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Investigation of dynamical properties in nonlinear systems

Abstract

The subject of investigation of this project is the study of the several dynamical properties present in nonlinear systems. In dynamical systems described either by differential equations or discrete mappings, quite often we find observables that are described by a power law. Examples include Lyapunov exponents, diffusion coefficient, quadratic mean velocity, periodic structures in the parameter plane producing objects called as shrimps, distance from the attractor, chaotic transient, the attractor itself either periodic or chaotic, among many others. When such measurable quantities are also scaling invariant, generally made via a control parameter or change in the initial condition, one can find a set of critical exponents that describe the dynamics of the observable by using scaling transformations. The main phenomenology to describe this property uses a set of scaling hypotheses as well as a generalized homogeneous function. From them, it is possible to find an analytic relation for the exponents leading to a scaling law. Indeed, scaling laws are much useful in the characterization and definition of classes of universality and can be proved either using numerical simulations or analytic descriptions. When the characterization is not given in terms of power, as it is the case of the anomalous diffusion in chaotic systems, often the properties are characterized by other laws including exponentials, stretched exponential among many others. Following this thematic, the most distinct dynamical properties in several nonlinear dynamical systems either described by ordinary differential equations or by mappings, will be investigated. Some of them include the characterization of chaotic seas, chaotic transport, the transition from integrability to no integrability, time-dependent billiards among others. (AU)

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Scientific publications (8)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
HANSEN, MATHEUS; LANES, GABRIEL C.; BRITO, VINICIUS L. G.; LEONEL, EDSON D. Investigation of pollen release by poricidal anthers using mathematical billiards. Physical Review E, v. 104, n. 3 SEP 13 2021. Web of Science Citations: 0.
DA SILVA, V. B.; VIEIRA, J. P.; LEONEL, EDSON D. Fisher information of the Kuramoto model: A geometric reading on synchronization. PHYSICA D-NONLINEAR PHENOMENA, v. 423, SEP 2021. Web of Science Citations: 0.
SILVEIRA, FELIPE AUGUSTO O.; ALVES, SIDINEY G.; LEONEL, EDSON D.; LADEIRA, DENIS G. Dynamical aspects of a bouncing ball in a nonhomogeneous field. Physical Review E, v. 103, n. 6 JUN 3 2021. Web of Science Citations: 0.
VELOSO HERMES, JOELSON D.; LEONEL, EDSON D. Characteristic Times for the Fermi-Ulam Model. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, v. 31, n. 2 FEB 2021. Web of Science Citations: 0.
DE OLIVEIRA, JULIANO A.; PERRE, RODRIGO M.; MENDEZ-BERMUDEZ, J. A.; LEONEL, EDSON D. Leaking of orbits from the phase space of the dissipative discontinuous standard mapping. Physical Review E, v. 103, n. 1 JAN 13 2021. Web of Science Citations: 0.
DA FONSECA, JULIO D.; LEONEL, EDSON D.; CHATE, HUGUES. Instantaneous frequencies in the Kuramoto model. Physical Review E, v. 102, n. 5 NOV 23 2020. Web of Science Citations: 0.
LEONEL, EDSON D.; KUWANA, CELIA MAYUMI; YOSHIDA, MAKOTO; DE OLIVEIRA, JULIANO ANTONIO. Chaotic diffusion for particles moving in a time dependent potential well. Physics Letters A, v. 384, n. 28 OCT 9 2020. Web of Science Citations: 0.
LEONEL, EDSON D.; YOSHIDA, MAKOTO; DE OLIVEIRA, JULIANO ANTONIO. Characterization of a continuous phase transition in a chaotic system. EPL, v. 131, n. 2 JUL 2020. Web of Science Citations: 0.

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