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Statistical and dynamical properties of time-dependent systems

Grant number: 13/01449-1
Support Opportunities:Research Grants - Visiting Researcher Grant - International
Duration: March 01, 2013 - April 30, 2013
Field of knowledge:Physical Sciences and Mathematics - Physics - General Physics
Principal Investigator:Edson Denis Leonel
Grantee:Edson Denis Leonel
Visiting researcher: Diego Fregolente Mendes de Oliveira
Visiting researcher institution: Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Germany
Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil


We study some dynamical and statistical properties of classical time-dependent systems. It is well known that the structure of the phase space of Hamiltonian systems depends on the individual characteristics of each system. Basically they can be divided in three groups: (i) integrable, (ii) ergodic and (iii) mixed. In case (i) the phase space consists of invariant tori filling the entire phase space. In case (ii) the time evolution of a single initial condition is enough to fill the phase space. In case (iii), one important property in the mixed phase space is that chaotic seas are generally surrounding Kolmogorov-Arnold-Moser (KAM) islands which are confined by invariant spanning curves. In particular such curves can cross the phase plane and partition it into several separated and disconnected portions of the phase space. Sometimes, the existence of invariant spanning curves can also prevent the unlimited energy growth of a particle. One of the main point of our work is to study some statistical properties of systems with integrable, ergodic and mixed phase space close to the transition from integrability to nonintegrability and unlimited to limited energy growth using scaling arguments. Once the critical exponents are known, classes of universality can be defined. On the other hand, the introduction of inelastic collisions on this model is enough to destroy such a mixed structure and the system exhibits attractors. Depending on the initial conditions and control parameters, one can observe a chaotic attractor characterized by a positive Lyapunov exponent. By a suitable control parameter variation, the chaotic attractor might be destroyed via a crisis event. After the destruction, the chaotic attractor is replaced by a chaotic transient. Additionally, by changing simultaneously the dissipation parameter and the parameter which controls the transition from integrability to non-integrability and by using the Lyapunov exponents in order to classify regions with chaotic and regular behavior we have also studied the structure of the parameter-space. (AU)

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Scientific publications
(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)
OLIVEIRA, DIEGO F. M.; SILVA, MARIO ROBERTO; LEONEL, EDSON D.. A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping. PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS, v. 436, p. 909-915, . (13/01449-1, 12/23688-5)
HANSEN, MATHEUS; DA COSTA, DIOGO R.; OLIVEIRA, DIEGO F. M.; LEONEL, EDSON D.. Statistical properties for a dissipative model of relativistic particles in a wave packet: A parameter space investigation. Applied Mathematics and Computation, v. 238, p. 387-392, . (13/01449-1, 12/23688-5, 12/18962-0, 13/22764-2)
DA COSTA, DIOGO RICARDO; OLIVEIRA, DIEGO F. M.; LEONEL, EDSON D.. Dynamical and statistical properties of a rotating oval billiard. COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION, v. 19, n. 6, p. 1926-1934, . (12/18962-0, 13/01449-1, 12/23688-5)

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