C. Correia Ramos

Orbit Representations and Circle Maps

We yield C*-algebras representations on the orbit spaces from the family of interval maps f(x) = βx+α (mod 1) lifted to circle maps, in which case β ∈ N.

Iteration of Differentiable Functions under m-Modal Maps with Aperiodic Kneading Sequences

We consider the dynamical system (𝒜, 𝑇), where 𝒜 is a class of differentiable functions defined on some interval and 𝑇 : 𝒜 → 𝒜 is the operator 𝑇𝜙∶=𝑓∘𝜙, where 𝑓 is a differentiable m-modal map. Using an algorithm, we obtained some numerical and symbolic results related to the frequencies of occurrence of critical values of the iterated functions when the kneading sequences of 𝑓 are aperiodic. Moreover, we analyze the evolution as well as the distribution of the aperiodic critical values of the iterated functions.

ITERATION OF QUADRATIC MAPS ON MATRIX ALGEBRAS

We study the iteration of a quadratic family in the algebra of 2 × 2 real matrices, parameterized by a matrix C. We analyze and classify the existing cycles (periodic orbits) and their dependence on the parameter matrix. We discuss how new dynamical phenomena occur as a consequence of the noncommutativity of the matrix product. In particular, we show that the commutator of the initial condition with parameter matrix C has a decisive role in the overall dynamics.

Conditions for the formation of clusters depending on the conductance and the coefficient of clustering

One of the ultimate goals of researches on complex networks is to understand how the structure of complex networks affects the dynamical process taking place on them, such as traffic flow, epidemic spread, cascading behavior, and so on. In previous works [1] and [2], we have studied the synchronizability of a network in terms of the local dynamics, supposing that the topology of the graph is fixed. Now, we are interested in studying the effects of the structure of the network, i.e., the topology of the graph on the network synchronizability. The synchronization interval is given by a formula relating the first non zero and the largest eigenvalue of the Laplacian matrix of the graph with the maximum Lyapunov exponent of the local nodes. Our goal is to understand under what conditions can ensure the formation of clusters depending on the conductance and the coefficient of clustering.

Optimal homotopy analysis of a chaotic HIV-1 model incorporating AIDS-related cancer cells

The studies of nonlinear models in epidemiology have generated a deep interest in gaining insight into the mechanisms that underlie AIDS-related cancers, providing us with a better understanding of cancer immunity and viral oncogenesis. In this article, we analyze an HIV-1 model incorporating the relations between three dynamical variables: cancer cells, healthy CD4 + T lymphocytes, and infected CD4 + T lymphocytes. Recent theoretical investigations indicate that these cells interactions lead to different dynamical outcomes, for instance to periodic or chaotic behavior. Firstly, we analytically prove the boundedness of the trajectories in the system’s attractor. The complexity of the coupling between the dynamical variables is quantified using observability indices. Our calculations reveal that the highest observable variable is the population of cancer cells, thus indicating that these cells could be monitored in future experiments in order to obtain time series for attractor’s reconstruction. We identify different dynamical behaviors of the system varying two biologically meaningful parameters: r1, representing the uncontrolled proliferation rate of cancer cells, and k1, denoting the immune system’s killing rate of cancer cells. The maximum Lyapunov exponent is computed to identify the chaotic regimes. Considering very recent developments in the literature related to the homotopy analysis method (HAM), we calculate the explicit series solutions of the cancer model and focus our analysis on the dynamical variable with the highest observability index. An optimal homotopy analysis approach is used to improve the computational efficiency of HAM by means of appropriate values for the convergence control parameter, which greatly accelerate the convergence of the series solution. The approximated analytical solutions are used to compute density plots, which allow us to discuss additional dynamical features of the model.

KLEINIAN GROUPS AND HOLOMORPHIC DYNAMICS

It is known as a correspondence between iteration of rational maps and Kleinian groups, and is usually designated as the Sullivans dictionary. This dictionary enumerates analogies between iteration of holomorphic endomorphisms and action of Kleinian groups. We propose to study explicitly examples establishing the correspondence between the two theories: iteration theory and Kleinian groups.

Numerical semigroups and periodic orbits for Markov interval maps

Abstract Interval maps constitute a very important class of discrete dynamical systems with a well developed theory. Our purpose in this paper is to study a particular class of interval maps for which the set of periods is a numerical semigroup.

Asymptotic Behaviour in a Certain Nonlinearly Perturbed Heat Equation: Non Periodic Perturbation Case

We consider a system described by the linear heat equation with adiabatic boundary conditions. We impose a nonlinear perturbation determined by a family of interval maps characterized by a certain set of parameters. The time instants of the perturbation are determined by an additional dynamical system, seen here as part of the external interacting system. We analyse the complex behaviour of the system, through the scope of symbolic dynamics, and the dependence of the behaviour on the time pattern of the perturbation, comparing it with previous results in the periodic case.

Orbit Representations from Linear mod 1 Transformations

We show that every point x0 2 (0; 1) carries a representation of a C -algebra that encodes the orbit structure of the linear mod 1 interval map f; (x) = x + . Such C -algebra is generated by partial isometries arising from the subintervals of monotonicity of the underlying mapf; . Then we prove that such representation is irreducible. Moreover two such of representations are unitarily equivalent if and only if the points belong to the same generalized orbit, for every 2 (0; 1( and 1.