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Scaling investigation in dynamical systems

Abstract

The subject of scaling laws define the main research line investigation of the present project. 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, among many others. When such measurable quantities are also scaling invariant, in other words, when they are invariant by a reduction or amplification, 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. Following this thematic, we shall investigate some dynamical systems that may exhibit chaos focusing in the characterization of chaotic seas, chaotic transport, transition from integrability to no integrability, time dependent billiards among others. (AU)

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Scientific publications (17)
(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; CIRO, DAVID; CALDAS, IBERE L.; LEONEL, EDSON D.. Explaining a changeover from normal to super diffusion in time-dependent billiards. EPL, v. 121, n. 6, . (17/14414-2, 11/19296-1)
OLIVEIRA, DIEGO F. M.; CHAN, KEVIN S.; LEONEL, EDSON D.. Scaling invariance in a social network with limited attention and innovation. Physics Letters A, v. 382, n. 47, p. 3376-3380, . (17/14414-2)
LIVORATI, ANDRE L. P.; KROETZ, TIAGO; DETTMANN, CARL P.; CALDAS, IBERE L.; LEONEL, EDSON D.. Transition from normal to ballistic diffusion in a one-dimensional impact system. Physical Review E, v. 97, n. 3, . (14/25316-3, 17/14414-2, 15/26699-6)
DE OLIVEIRA, JULIANO A.; DE MENDONCA, HANS M. J.; DA SILVA, ANDERSON A. A.; LEONEL, EDSON D.. Critical Slowing Down at a Fold and a Period Doubling Bifurcations for a Gauss Map. Brazilian Journal of Physics, v. 49, n. 6, p. 923-927, . (18/14685-9, 15/22062-3, 14/18672-8, 17/14414-2, 12/23688-5)
HANSEN, MATHEUS; CIRO, DAVID; CALDAS, IBERE L.; LEONEL, EDSON D.. Dynamical thermalization in time-dependent billiards. Chaos, v. 29, n. 10, . (18/03211-6, 17/14414-2)
DA COSTA, DIOGO RICARDO; SILVA, MARIO R.; LEONEL, EDSON D.; MENDEZ-BERMUDEZ, J. A.. Statistical description of multiple collisions in the Fermi-Ulam model. Physics Letters A, v. 383, n. 25, p. 3080-3087, . (17/14414-2)
PALMERO, MATHEUS S.; DIAZ, GABRIEL I.; MCCLINTOCK, PETER V. E.; LEONEL, EDSON D.. Diffusion phenomena in a mixed phase space. Chaos, v. 30, n. 1, . (16/15713-0, 14/27260-5, 17/14414-2)
PALMERO, MATHEUS S.; LIVORATI, ANDRE L. P.; CALDAS, IBERE L.; LEONEL, EDSON D.. Ensemble separation and stickiness influence in a driven stadium-like billiard: A Lyapunov exponents analysis. COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, v. 65, p. 248-259, . (14/25316-3, 11/19296-1, 12/23688-5, 17/14414-2, 15/26699-6, 12/00556-6)
DE OLIVEIRA, JULIANO A.; MONTERO, LEONARDO T.; DA COSTA, DIOGO R.; MENDEZ-BERMUDEZ, J. A.; MEDRANO-T, RENE O.; LEONEL, EDSON D.. An investigation of the parameter space for a family of dissipative mappings. Chaos, v. 29, n. 5, . (14/18672-8, 17/14414-2, 15/50122-0, 18/14685-9)
DIAZ, GABRIEL; YOSHIDA, MAKOTO; LEONEL, EDSON D.. A Monte Carlo approach for the bouncer model. Physics Letters A, v. 381, n. 42, p. 3636-3640, . (17/14414-2)
DE OLIVEIRA, JULIANO A.; RAMOS, LARISSA C. N.; LEONEL, EDSON D.. Dynamics towards the steady state applied for the Smith-Slatkin mapping. CHAOS SOLITONS & FRACTALS, v. 108, p. 119-122, . (17/14414-2, 17/17294-8, 14/18672-8)
DA COSTA, DIOGO R.; MENDEZ-BERMUDEZ, J. A.; LEONEL, EDSON D.. Scaling and self-similarity for the dynamics of a particle confined to an asymmetric time-dependent potential well. Physical Review E, v. 99, n. 1, . (17/14414-2)
DE OLIVEIRA, JULIANO A.; DE MENDONCA, HANS M. J.; DA COSTA, DIOGO R.; LEONEL, EDSON D.. Effects of a parametric perturbation in the Hassell mapping. CHAOS SOLITONS & FRACTALS, v. 113, p. 238-243, . (15/22062-3, 14/18672-8, 17/14414-2, 12/23688-5)
DIAZ, I, GABRIEL; PALMERO, MATHEUS S.; CALDAS, IBERE LUIZ; LEONEL, EDSON D.. Diffusion entropy analysis in billiard systems. Physical Review E, v. 100, n. 4, . (18/03211-6, 17/14414-2, 18/03000-5)
PERRE, RODRIGO M.; CARNEIRO, BARBARA P.; MENDEZ-BERMUDEZ, J. A.; LEONEL, EDSON D.; DE OLIVEIRA, JULIANO A.. On the dynamics of two-dimensional dissipative discontinuous maps. CHAOS SOLITONS & FRACTALS, v. 131, . (18/14685-9, 17/14414-2, 19/06931-2, 14/18672-8)
LEONEL, EDSON D.; KUWANA, CELIA M.. An Investigation of Chaotic Diffusion in a Family of Hamiltonian Mappings Whose Angles Diverge in the Limit of Vanishingly Action. Journal of Statistical Physics, v. 170, n. 1, p. 69-78, . (17/14414-2, 12/23688-5)
HANSEN, MATHEUS; DA COSTA, DIOGO RICARDO; CALDAS, IBERE L.; LEONEL, EDSON D.. Statistical properties for an open oval billiard: An investigation of the escaping basins. CHAOS SOLITONS & FRACTALS, v. 106, p. 355-362, . (13/22764-2, 11/19296-1, 14/00334-9, 12/23688-5, 17/14414-2)

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