18 .…”

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confidence: 99%

Chaotic, informational and synchronous behaviour of multiplex networks

Baptista

Szmoski

Pereira

et al. 2016

Sci Rep

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The understanding of the relationship between topology and behaviour in interconnected networks would allow to charac- terise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex net- works with constant Jacobian. We also study this relationship numerically in multiplex networks of Hindmarsh-Rose neurons. Whereas behaviour in the analytically tractable network is a direct but not trivial consequence of the spectra of eigenvalues of the Laplacian matrix, where behaviour may strongly depend on the break of symmetry in the topology of interconnections, in Hindmarsh-Rose neural networks the nonlinear nature of the chemical synapses breaks the elegant mathematical connec- tion between the spectra of eigenvalues of the Laplacian matrix and the behaviour of the network, creating networks whose behaviour strongly depends on the nature (chemical or electrical) of the inter synapses.

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confidence: 99%

Chaotic, informational and synchronous behaviour of multiplex networks

Baptista

Szmoski

Pereira

et al. 2016

Sci Rep

Self Cite

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“…We return to this in Remark 3.10 after the proof of Theorem 3, where we derive an approximate formula for φ that yields a value twice as large as that used in [22,23,26]. This might explain why not only in [22, Figure 13], but also in [26, Figure 6b (small κ)], the numerically observed values for φ are roughly twice as large as the values suggested by the formula used there.…”

Section: The Uncertainty Exponentmentioning

confidence: 89%

“…We compare the exponent φ with the corresponding one from [22,23] (also called φ in those papers), which is derived there for the case of riddled basins. That formula is taken as a reference in [26], although the authors of that paper study numerically the model from [14], which has intermingled basins. So we take the chance to derive an approximate formula in the style of [22,23] for the model studied in [14,26], although it has the full logistic map as its driving system.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

Keller

2017

Discrete &Amp; Continuous Dynamical Systems - S

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Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins [1,5,16,30]. To quantify the degree of intermingledness the uncertainty exponent [23] and the stability index [29,20] were suggested and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

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“…For hyperbolic chaotic systems the time of shadowing is unlimited [16,17], and most of chaotic systems with homoclinic tangencies also have a reasonably long shadowing time [18]. For systems with UDV the shadowing time can be small which is a serious obstacle for computer modelings [14,12,19]. In [14] the estimates for shadowing time and distance are found to be functions of computer round-off error and variance of the sing-changing FTLE.…”

Section: Introductionmentioning

confidence: 99%

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

Kuptsov¹

2013

J. Phys. A: Math. Theor.

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Abstract. We consider a chain of oscillators with hyperbolic chaos coupled via diffusion. When the coupling is strong the chain is synchronized and demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent. With the decay of the coupling the second and the third Lyapunov exponents approach zero simultaneously. The second one becomes positive, while the third one remains close to zero. Its finite-time numerical approximation fluctuates changing the sign within a wide range of the coupling parameter. These fluctuations arise due to the unstable dimension variability which is known to be the source for non-hyperbolicity. We provide a detailed study of this transition using the methods of Lyapunov analysis.

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Periodic-orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system

Cited by 10 publications

References 41 publications

Chaotic, informational and synchronous behaviour of multiplex networks

Chaotic, informational and synchronous behaviour of multiplex networks

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors

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