Fractional dynamics in DNA

This paper addresses the DNA code analysis in the perspective of dynamics and fractional calculus. Several mathematical tools are selected to establish a quantitative method without distorting the alphabet represented by the sequence of DNA bases. The association of Gray code, Fourier transform and fractional calculus leads to a categorical representation of species and chromosomes.

Fractional dynamics in the trajectory control of redundant manipulators

Under the pseudoinverse control, robots with kinematical redundancy exhibit an undesirable chaotic joint motion which leads to an erratic behavior. This paper studies the complexity of fractional dynamics of the chaotic response. Fourier and wavelet analysis provides a deeper insight, helpful to know better the lack of repeatability problem of redundant manipulators. This perspective for the study of the chaotic phenomena will permit the development of superior trajectory control algorithms.

Complex-order dynamics in hexapod locomotion

This paper studies the dynamics of foot–ground interaction in hexapod locomotion systems. For that objective the robot motion is characterized in terms of several locomotion variables and the ground is modelled through a non-linear spring-dashpot system, with parameters based on the studies of soil mechanics. Moreover, it is adopted an algorithm with foot-force feedback to control the robot locomotion. A set of model-based experiments reveals the influence of the locomotion velocity on the foot–ground transfer function, which presents complex-order dynamics.

Mathematical model for HIV dynamics in HIV-specific helper cells

In this paper we study a delay mathematical model for the dynamics of HIV in HIV-specific CD4 + T helper cells. We modify the model presented by Roy and Wodarz in 2012, where the HIV dynamics is studied, considering a single CD4 + T cell population. Non-specific helper cells are included as alternative target cell population, to account for macrophages and dendritic cells. In this paper, we include two types of delay: (1) a latent period between the time target cells are contacted by the virus particles and the time the virions enter the cells and; (2) virus production period for new virions to be produced within and released from the infected cells. We compute the reproduction number of the model, R0, and the local stability of the disease free equilibrium and of the endemic equilibrium. We find that for values of R0<1, the model approaches asymptotically the disease free equilibrium. For values of R0>1, the model approximates asymptotically the endemic equilibrium. We observe numerically the phenomenon of backward bifurcation for values of R0⪅1. This statement will be proved in future work. We also vary the values of the latent period and the production period of infected cells and free virus. We conclude that increasing these values translates in a decrease of the reproduction number. Thus, a good strategy to control the HIV virus should focus on drugs to prolong the latent period and/or slow down the virus production. These results suggest that the model is mathematically and epidemiologically well-posed.

Fractional Dynamics in the Rayleigh's Piston

This paper studies the dynamics of the Rayleigh piston using the modeling tools of Fractional Calculus. Several numerical experiments examine the effect of distinct values of the parameters. The time responses are transformed into the Fourier domain and approximated by means of power law approximations. The description reveals characteristics usual in Fractional Brownian phenomena.

Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

We study the peculiar dynamical features of a fractional derivative of complex-order network. The network is composed of two unidirectional rings of cells, coupled through a "buffer" cell. The network has a Z3 × Z5 cyclic symmetry group. The complex derivative Dα±jβ, with α, β ∈ R+ is a generalization of the concept of integer order derivative, where α = 1, β = 0. Each cell is modeled by the Chen oscillator. Numerical simulations of the coupled cell system associated with the network expose patterns such as equilibria, periodic orbits, relaxation oscillations, quasiperiodic motion, and chaos, in one or in two rings of cells. In addition, fixing β = 0.8, we perceive differences in the qualitative behavior of the system, as the parameter c ∈ [13, 24] of the Chen oscillator and/or the real part of the fractional derivative, α ∈ {0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, are varied. Some patterns produced by the coupled system are constrained by the network architecture, but other features are only understood in the light of the internal dynamics of each cell, in this case, the Chen oscillator. What is more important, architecture and/or internal dynamics?