Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

Liberalquino, Ricardo Bioni; Monge, Maurizio; Galatolo, Stefano; Marangio, Luigi

doi:10.3390/math6030039

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…Such parts have been optimized to use high performance computing; even so each contraction test has required a time of the of order of a week. 3 The code we used can be found at http://im.ufrj.br/~maurizio.monge/wordpress/rigorous_computation_dyn/ 1.…”

Section: Computation Details and Resultsmentioning

confidence: 99%

“…This is the main part of the mathematical work contained in what follows and can be applied to many other dynamical systems perturbed by noise. The algorithm we develop in this work was indeed already used in [9] and [3] for the study of other dynamical systems with additive noise considered as models of certain phenomena in climate science and neuroscience.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence of Noise Induced Order, a Computer Aided Proof

Galatolo¹,

Monge²,

Nisoli³

2017

Preprint

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We prove the existence of Noise Induced Order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belosouv-Zabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certified approximation of the stationary measure in the L 1 norm. This is realized by explicit functional analytic estimates working together with an efficient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure of the system varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases.

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Section: Computation Details and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Existence of Noise Induced Order, a Computer Aided Proof

Galatolo¹,

Monge²,

Nisoli³

2017

Preprint

Self Cite

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“…Itinerancy: for an instance of its specialized usage in mathematical considerations in neuroscience, see [177].…”

Section: Footnotesmentioning

confidence: 99%

"Snakes and ladders" in Paleoanthropology: From Cognitive Surprise to Skillfulness a Million Years Ago"

Manrique¹,

Friston²,

Walker³

2022

Preprint

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A scientific paradigmatic account suffices to interpret behavioral evolution in early Homo. Cognitive surprises, favoring anomalous behavioral propensities to sporadic expression, can explain “snakes-and-ladders” appearances and disappearances of Paleolithic skills in the Early and Middle Pleistocene record. The account applies the principle of stationary action, which underpins the free energy principle, to self-organizing systems at an evolutionary timescale. Unusual personal attainments, often explained by invoking progressive ascent of evolutionary phylogenetic “ladders” of cognitive and technical abilities, could be disregarded in a hominin community that failed to imagine or articulate possible advantages for its survivability. Such failure, as well as diverse fortuitous demographic accidents, could erase from collective memory the recollection of exceptional individual conduct which disappeared down a “snake”, so to speak, of the human evolutionary “puzzle”. The puzzle discomforts paleoanthropologists. Some explain it away with the self-justifying assertion that separate paleospecies of Homo differentially possessed cognitive abilities that allegedly underlay the differential presence or absence in the Pleistocene archeological record of traces of particular behavioral outcomes or skills. An alternative methodological perspective, grounded in the fundamental relationships between organisms and their environments, affords a parsimonious, prosaic, deflationary account for appearances and disappearances of behavioral outcomes and skills.

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“…This is the main part of the mathematical work contained in what follows and can be applied to many other dynamical systems perturbed by noise. The algorithm we develop in this work was indeed already used in [4,10] for the study of other dynamical systems with additive noise considered as models of certain phenomena in climate science and neuroscience (other applications to linear response appear in [12]).…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the ndings of the present work applied to the Belousov-Zhabotinsky model de ned above allow to state the following Theorem 2. Let λ ξ be the Lyapunov expontent of the system de ned above 4 with noise of size ξ. For each α < 1, λ ξ is α−H őlder continuous as a function of ξ when ξ ∈ [ ], hence it is rigorously certi ed to be negative.…”

Section: Introductionmentioning

confidence: 99%

Existence of noise induced order, a computer aided proof

Galatolo

Monge²,

Nisoli³

2020

Nonlinearity

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We prove the existence of noise induced order in the Matsumoto-Tsuda model, where it was originally discovered in 1983 by numerical simulations. This is a model of the famous Belousov-Zhabotinsky reaction, a chaotic chemical reaction, and consists of a one dimensional random dynamical system with additive noise. The simulations showed that an increase in amplitude of the noise causes the Lyapunov exponent to decrease from positive to negative; we give a mathematical proof of the existence of this transition. The method we use relies on some computer aided estimates providing a certi ed approximation of the system's stationary measure in the L 1 norm. This is realized by explicit functional analytic estimates working together with an ef cient algorithm. The method is general enough to be adapted to any piecewise differentiable dynamical system on the unit interval with additive noise. We also prove that the stationary measure varies in a Lipschitz way if the system is perturbed and that the Lyapunov exponent of the system varies in a Hölder way when the noise amplitude increases.

show abstract

Chaotic Itinerancy in Random Dynamical System Related to Associative Memory Models

Cited by 4 publications

References 21 publications

Existence of Noise Induced Order, a Computer Aided Proof

Existence of Noise Induced Order, a Computer Aided Proof

"Snakes and ladders" in Paleoanthropology: From Cognitive Surprise to Skillfulness a Million Years Ago"

Existence of noise induced order, a computer aided proof

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