This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >

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Chaos in a fractional order Chua's system

Many real dynamic systems are better characterized using a non-integer order dynamic model based on fractional calculus or, differentiation or integration of non-integer order. Traditional calculus is based on integer order differentiation and integration. The concept of fractional calculus has tremendous potential to change the way we see, model, and control the nature around us. Denying fractional derivatives is like saying that zero, fractional, or irrational numbers do not exist. In this paper, we offer a tutorial on fractional calculus in controls. Basic definitions of fractional calculus, fractional order dynamic systems and controls are presented first. Then, fractional order PID controllers are introduced which may make fractional order controllers ubiquitous in industry. Additionally, several typical known fractional order controllers are introduced and commented. Numerical methods for simulating fractional order systems are given in detail so that a beginner can get started quickly. Discretization techniques for fractional order operators are introduced in some details too. Both digital and analog realization methods of fractional order operators are introduced. Finally, remarks on future research efforts in fractional order control are given.

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Fractional order control - A tutorial

The dynamics of fractional-order systems have attracted increasing attentions in recent years. In this paper, we numerically study the chaotic behaviors in the fractional-order Rossler equations. We found that chaotic behaviors exist in the fractional-order Rossler equation with orders less than 3, and hyperchaos exists in the fractional-order Rossler hyperchaotic equation with order less than 4. The lowest orders we found for chaos and hyperchaos to exist in such systems are 2.4 and 3.8, respectively. Period doubling routes to chaos in the fractional-order Rossler equation are also found.

Chaos and hyperchaos in the fractional-order Rössler equations

Impedance of rough capacitive electrodes

In this letter, we study the chaotic behaviors in the fractional order Chen system. We found that chaos exists in the fractional order Chen system with order less than 3. The lowest order we found to have chaos in this system is 2.1. Linear feedback control of chaos in this system is also studied.

Chaos in the fractional order Chen system and its control

A fractional slope on the log log Bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole. In order to represent and study its dynamical behavior, a singularity function method is presented which consists of cascaded branches of a number of pole-zero (negative real) pairs or simple RC section. The distribution spectrum of the system can also be easily calculated, and its accuracy depends on a prescribed error specified in the beginning. The method is then extended to a multiple-fractal system which consists of a number of fractional power poles. The result can be simulated by a combination of singularity functions, each representing a single-fractal system. >

Fractal system as represented by singularity function

The driving point immittance (impedance or admittance) function is commonly used in electrical characterization of polarized materials and interfaces. The immittance function typically attenuates following a power function dependence on frequency. This fact has been recognized as a macroscopic dynamical property manifested by strongly interacting dielectric, viscoelastic and magnetic materials and interfaces between different conducting substances. Linear interfacial polarization processes which occur at metal electrode-electrolyte interfaces have been represented by the Fractional Power Pole [FPP] function in single or multiple stages. The FPP function is referred to as the Davidson-Cole function in the dielectrics literature. A related function widely used in mathematical modeling of dielectric and viscoelastic polarization dynamics is the Cole-Cole function. The fractional power factor which parametrizes the FPP or the Davidson-Cole function has been shown earlier to equal the logarithmic ratio of the locations of the pole-zero singularities. In this paper we first review a modified form of the singularity decomposition of the FPP function accomplished within a prescribed error range. The distribution spectrum and the corresponding simulation by a cascadeR-C network, as opposed to the synthesis by a ladderR-C network, are readily obtained as the next step in the simulation. The method is then applied to decompose the Cole-Cole function; the pole-zero placement of the singularity function is determined and the equivalent cascadeR-C network is synthesized.

Analysis of polarization dynamics by singularity decomposition method.

The magnitude spectral density of many physical phenomena such as electrical noise, the relaxation of polarized dielectrics, viscous and magnetic materials, and the interface between two dissimilar conducting materials attenuate following a fractional power function dependence on frequency. Such systems are recognized as fractal systems distinguished by the 1/f-type attenuation in the magnitude spectrum and characterized by global parameters such as fractal dimension and global corners. Fractal systems which relax to steady state by a distribution of purely real exponentials are recognized as fractal relaxation systems. This is the first part of a series of planned articles, each focusing on a particular aspect of fractal relaxation systems. In this part, the singularity structure model is proposed to represent the steady state frequency response of fractal relaxation systems in the linear range. The singularity structure model of fractal relaxation systems can be mathematically represented by a rational ...

FRACTAL RELAXATION SYSTEMS. Part I: Singularity Structure Analysis

This study is concerned with mathematical modelling of the fundamental relationship which exists between the current density and the overpotential across the metalsolution interface in the linear range using methods of system theory enhanced by ‘fractal’ concepts. A primer for both 1/f-type scaling and ‘anomalous’ relaxation/dispersion concepts is provided, followed by a brief review of the research history pertinent to the metal electrode polarization dynamics. Next, the ‘fractal relaxation systems’ approach is introduced to characterize, systems which attenuate with a fractional power-low dependence on frequency through a ‘scaling exponent’. The ‘singularity structure’ which is a scaling, rational system function is proposed to expand fractal systems in terms of basic subsystems individually representing elementary exponential relaxations and collectively exhibiting scaling properties. We stress that the ‘singularity structure’ carries scaling information identical to the conventional ‘distribution of relaxation times’ function. ‘Structure scale’ and ‘view scale’ concepts are presented in the due course to streamline the analysis of scaling phenomena in general and the polarization impedance in particular. System theory-wise, the notable result is that the fractional power function attenuation, or equivalently, the logarithmic nature of the distribution function translates into the ‘self-similar’ pattern replication of the system singularities in the s-plane. The singularity arrangement is governed by a recursive rule solely based on the knowledge of the fractional power factor or the scaling exponent.

Fractal Dynamics of Polarized Bioelectrodes

A system is recognized as fractal if its magnitude frequency spectrum attenuates following a power function dependence on the frequency and exhibits a fractional slope over a broad band of frequencies. If the transient response of the fractal system consists solely of real exponentials, the system is classified as a fractal relaxation system. The relaxation elements of the system reach steady state at different rates, i.e. different relaxation times or time constants. The system can therefore can be characterized by a distribution of the relaxation times. The distribution of relaxation times completely defines the system, thus is equivalent to the system function in the relaxation domain. This is the second part of a series of articles, each focusing on a particular aspect of fractal relaxation systems. In the first part, the singularity structure model of the fractal relaxation systems has been proposed. In this part, the concept of distribution of relaxation times of fractal relaxation systems is introd...

Yuan-Ying Tsao

Papers

Fractal system as represented by singularity function

Analysis of polarization dynamics by singularity decomposition method.

FRACTAL RELAXATION SYSTEMS. Part I: Singularity Structure Analysis

Fractal Dynamics of Polarized Bioelectrodes

FRACTAL RELAXATION SYSTEMS. Part II: Distribution of Relaxation Times