Diogo Ricardo da Costa

This paper presents a recursive method for identifying extreme and superstable curves in the parameter space of dissipative one-dimensional maps. The method begins by constructing an Archimedean spiral with a constant arc length. Subsequently, it identifies extreme and superstable curves by calculating an observable ψ. The spiral is used to locate a region where ψ changes sign. When this occurs, a bisection method is applied to determine the first point on the desired superstable or extreme curve. Once the initial direction is established, the recursive method identifies subsequent points using an additional bisection method, iterating the process until the stopping conditions are met. The logistic-Gauss map demonstrates each step of the method, as it exhibits a wide variety of periodicity structures in the parameter space, including cyclic extreme and superstable curves, which contribute to the formation of period-adding structures. Examples of extreme and superstable curves obtained by the recursive method are presented. It is important to note that the proposed method is generalizable and can be adapted to any one-dimensional map.

This study focuses on the analysis of a unique composition between two well-established models, known as the Logistic–Gauss map. The investigation cohesively transitions to an exploration of parameter space, essential for unraveling the complexity of dissipative mappings and understanding the intricate relationships between periodic structures and chaotic regions. By manipulating control parameters, our approach reveals intriguing patterns, with findings enriched by extreme orbits, trajectories that connect local maximum and minimum values of one-dimensional maps. This theory enhances our perception of structural organization and offers valuable perceptions of the system behaviors, contributing to an expanded understanding of chaos and periodicity in dynamic systems. The analysis reveals Complex Sets of Periodicity (CSP) in the parameter space, characterized by superstable curves that traverse their main bodies. The exploration of different combinations of parameters shows cascades of CSP structures with added periods and are organized based on extreme curves. This investigation offers valuable discoveries of the dynamics of dissipative mappings, opening avenues for future explorations in chaotic systems.

We investigate some statistical properties of escaping particles in a billiard system whose boundary is described by two control parameters with a hole on its boundary. Initially, we analyze the survival probability for different hole positions and sizes. We notice that the survival probability follows an exponential decay with a characteristic power-law tail when the hole is positioned partially or entirely over large stability islands in phase space. We find that the survival probability exhibits scaling invariance with respect to the hole size. In contrast, the survival probability for holes placed in predominantly chaotic regions deviates from the exponential decay. We introduce two holes simultaneously and investigate the complexity of the escape basins for different hole sizes and control parameters by means of the basin entropy and the basin boundary entropy. We find a non-trivial relation between these entropies and the system’s parameters and show that the basin entropy exhibits scaling invariance for a specific control parameter interval.

In this work, we use the finite-time Lyapunov exponent (FTLE) for a version of the potential well model with an oscillating bottom. This model is described by a conservative two-dimensional mapping that exhibits a mixed phase space, namely it contains a sea of chaos and the regions of stability. Via FTLE, we show that throughout its orbit, an initial condition in the chaotic sea can become trapped in a certain region. We construct and analyze phase spaces for different ranges of Lyapunov exponents in finite time, highlighting these traps. For smaller and larger values of FTLE, we identified a very regular phase space and a phase space with more chaos. By means of low FTLE values, we observe some small regions in the phase space that present regularities. Therefore, we demonstrate that FTLE can be useful to identify the stickiness effect in a time-dependent potential well.

In this paper, we consider the evolution of an ensemble of noninteracting classical particles confined in a closed region. Each one of these particles experiences several elastic collisions with a rigid-smooth oval-like boundary. As known in the literature, numerical simulations for this kind of system demand too much time, mostly because of the time spent in solving transcendental equations. To deal with that, we highlight an alternative approach to speed up the search of solutions for the transcendental equations via the tangent method, which has proved to be a faster way to solve this problem when compared with other methods, like the bisection one. The methods applied here are generic and can be used or adapted for other billiard boundaries.

The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.