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Transport properties and bifurcation analysis in nonlinear dynamical systems

Abstract

In this project we consider as main line the investigation of the effects and consequences of dissipation and scaling laws in nonlinear mappings. Generally in dynamic systems we find dissipative physical problems that obey power laws and have periodic structures in the parameter space, which can be determined by the Lyapunov exponents. When these quantities are measurable variables are also scaling invariants and can find critical exponents that describe the system. Phenomenology to describes the scaling laws rely on the assumption of hypothesis scaling and a generalized homogeneous function. The scaling laws are very importants to characterize and define universality classes. In this project we propose to consider the Gauss map, the logistic-like map with parametric perturbation and the squared cosine logistic map that belong to a set of discrete one-dimensional maps. We intend to build the bifurcation diagrams to analyze the dynamical systems. In the Gauss map, we propose to investigate the scaling laws in the decay of orbits for the steady state in the tangent and period doubling bifurcations. We advance our studies considering the logistic-like map with periodic control parameter. We intend to consider small perturbations so that new attractors can be observed. Our goal is to investigate the transient in the change of the basin of attraction characterizing it by a power law. In the squared cosine logistic map, when we vary the parameters, we can observe the appearance of boundary crisis in the bifurcation diagram. We intend to determine the transient exponents in boundary crisis. In addition, we hope to illustrate the sensitivity of the system to the initial conditions in the study of the Lyapunov exponents in order to investigate the periodic structures in the parameter space. We propose to continue our studies in the investigation of the periodic structures in the parameter space of a family of dissipative two-dimensional mappings defined in the moment and angle variables. We propose to calculate the Lyapunov exponents in order to investigate the periodic structures producing the objects known as shrimps. In the parameter spaces complex structures such as homoclinic tangents can be explored. Finally, we intend to investigate the transport properties in the dissipative discontinuous standard mapping. For a specific choice of control parameters chaotic attractors can be observed in phase space. In this case our focus is to explore the formalism of particle transport and to determine exponents from scape of the particles characterizing them by a power law. (AU)

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Scientific publications (12)
(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)
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, . (19/14038-6, 18/14685-9)
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)
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)
LEONEL, EDSON D.; MAYUMI KUWANA, CELIA; YOSHIDA, MAKOTO; ANTONIO DE OLIVEIRA, JULIANO. Application of the diffusion equation to prove scaling invariance on the transition from limited to unlimited diffusion. EPL, v. 131, n. 1, p. 5-pg., . (19/14038-6, 18/14685-9)
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)
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, 18/14685-9, 17/14414-2, 15/50122-0)
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, . (18/14685-9, 19/06931-2, 19/14038-6)
LEONEL, EDSON D.; YOSHIDA, MAKOTO; DE OLIVEIRA, JULIANO ANTONIO. Characterization of a continuous phase transition in a chaotic system. EPL, v. 131, n. 2, . (19/14038-6, 18/14685-9)
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)
MIRANDA, LUCAS K. A.; KUWANA, CELIA M.; HUGGLER, YONA H.; DA FONSECA, ANNE K. P.; YOSHIDA, MAKOTO; DE OLIVEIRA, JULIANO A.; LEONEL, EDSON D.. A short review of phase transition in a chaotic system. European Physical Journal-Special Topics, . (20/10602-1, 18/14685-9, 19/14038-6, 20/07219-1)
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, p. 6-pg., . (19/14038-6, 18/14685-9, 19/06931-2)

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