Universal pattern for homoclinic and periodic points

Bevilaqua, Diego Vaz; Matos, M. Basílio de

doi:10.1016/s0167-2789(00)00088-9

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…I am to focus on Tsuda's concept of chaotic itinerancy which appears very similar to heteroclinic cycles or cycling chaos and associated research that may provide better definition (Armbruster & Guckenheimer 1988;Ashwin 1997;Buono et al 1999;Field 1980;Guckenheimer & Holmes 1988). Also, Tsuda's mention of Cantor set is more synonymous with the "simpler" homoclinic cycle (Bevilaqua & de Matos 2000;Guckenheimer & Holmes 1983) rather than chaotic itinerancy. I present a brief description and comparison of cycling chaos and mention its relevance to memory.…”

Section: Dynamic Neural Activity As Chaotic Itinerancy or Heteroclinimentioning

confidence: 99%

Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems

Tsuda¹

2001

Behav Brain Sci

439

268

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Using the concepts of chaotic dynamical systems, we present an interpretation of dynamic neural activity found in cortical and subcortical areas. The discovery of chaotic itinerancy in high-dimensional dynamical systems with and without a noise term has motivated a new interpretation of this dynamic neural activity, cast in terms of the high-dimensional transitory dynamics among "exotic" attractors. This interpretation is quite different from the conventional one, cast in terms of simple behavior on low-dimensional attractors. Skarda and Freeman (1987) presented evidence in support of the conclusion that animals cannot memorize odor without chaotic activity of neuron populations. Following their work, we study the role of chaotic dynamics in biological information processing, perception, and memory. We propose a new coding scheme of information in chaos-driven contracting systems we refer to as Cantor coding. Since these systems are found in the hippocampal formation and also in the olfactory system, the proposed coding scheme should be of biological significance. Based on these intensive studies, a hypothesis regarding the formation of episodic memory is given.

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Section: Dynamic Neural Activity As Chaotic Itinerancy or Heteroclinimentioning

confidence: 99%

Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems

Tsuda¹

2001

Behav Brain Sci

439

268

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“…As numerically demonstrated by Eq. ( 10) in [48], such asymptotic scaling relations exist inside every family of homoclinic points. A concrete mathematical description of this phenomenon is given by Lemma 2 in Appendix.…”

Section: B Asymptotic Accumulation Of Homoclinic Pointsmentioning

confidence: 87%

“…It is possible to construct the complete set of homoclinic orbit relative actions arising from horseshoeshaped homoclinic tangles in terms of the primitive orbits' relative actions and an exponentially decreasing set of parallelogram-like areas bounded by stable and unstable manifolds. Important constraints exist on the distribution of homoclinic points [48,49], which are imposed by the topology of the homoclinic tangle. This enables an organizational scheme for the orbits by their winding numbers and assigns binary symbolic codes to each of them.…”

Section: Discussionmentioning

confidence: 99%

“…The infinite set of homoclinic orbits can be put into a hierarchical structure, organized using a winding number [47,48] that characterizes the complexity of phasespace excursion of each individual orbit. The winding number of a homoclinic point h is defined to be the number of single loops (i.e., loops with no self-intersection) that the loop U S[x, h] can be decomposed into [48]. The primary homoclinic points h 0 and g 0 points in Fig.…”

Section: A Winding Numbers and Transit Timesmentioning

confidence: 99%

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Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction

Tomsovic

2019

Phys. Rev. E

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Homoclinic and heteroclinic orbits provide a skeleton of the full dynamics of a chaotic dynamical system and are the foundation of semiclassical sums for quantum wave packet, coherent state, and transport quantities. Here, the homoclinic orbits are organized according to the complexity of their phase-space excursions, and exact relations are derived expressing the relative classical actions of complicated orbits as linear combinations of those with simpler excursions plus phase-space cell areas bounded by stable and unstable manifolds. The total number of homoclinic orbits increases exponentially with excursion complexity, and the corresponding cell areas decrease exponentially in size as well. With the specification of a desired precision, the exponentially proliferating set of homoclinic orbit actions is expressible by a slower-than-exponentially increasing set of cell areas, which may present a means for developing greatly simplified semiclassical formulas.

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Homoclinic bifurcations in reversible Hamiltonian systems

Francisco¹,

Fonseca²

2006

Applied Mathematics and Computation

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Universal pattern for homoclinic and periodic points

Cited by 4 publications

References 17 publications

Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems

Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems

Exact decomposition of homoclinic orbit actions in chaotic systems: Information reduction

Homoclinic bifurcations in reversible Hamiltonian systems

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