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

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.

Tsugio Kojima

Papers

An analog simulation of non-integer order transfer functions for analysis of electrode processes

Application of pseudo-random signals and cross-correlation techniques in electroanalytical chemistry

Correlation between electron capture response and chemical structure for benzyl chlorides.

Application of Laser to Pyrolysis Gas Chromatography I

Application of pattern recognition techniques to qualitative electrochemical analysis