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Investigation of dynamical properties in nonlinear systems

Grant number: 19/14038-6
Support type:Regular Research Grants
Duration: December 01, 2019 - November 30, 2021
Field of knowledge:Physical Sciences and Mathematics - Physics
Principal Investigator:Edson Denis Leonel
Grantee:Edson Denis Leonel
Home Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil

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)