Fractional generalization of memristor and higher order elements

Fractional calculus generalizes integer order derivatives and integrals. Memristor systems generalize the notion of electrical elements. Both concepts were shown to model important classes of phenomena. This paper goes a step further by embedding both tools in a generalization considering complex-order objects. Two complex operators leading to real-valued results are proposed. The proposed class of models generate a broad universe of elements. Several combinations of values are tested and the corresponding dynamical behavior is analyzed.

Fractional dynamics of a system with particles subjected to impacts

This paper studies the dynamics of a system composed of a collection of particles that exhibit collisions between them. Several entropy measures and different impact conditions of the particles are tested. The results reveal a Power Law evolution both of the system energy and the entropy measures, typical in systems having fractional dynamics.

Calculation of fractional derivatives of noisy data with genetic algorithms

This paper addresses the calculation of derivatives of fractional order for non-smooth data. The noise is avoided by adopting an optimization formulation using genetic algorithms (GA). Given the flexibility of the evolutionary schemes, a hierarchical GA composed by a series of two GAs, each one with a distinct fitness function, is established.

Fractional control of heat diffusion systems

The concept of differentiation and integration to non-integer order has its origins in the seventeen century. However, only in the second-half of the twenty century appeared the first applications related to the area of control theory. In this paper we consider the study of a heat diffusion system based on the application of the fractional calculus concepts. In this perspective, several control methodologies are investigated and compared. Simulations are presented assessing the performance of the proposed fractional-order algorithms.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

Fractional Dynamics of Computer Virus Propagation

We propose a fractional model for computer virus propagation. The model includes the interaction between computers and removable devices. We simulate numerically the model for distinct values of the order of the fractional derivative and for two sets of initial conditions adopted in the literature. We conclude that fractional order systems reveal richer dynamics than the classical integer order counterpart. Therefore, fractional dynamics leads to time responses with super-fast transients and super-slow evolutions towards the steady-state, effects not easily captured by the integer order models.

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?

On a generalized Laguerre operational matrix of fractional integration

A new operationalmatrix of fractional integration of arbitrary order for generalized Laguerre polynomials is derived.The fractional integration is described in the Riemann-Liouville sense.This operational matrix is applied together with generalized Laguerre tau method for solving general linearmultitermfractional differential equations (FDEs).Themethod has the advantage of obtaining the solution in terms of the generalized Laguerre parameter. In addition, only a small dimension of generalized Laguerre operational matrix is needed to obtain a satisfactory result. Illustrative examples reveal that the proposedmethod is very effective and convenient for linear multiterm FDEs on a semi-infinite interval.

A fractional approach for the motion planning of redundant and hyper-redundant manipulators

The trajectory planning of redundant robots through the pseudoinverse control leads to undesirable drift in the joint space. This paper presents a new technique to solve the inverse kinematics problem of redundant manipulators, which uses a fractional differential of order α to control the joint positions. Two performance measures are defined to examine the strength and weakness of the proposed method. The positional error index measures the precision of the manipulator's end-effector at the target position. The repeatability performance index is adopted to evaluate if the joint positions are repetitive when the manipulator execute repetitive trajectories in the operational workspace. Redundant and hyper-redundant planar manipulators reveal that it is possible to choose in a large range of possible values of α in order to get repetitive trajectories in the joint space.

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.

Time domain design of fractional differintegrators using least-squares

In this paper we propose the use of the least-squares based methods for obtaining digital rational approximations (IIR filters) to fractional-order integrators and differentiators of type sα, α∈R. Adoption of the Padé, Prony and Shanks techniques is suggested. These techniques are usually applied in the signal modeling of deterministic signals. These methods yield suboptimal solutions to the problem which only requires finding the solution of a set of linear equations. The results reveal that the least-squares approach gives similar or superior approximations in comparison with other widely used methods. Their effectiveness is illustrated, both in the time and frequency domains, as well in the fractional differintegration of some standard time domain functions.

Fractional central pattern generators for bipedal locomotion

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots. In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative D α f(x), with α non-integer, is a generalization of the concept of an integer derivative, where α=1. The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators which has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs’ rhythms in bipeds, occur, for 0<α≤1. We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to unity.

Rhapsody in fractional

This paper studies several topics related with the concept of “fractional” that are not directly related with Fractional Calculus, but can help the reader in pursuit new research directions. We introduce the concept of non-integer positional number systems, fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities, fractional delay discrete-time linear systems, and fractional Fourier transform.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.

Riesz potential versus fractional Laplacian

This paper starts by introducing the Grünwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian.