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Characterization of phase transitions in nonlinear systems

Abstract

We want to investigate and characterize different phase transitions observed in nonlinear dynamical systems due to the variation of control parameters. It is known in the literature that at a second order phase transition, also called as continuous phase transition, the dynamical variable identifying the order parameter approaches zero continuously at the same time that the susceptibility of the order parameter diverges. Near a phase transition, the observables characterizing the dynamics are described by power laws leading the dynamics to be scaling invariant, which is a characteristic of a continuous phase transition. 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. Although much is known about scaling, it is yet unknown on the type of the transition observed in chaotic systems. Nonetheless, it is known what are the parameters identifying the order and its corresponding susceptibility in such transitions. These are the main goals of the project and mark our original contribution to the area. We plan to study, understand and whenever possible identify their observables determining the parameter which defines the order (symmetry) and the equivalent susceptibility of the order parameter in the dynamical systems presenting the phase transitions object of this project. Among them include: (1) transition from integrability to non integrability (observed in nonlinear mappings); (2) transition from limited to unlimited chaotic diffusion (in dissipative mappings); (3) transition from limited to unlimited energy gain (in time dependent billiards) which turns to be the main focus of this project. (AU)

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Scientific publications (4)
(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)
BORIN, DANIEL; LIVORATI, ANDRE LUIS PRANDO; LEONEL, EDSON DENIS. An investigation of the survival probability for chaotic diffusion in a family of discrete Hamiltonian mappings. CHAOS SOLITONS & FRACTALS, v. 175, p. 8-pg., . (22/03612-6, 21/09519-5)
GRACIANO, FLAVIO HELENO; DA COSTA, DIOGO RICARDO; LEONEL, EDSON D.; DE OLIVEIRA, JULIANO A.. Multiple Reflections for Classical Particles Moving under the Influence of a Time-Dependent Potential Well. Entropy, v. 24, n. 10, p. 15-pg., . (17/14414-2, 21/09519-5, 19/14038-6, 20/02415-7, 12/23688-5, 18/14685-9)
DE OLIVEIRA, JULIANO A.; DE MENDONCA, HANS M. J.; FAVARIM, VITOR A.; DE CARVALHO, R. EGYDIO; LEONEL, EDSON D.. Boundary crises and supertrack orbits in the Gauss map. European Physical Journal-Special Topics, v. 231, n. 3, p. 4-pg., . (21/09519-5, 12/23688-5, 19/07329-4, 18/14685-9, 15/22062-3, 19/14038-6)
MIRANDA, LUCAS KENJI ARIMA; MORATTA, RAPHAEL; KUWANA, CELIA MAYUMI; YOSHIDA, MAKOTO; DE OLIVEIRA, JULIANO ANTONIO; LEONEL, EDSON DENIS. A second order phase transition characterized in the suppression of unlimited chaotic diffusion for a dissipative standard mapping. CHAOS SOLITONS & FRACTALS, v. 165, p. 4-pg., . (20/10602-1, 18/14685-9, 21/09519-5, 19/14038-6)

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