Complex behaviors in one dimensional dynamic systems

Abstract

Behaviors of nonlinear phenomena are observed in engineering, medicine, biology, ecology, chemistry, and in several areas of science. Nevertheless, the evolution processes of these systems develop dynamic completely equivalent. Routes to chaos, creation and annihilation of periodic orbits, and the emergence of equilibrium points are examples of behaviors that have similar characteristics in different systems. In this context, understanding dynamic phenomena with universal characteristics plays an important role leading to a more comprehensive understanding of complex dynamics. Equilibrium points and periodic orbits, quasi-periodic and chaotic manifestations are common in dynamic systems and can be studied by varying the control parameters. In the 1960s and 1980s, structures of isoperiodicity were identified in two-dimensional parameter spaces now known as Arnold tongues and shrimps, respectively. These structures are continuous sets of periodicity immersed in regions of the parameter space where the behavior is quasi-periodic (Arnold tongues) or chaotic (shrimps). Since chaos is more common than quasi-periodic behaviors, shrimps are present in a wide variety of systems including systems with Arnold tongues. Currently, shrimp is an effervescent topic with intense study focused on their emergence and internal structural feature. This project proposes to investigate the chaotic regions surrounding the shrimp in an one-dimensional map where the presence of shrimp strongly influences the dynamics of the system. (AU)